1hf~ bt>oK helo~s-tz,Arnj b 'DJ]at< t.411<a)(. nt- S/lf/2.D. £•

. ..

..

•

------­Physics for Everyone

Book2

~I

L,D, LandauA,I,Kitaigorodsky

MOLECULESTranslated from

the Russian

by Martin Greendlinger,D. Sc. (Math.)

M:r Publishers Moscow

([lll:!J'llW ;\."[1\ BCCX

l\[I1ua 2

Jl. it. Jlau/\ay1\. 11. hHTaiiropll)J,CI;llil

MII,IICI;y:II,I

!!:l/\aTl'.II,cTBIl «HaYI;a,)

Fir,;\ jJllhli ...;!Icd 1'1/0~eClilld pdilioll I'I:';()

Jla ([!I.',IU/'Ch'O.H RJI)l1;(

lC'1 Il:l;~aTe:ll)cTB(l <dlayHa», 1\l7K,<) j':I'gli,dl I ral1slatilll1.

\lil' I'llbli:-;!Jel':-', l!IRII

PREFACE TO THE FOURTHRUSSIAN EDITION

This book has been nalllOd Jlo7ec1I7es. "rany chaptersfrom tbe second half of a previous book, Physics torEveryone, by Lev Landau and Alexander Kitaigorodsky,have been included without revision.

The book is clDvoled mninl v to a studv of the structureof matter dealt with from ~arious aSI~ects. The atom,however, remains. for the time being. Lhe indivisibleparticle conceived hv DeTIJocritns of ancient Greece.Problems related 10 "he motion of molecules are con­siderod, of COll[,~O. bocause Ihey are the basis for 0111'

modern knO\dedge of thermal motion. AttenLion ha"b.Dl:n g'iv8n, as IYelL 10 problems concC'rning' phas£' tran­~ntlOns.

In ,the years si IICO the preceding Ddi Lion of Physicsfor lweryolle was rpubli,;]wd, 0111' informaLioli on thestrllcLnre of moiecliles and thuir interaction has boenCOllsidurably snpplplllellted. \Ianv discoveries Iiave beenIll.ado Lila!' bridgp llip gaps "PtWO~'11 till' problolns dealingw1th tho Illoiecli LII' ,~I I'llclurp of ~II ""Lances and tliei'rproporLies.Tbis IliI' Illdlll'cd Jill' 10 add iI slll>stantialamollnt oj' /le\\ Illall'l'jal

A 10/lll OV('I'I!ll(' " . 1 I1'1'."' , !llcasuJ'('. III llJY O]lJIIJOIl, I~ 11(' ilf-l[lOll to "t'lIlli'II'1 I'll I j' I' f .

I '" ( 1'\ 100 \~ 0 "I'III'ra III Ul'lilatioll 011lila eculo..; L1' t . hf .' [,I. ,11'(' !llOl'l' ('()JII!'l('.\' tllilll til(' 1l1OIccuiesa tox?gen, llitl'ogl'll alld (',ll'huu diu.\'idp. { ]l !'u the pres-en time th" II '

t .' L.tU IOI'S o! IU'l.~t ('OIIl'Sl'." iJi ]Jhysics haveno cons1dered 'L '" .',

I IIl)Ce""ary to dual \llth 11101'8 compll··

cated combinations of atom.s. Bl1t giant molecn!es havebecome extremely common in our everyday life in theform of a great diversity of synthetic. materials. A newscience, molecnlar biology. has been fou nded to ex­plain the phenomena of living matter, using the languageof protein molecl1les and nucleic Rcids.

Likewise undeservedly omitted. as a rule, ilre problemsconcerning chemical r~Rctions. Such nlilctions belong,however, to the physical process of the collision of mol­ecules, accompanying their rearrangement. It provesmuch simpler to explain the essence of nuclear reactionsto a student or reader who is already acquainted withentirelv similar behaviour of molecules.

It w;s found expedient in revising the book to transfercertain parts of the previollS Physics for Etwyone to thesubsequent Looks of this series, 1t \\as considered fea­sible, for instance, to refer only briefly to sound in thechapter on moleeular mechanies.

It was found aJvisable. in the same manner. 10 deferthe diseussion on the features of wave motion to thetreatment of eleetromagnetic phenomena.

As a whole. the foUl' Looks of the new edition of Physicsfor Everyone (Physical Bodies, Jl1olecules. Xlectrons, and'PJlOtons and Nuclei) cover the fundamentals of physics.

Preface to the Fourth Russian Edition

April 1978

6

/1. I. !( itai[.[orodsky('

CONTENTS

Preface to the Fourth Russian Edition1. Building Blocks of the UniverseElements 9. Atoms and Molecules 12. What Heat Is 18.Energy Is Conserved Forever 20. Calorie 23. SomeHistory 24.

2. Structure of MatterIntramolecular Bonds 29. Physical and Chemical Mole·cules 35. Interaction of Molecules 36. What ThermalMotion looks like 38, Compressibility of Bodies 40.Surface Tension 43. Crystals and Their Shape 47. Struc­ture of Crystals 54. Polycrystalline Substances 68.3. TemperatureThermometer 72. Ideal Gas Theory 78. Avogadro'slaw 81. Molecular Velocities 82. Thermal Expansion 86.Heat Capacity 88. Thermal ConductiVity 89. Convec­tion 93.

4. States of MatterIron Vapour and Solid Air 96. Boiling 97. Dependence01 B T .

01 mg Pomt on Pressure 98. Evaporation 102.Critical Tempe t . ,ra ure 105. Obtaining low Tempera-tures 109 S I .. . upercoo ed Vapours and Superheated liq-Uids 112 Melf 1I II . Ing 13. How to Grow a Crystal 117.n uen,ce 01 Pressure on Melting Point 126. Evaporation~~llSOhdS 127. Triple POint 129. The Same Atoms but

I erent Crystals 132. An Amiuing Liquid 137.S. SolutionsWha1 a Solution is 141.Glses 142 S I'd ,. Solutions 01 liquids and

• 0 I SOlutions 144. How Solutions Freeze

Building Blocks

UniverseJ.of the

Elements

\Vhat is the world SlllTOlHldilig lIS made of? The firstanswers to this ([nes! ion which llave reached llS originat~

ed in Ancient Grecce more than 2.1 celltlll'ies ago.At fIrst glance. thE' answers seem as strange as can

be, and we wOlllf] have to waste a lot of paper in orderto explain to the reader the logic of the ancient sages­Thales, having asserted that everytIling consists ofwater. Anaxirnander. having said that the world ismade of air. or Heraclitus. in whose opinion every~thing consisls of nrC', '

The incongrllity of such explanations forced laterGreek "lovors of wisdom" (that's how the word "philos~oph " ' 1er lsranslatod) to increase tho number of funda~

li,e,ntal principll,'s or, as thoy were callod in antiqnitye emc t D J ~' , ,

11 s, I',rnpelocles assortod that thero are fOllr elo~monts' ("\I'll "I' . j f' \. 1 .tl) .' , ,J. \\,101. all' alll Ire. j "ls1ot 0 cOlltnbuted. ll. ,fJlla~ (101' ;1 very long lillie) corred iOlls to this iIl~vestlgalloll.

Accol'dill<l t A " 't~11 11 I I' . ftJ ~ 0 :~IISO e, a )Ol10S consls1 00110 ,IIII!

d ~ffe S;1I11(' SllhslallcP. hut this suhsl'\1lcP C"111 'ISSII11'1('1 erelll . , ' " " "

cold I ,qllal~tIPs. There are fonr immaterial eloillents:imp' t

lo1t. mOls1 and llr~r, Comhining' in pair" and hping

ar el to 'I "1 I t '\ '" I' I .I ' ,1 )"aIlCe, 1 n"tot e " P ernen1s form tho

e ements of Fj I r

Yiele]' tl ,mpe( oC,es. filii", a dry and cold suhstance, s ear I' dry ~ ] I I' ,and nnall' '. ;ulllOt, 11'0; mOist alld cold, water'

, y, mOIst and hot, air,

8Contents

146. Boiling of Solutions 148. How Liquids Are Freedof Admixtures 149. Purification of Solids 153. Adsorp­

tion 154. Osmosis 156.

6. Molecular MechanicsFrictional Forces 159. Viscous Friction in Liquids andGases 164. Forces of Resistance at High Speeds 166.Streamline Shape 169. Disappearance of Viscosity 171,Plasticity 176. Dislocations 179. Hardness 184. SoundVibrations and Waves 186. Audible and Inaudible

Pitches 195.

7. Transformations of MoleculesChemical Reactions 197. Combustion and Explosion 200,Engines Operated by Transformations of Molecules 206,

8. Laws of ThermodynamicsConservation of Energy at the Molecular Level 215. HowHeat Is Converted into Work 218. Entropy 221, Fluc­tuations 225. Who Discovered the Laws of Thermo­

dynamics! 227.

9. Giant MoleculesChains of Atoms 231. Flexibility of Molecules 234. Glob-ular Crystals 236. Bundles of Molecules 238. MuscularContraction 243,

~~~~--~-------

Molecules

)

10 1. Building Blocks of the Universe 11

However, in view of the difficulty involvrd in an­swering many questions, ancient philosophers addeda "divine quintessence" to the four elements. This wasa kind of god-cook, cooking the various elements together.Of course, it isn't hard to explain away any perplexityby reference to a god.

But for a very long time-·almost up to the 18th cen­tury-few dared be perplexed and ask questions, Aris­totle's teachings were avowed by the Church, and anydoubt in their validity was a heresy.

But these doubts arose anyway. They were engenderedby alchemy,

'In the distant past, into the heart of which we canlook by reading ancient manuscripts, people knew thatall bodies surrounding us were capable of being trans­formed into others. Combustion, sinte'ring. the meltingof metals-all these phenomena were well known.

This. it wOllld !'eem, did not contradict Aristotle'steaching. The so-called "dosage" of the elernenh changedduring any transformation. If the whole world consistsof only foUl' elements, the possibilities of transformingbodies should be very great. I t is merel y neceosary tofmd the secret of what to do in o[~le[' that <tJlY bodymight be obtained from any other one.

How tempting is the problem of making gold or fiud­ing a special, extraonlinary "philosophers' stone", givingits possessor weal th, power and eternal yout h. Thescience of manufacturing gold and a philosophers' stone,of transforming any body into any other one was calledalchem v bv the ancient Arabs.

The 1abo'1IJ' of people devoting themselves to the so­lution of this problem continlled for centuries. Alchemistsdid not learn how to make gold, did not lind a philos­ophers' stone, hut made up for this by collecting manyvaluable facts about the transformation of bodies. Inthe final analysis, these facts served as the death sen-

tence for alchemy. In the 17th century, it became ob­vious to many people that the number of basic sub­stances-elements---is incomparably greater than four.Mercury, lead, sulphur, gold and antimony turned outto be undecomposahle substances; one could no longersay that these substances were made out of elements.On the contrary, one had to rank them among the ele­ments of the world.

In 1661 in England. Robert Boyle (1627-1691) pub­lished the book The Sceptical Chemist. Here we finda completely new del'lllition of an element, This is nolonger the elusive, mysterious immaterial element ofalchemists. An element is now a substance, a componentpart of a body. This is consistent with the modern def­inition of the ('oneHpt of an element.

Boyle's list of pjements was not very large. He added~re to a correct iiQt. Incidentally, the ideil of elementslIved on even ;lfter Boyle. Even in a list of the great~renchman Antoinu Laurent Lavoisier (174.3-17D4), whoIS, regarded as the founder of cl18mistry, side by sidewIth real elements there also appear imponderableelements: a heat-p,'oducing and a light-producinff sub-stances. "'"

kIn the first halt' of tl10 18th century there were 15

nOWll element I ! . '1 . .s, aJl( t lelr number rose to 35 by theeeTll( of the clmtllry. True, only 23 of them were real

ements but tl·. . . .or I" _,' " ll~ le"t were eIther non-eXIstent elementse ,H Ciuhstances' 1'1 . d d 'who 1 , ", 1 \.e caustIc so a an caustIc potash

I~c I turned out to be compounds.'y the mid II f J

d ( e 0 t wIDth century more than 50un ecompo"'d! ,.. 'h db " J e :·;u Jst ,lnces were descrihed in chemicalan ooks.

The periodic 1, 'j' 1,1 '.Me d I. '. d\\ () 18 (IT.pat f{ Il"SJaIl ch"rnI'st DmI'tri'

Il e eev (1N')/ (~ "'. . " , "scious s 'l(·:ll.lUI) provHle.d tho !'timuh.Is for a con­

ealCl lor 1 .[.[ Jearly to . k lIle Iscover8< e ernents. It is still toospea about this law here. Let us merely say

Molecules 12 1. Building Blocks of the Universe

that by means of his Jaw Mendeleev showed how onemust Jook for the elements which had not yet Iwen dis­covered.

Almost all the elements occurring in nature werediscovered by the beginning of the 2Uth century.

Atoms and Molecules

Ahout 20UU years ago, an original poem was writtenin Ancient Homp. Its author was the Homan poet Lucre­tius. His poem was called Oil the Nature uf Things.

\Vith sonorous lines Lucretius told of the ancientGreek philosopher [)emocritus' views on the worldin his poetic work.

\Vhat views were these? These were teachings aboutthe minutest, invisible particles which our whole worldis made of. Having observed various phenomena, De­mocritus tried to give them an explanation.

Take wator, for ('samplo. \VlJen suff'lcieutJy heated,it eVapOI'Cltes and disappears. TIow can this he explained?It is clear tit'll "liCit a property of water is related toits internal strllctlln'.

Or why, for example, do we .!lereeive the scent off1ower~ at. a dista~lc~? . , .

JVledltatlllg on sundar questIOns. Democl'Itus becamecOl\vinced tltat bodies only seem to be solid. but illfact consist of the Illinute'st partieles. These particlesare dif[erent in forlll for dif[ment bodips, hut they meso slllall that the~' c,lnnot be .~()('[1. Tlla! is wlty all hudi('ssoom to us to bp ~olid.

Domocritus called slich n~ry tin~' particles whichcannot be fnrtbpl' dividpd and of which wator and allother bodies consi~t aLums (derived from tho (;reekaLomos meaning "indivisible").

This remarkable glless of ancient Greek thinkers,born 24 centuries ilg·U. was later long' forgotten. Aristotle's

erroneous teaching exercised complete sway over theSCiClilii"lc world for more than (1 thollsand years.

AC'~erting that all substances art) III II tuaU y trmlS­Illiitable, Aristotle categorically dellied the existence"I' atoms. Any hocly can be illflTlitel~' divided, taught.\ristotle.

In 1G47, the Frenchman Pierre CassPlldi (1;)~J2-Hi55)

puhli~hed a book in which Iw ('ourageonsly denied Aris­totle's teaching and asserted that all substances in theworld consist of smalI indivisible particles-atoms..'\ toms differ froJIl each other ill shape, size and JIlass.

Agreeing with the teachings of the ancient atomists,(; as~ell di developed these teach ings further. He explainedl!ow the millions of diverse bodies of nature can andtI'l arise in the world. For this, he assertpd, a large numberof different atoms is not necessary. For an atom is the~,1I1]() thing as building materials for houses. It is pos­~i ble to construct an enormous number of the most di­\ ('rse houses from three different kinds of building mate­rial-- bricks. boards and logs. In precisely the sameway. nature can create thousands of the most diversebodip" from several tens of different atoms. Moreover,I" paeh bod y various atoms are united in small groups;(;a",.;endi called these groups molecules. i.e. "small!lIasses" (derived from the Latin moles meaning "mass").

\folecules of various bodies differ frOIll each otherIII the n 11m bet· and kind ("sort") of a toms belonging toth('Ill. It is not difficult to understand that an ill1mpnse11'"nher of dil"forent combinations of atoms, molecldes,,an he created from sevel'id (,f)ns of diffprent atoms.rhi,., is why there is s1lch a great variety in the bodies-lIlTolindilig liS.

1-I0\\8V8J'. Gassentli's views still ('onlaill8d IllllCh thatwas iIiCOLTOCt. Thus, he belipved that tllOre \I'('re s]wcial<llOIll" for Iwat. cold, taste and :mwll. As other scientists,II' that time he too could not cotllplet8ly fn·e himself

from Aristotle's influence, and recognized liifi immaterialelements.

The following ideas experimentally veril'led rnuch laterare contained in the writings of j\1. V. Lomonosov,the great enljo'htener and founder of science in Hussia.

Lomonofiov ~\Tites that molecules can be homogeneoufior heterogeneous. In the forUler case, fiimilar atomsare grouped in a molecule. In 018 latter, a moleculeconsists of atoms di ffering from each other. If someboLly is comjlOsed of homogeneous molecules, it 111 ustbe regarded as simple. II, on the contrary, a body con­sists of molecules built up from various atoms, Lomo­nosov calls it compound.

vVe now We'll know that lJature's various bodies haveprecisely fiuch a structure. In fact, take the gas oxygenfor exam pIe; two iden tical atoms of oxygen are containedin each of its molecules. This is a molecule of a simplesubstance. But if the atoms composing a molecule aredifferent, it is a chemical compound. Its moleculesconsist of atoms .of those c~lOmieal eleI1J8nts w~lich occurin the compC,sltlOn of tIllS compounJ. TIm; can alsobe saiLl othe:'\\ise: each simple substance qonsists ofatoms of one chemical element; a compound. containsatoms of two or more elements.

A number ,jf thinkers spoke about atoms, adducinglogical arguments in favour of their existence. TheEnglish scientist John Dalton (17GG-lo44) introducedatoms into SClence in tile right way and maLle them anobject of research. Dalton showed that there exist chemicalregularities which can be explained in a natural manneronly by making use of the idea of an atom.

After Dalton, atoms lirmly entered science. However,for a very lam;' time there still were scielltisLi who did llotbelieve in atoms. Even at the very elld of tIw last century,one of thelll.vI'ote that after several decades it wouldbe possible to End atoms ollly ill tIll' dust of libraries.

..

Molecules 14 1. Building Blocks of the Universe 15

Such reasoning seems funny now. \Ve know now soIllany Lletails about the "life" of an atom that to doubtits existence is the same thing as to doubt the realityof the Black Sea.

The relative masses of atoms \vere determined bychemists. At first the mass of a hydrogen atom was takenas the atoIllic mass unit. The relative atomic mass ofnitrogen turned out to be approximately equal to 14,oxygen, approximately 16, chlorine, approximately 35.5.A somewhat different choice of the relative atomicmass units was later made, for which the number Hi-OOOOwas assigned to oxygen. The atomic mass of hydrogenturned out equal to 1.008 in this scale.

Of interest, of course, is the absolute mass of atomsand not onl y their relative mass. It is su fficien t, forthis purpose, to measure the absolute mass of an atomof anyone kind. Taken as the basis today is carbonrather than oxygen or hydrogen. Up to the present time,investigators regarded measlll'ements of absolute massesof atoms with distrust and proceeded as follows. Theytook the IIlass of the carbon isotope He to be equal toexactly twelve atomic mass units (amu). Then, payingno attention to the accuracy of measurement of theabsolute masses of atoms, they assumed that

1 amu = 1.6G2 X 10-2" g

In auy case, this value does not differ appreciablyfrom the true one. Perhaps they are overcautious, how­ever, since the precision of measurement today is withina fraction of about one-millionth. :\Teasuring techniqueshave advanced greatly during the last century. In 1875,the known value of 1 amu was accurate within about:3U per cen t.

How do we measure the mass of the atom in grams?No scales have been constructed, of course, on whicha physicist could place a single atom and then balance

it with a tiny weight. Like a hundred years ago, physi­cists use indirect experiments for this purpose todayas well. They arc not, however, in any way less reliablethan direct weiglling would be. nut we cannot do withoutany weighing at all. \Ve put a solid ball of carbon 12C

on the scale rather than a single atom (actually we pro­ceed in a somewhat difl'Nent way, but the point is to ex­plain the idea of weighing and so we hope the informedreader forgives our simplifIed description). \Vhen themass of the ball and its size are known, we can determineits density. The substance being weighed must be aperfect crystal. This is not easy to achieve, but moreor less feasible. Hence, we can evidently write the fol­lowing for the density found in the experiment.

ZillAp=V

where mA=' mass of the atom in amuV ~= volume of a unit cell of the crystalZ number of atoms per unit cell (see p. 61).

The last two quantities can be determined by X-raystructure anal,\isis which is to be discussedi n the fourthbook.

The reader should not resent the fact that I seem tobe getting ahead of my "tory. Books on physics shouldbe read at least twice.

Employing this method, we can determine the atomicmass unit with exceptionally great precision. The mostreliable value today is

1 amu = (1.6IlH3 + 0.000 :11) X 10-2 '. g

We now ask the reader to usc his imagination in orderto grasp the minuteness of this value. Assume that youdemand a thousand million molecules from each personOIl earth. How milch matter can you collect in tIl is way?Only several thousand ths of a million til of a gram.

Molecules l(i

I

II

iL

1. Building Blocks of the Universe 1i

Or another such (;omparison: tho oarth is as manytill]()" /wavier tltall an apple a" all apple is heavier thana hydrogell atom.

The reciprocal of tho atomic mass uni t is called A vo­gadro's number:

N A = _1_ = (i .022 091:1 >( 10231 amll

This onormous number has tho following significance.Let us take a substance of an amount in grams equalto the relative mass 111 of its atoms or molecules, forexamplo, 12 grams of the carbon isotope 12C. This canbe expressed more concisely: lot us take one malo ofa substance (check, please, with the definition of thomole given in the first book, where wo introduced theInternational System of Units-SI units). The mass ofone mole of a substance is equal to Jlm\. Consequently,the numbor of carbon atoms in 12 grams of carbon, aswell as tho number of atoms, molecules or any otherpalticles in a mole af'"om hly of thef'e particlof', equals

JI ~T---- = lv AJ! iliA

which if' Avogadro's number.For a long time physicif'ts found no ncccsf'ity for using

lhe concept of the "amount of substance". As long aswo dealt only with atoms and molocules it was quitesuitablo to defJne the mole as tile molecular (01' atomic)weight expref'scd in gramf'.

HIlt then iOIlS, elee,troll". me"OIlf' and many, manyilion' pari ieil'" made II](lie apj>p,lrilllcP. Physicists camp10 L!w cOllclusion lhat it is lloL alwa\'s convPIliont tocharacterize an assembly of jJ;\I'ticl« by their masf'.This led to the establi"JlInenL of lhe unit for 1he amountof substanco-the mole. \Vhell w(~ f'peak of a mole ofelectrons, a mo](, of lead aLom Iluclei or a mole of pi­mosons, wo arc noL "pecifyillg Lho mass of t11Pf'8 particles2-038\)

~'I"'" -- ~-- U~

I

Molecules 18 t 8uilding 810cks of the Universe f9

(which, as you will find further on, depends upon theirvelocity), but only their number. The previous definitionof a mole is still valid, however, because N A atomsor molecules of any kind have a mass equal practicallyto the atomic or molecular mass expressed in grams.Neither has Avogadro's number changed its meaning;it simply has a new name: mole-I.

What Heat Is

How does a hot body differ frQm a cold oV? Up untilthe 19th century, this question was answered as follows:"A hot body contains more heat-producing matter (or'caloric') than a cold one, in exactly the sam~ senseas soup is saltier if it contains more salt." But whatis caloric? The following answer was given to this question:"Caloric is the matter of heat, it is the elementary fire."Mysterious and incomprehensible. And this answer isin essence the same as the following explanation of whata rope is: "A rope is simple 'ropeness'."

Along with the caloric theory, a different view onthe nature of heat had long been in existence. It wasbrilliantly advocated by many outstanding scientists ofthe 16-18th centuries.

Francis Bacon wrote in his book N ovum Organum:"Heat itself in its essence is nothing but motion .... Heatconsists of a variable motion of the minutest particlesof a body."

Robert Hooke asserted in his book Micrographia:"Heat is a continuous motion of the parts of a body ....There is no such body whose particles would b,e at rest."

We find particularly clear statements of this kindin Lomonosov's work (1745) Reflections on the Causeoj Heat and Cold. The existence of caloric is denied inthis work, where it is said that "heat consists of theinternal motion of particles of matter".

Count von Rumford put it very graphically at theend of the 18th century: "The more intensively thep~rticles composing a body move, the hotter the bodyWIll be, analogous to how the more vigorously a bellvibrates, the louder it rings."

In these remarkable guesses, far ahead of their timethe bases of our modern views on the nature of heaiare concealed.

There are sometimes quiet and clear days. The leaveslie still on the trees, not even a slight ripple disturbsthe glassy surface of water. The entire surroundingsh~~e frozen i.n strict, triumphant immobility. TheVISIble world IS at rest. But what is taking place inthe world of atoms and molecules?

Contemporary physicists can say much about this.Never, not under any circumstances, is there a cessation~o the invisible motion of the particles that the worldIS made of.

But why don't we see all these motions? Particlesmove, but the body is stationary. How is this possible?. Have you ever watched a swarm of midges? When thereIS no wind, the swarm appears to be suspended in air.But an intensive life is going on inside the swarm. Hun­dreds of insects flew off to the right, but just as manyflew off to the left at the same instant. The swarm as~ whole remained at the same place and did not changeItS form.

The invisible motions of atoms and molecules areIf the same chaotic, irregular nature. If some molecules~ave a volume, their place is occupied by others. But

~lllce the newcomers do not in the least differ from theAep~rted molecules, the body remains entirely as it was.thll Irregular, chaotic motion of particles does not change

? properties of the visible world.kHowever, isn't this idle talk?" the reader might

as us. In what sense are these arguments, however

20

.• i

Molecules

beautiful more convincing than the caloric theory? Hasanyone ~ctually seen the eternal thermal motion ofparticles of matter? ..

I t is possible to see the thermal motion o~ particlesand, moreover, with the aid of the simplest mICroscope.This phenomenon was first observed more than a hundredyears ago by the English botanist Robert Brown (1773-1858). . ....

Looking at the internal structure of a plant througha microscope, he noticed that tiny particles of mat.terfloating in the sap of the plant were continually movmgin all directions. The botanist became interested: whatforces made the particles move? Perhaps they wereliving beings of some kind? The scienti~t decided toexamine under a microscope small partIcles of claymaking some water turbid. But neither were these un­doubtedly lifeless particles at rest; they were engagedin a continual and chaotic motion. The smaller theparticles were, the faster they moved..The botani.stexamined this drop of water for a long time, but stIllhe couldn't see any end to the motion of the particles.Some invisible forces seemed to be constantly pushingthem.

The Brownian movement of particles is just a thermalmotion. Thermal motion is inherent in large and smallparticles, clots of molecules, individual molecules andatoms.

Energy Is Conserved Forever

Thus the world is composed of moving atoms. Atomspossess 'mass, moving atoms possess kinetic energy. Ofcourse the mass of an atom is unimaginably small, andso its ~nergy will also be minute, but there are millionsof millions of millions of atoms.

f. Building Blocks of the Universe 21

We now remind the reader that although we spokeof the law of conservation of energy, this was not asufficiently universal conservation law. Linear and angularmomenta were conserved experimentally, but energywas only conserved ideally-in the absence of friction.But as a matter of fact, energy always decreased.

But we did not say anything previously about theenergy of atoms. A natural idea arises: where at firstsight we noticed a decrease in energy, some energy wastransmitted to the atoms of a body in a manner whichis imperceptible to the naked eye.

Atoms are subject to the laws of mechanics. True(you will have to learn this from another book), theirmechanics is somewhat peculiar, but this does not changematters-with respect to the law of conservation ofmechanical energy, atoms do not differ at all from largebodies.

Hence, the complete conservation of energy will bedetected only when along with the mechanical energyof a body the internal energy of this body and the en­vironment is taken into account. Only in this case willthe law be universal.

What does the total energy of a body consist of? Wehave, in essence, already named its first component­it is the sum of the kinetic energies of all its atoms.But it must not be forgotten that atoms interact witheach other. Therefore, the potential energy of this in­teraction is added. Thus, the total energy of a bodyis equal to the sum of the kinetic energies of its particlesand the potential energy of their interaction.

I t is not difficult to comprehend that the mechanicalenergy of a body as a whole is only part of its total ener­gy. For when a body is stationary, its molecules donot stop moving and do not cease interacting with eachother. The energy of the thermal motion of particlesWhich remains in a stationary body and the energy

of the interaction he tween the part ides constitute theinternal energy of the hody. The toLal energy of a hodyis therefore equal to the snm of its mechaniral and in­ternal energies.

Tho gravitational energy of a body is n componentpart of its mechanical energy, i.e. the potenti,l1 energyof the inLeraction between its pmtic:les ,md the l':arth,

Considering internal energy, we no longer detectvanishing of energy. \Vhen ", consider nature throughglasses magnifying tllP WOrll' millions of times, thepicture seems to us to be of rare harmoniollsness. Thereare no losses of mechanical energy, but there is onlyits tnmsform ation in to the in ternal energy of a bodyor its sUl'l'ollndings. Has any work disappeared'? No!The energy went into an arco]rratioll of the relativemotioll of molecules or n clliwgC' in their mntnal distribu­tion.

Molecules obey the la \V 0 f conserva tion of mechanicalenergy. There are no frictional forces in the world ofmolecules; the world of molecu les is controlled by tra ns­formations of po ton ti al energy into kinetic one andv.ice versa. Only in the COi\rse world of large objects,whicJl does not notice molecules, doee; "energy vanish".

If 11Iec:lwnical prH'!";'y toUdly or partially vanishesduring sonw OCCl1ITeW,'O, tlte intern;11 enorgy of the bodiesand medii1 participating in tltis OCl:Ul'renco will growhy tlw sanw amount. ]JulLing it otlH'ndse, mcch;l!licnll'nprgy is tr;ln:,forllled \\ it!lout ,lily' loss whatsoever intotill' ClIl'rgy of nlOll'cli!l'" PI' Illoms.

'1'1](: Ll\Y of COIISPI'V,lliu!l of l'fL'l'gV i:" t he strictest"!Joold,pqH'I'" of pilysics. 'rbl' itll'ollle ,\lH[ ou Igo of energyshould eXilctly b,ilance dUl'ing ,Iny OCCUI'l'C'ftCC. If thisdill not takr placu in some experiment, this impliesthat sOltlethiftg import;mt escapeo OUI' ,lttefttion. In suclta case t lie la\\ or conserv;ll ion of ellf'l'g'~' [~'i\'es liS a sig­nal: researcher, repl:;I! the expel'illll'nl, ill('J'I',lse til(; IIC-

Molecules 22 1. Building Blocks of the Universe 23

curacy of your measurements, look for the cause of theloss! Physicists have repeatedly made new, importantd iscoveri es along these lines, convincing themselvestime and time again of the perfectly strict validity ofthis remarkable law.

Calorie

We already have two units of energy-the erg' andthe kilogram-farce-metre. It would seem that this isenough. However, it is traditional to employ yet a thirdunit-the calorie--in the study of thermal phenomena.

We shall see later that even with the calorie the listof units adopted for designating energies is not exhaust­ed.

It is possible that in each individual case the use ofits "own" unit of energy is convenient and expedient.But in any example which is the least bit complicated,dealing with the transformation of energy from oneform to another, an inconceivable mix-up with unitsarises.

In order to simplify computations, the system ofunits (81) provides for a single unit for work, energy'Ind an amount of heat--the joule. However, consideringIhe strength of tradition and the length of time whichwill be required for this system to become the onlysystem of units in general use, it is helpful to acquaintourselves more closely with the "departing" unit ofheat-the calol'ir.

The small ca/urie (cal) is the amount of heat rrquiredto raise the temperature of one gram of water from 14.5to 15.5 ac. The word "small" ill ust he mentioned be­canse one sometimes uses the "large" calorie, which is,1 thousand times as great as the chosen unit (the large('rzlorie is often denoted by kenl, which means "kilo­!·i1!orio").

Molecules 24 1. BUilding Blocks 01 the Universe 25The relationship between a calorie and mechanical

units of work, such as an erg or a kilogram-farce-metre,is found by heating water mechanically. Such experimentshave been performed repeatedly. I t is possible, for exam­ple, to raise the temperature of water by stirring itenergetically. The mechanical work expended for theheating can be evaluated with sufficient accuracy. Itwas found from such measurements that

1 cal == n.427 kgf-m ~~ 4.18 J

Since energy and work have units in common, itis also possible to measure work in calories. One mustexpend 2.35 calories in order to raise a kilogram weightby one metre. This sounds unusual, and it really isinconvenient to compare the raising of a load with theheating of water. Therefore, calories are not employedin mechanics.

.,Some History

The law of conservation of energy could only be for­mulated when the idea of the mechanical nature ofheat had become sufficiently clear and when technologyhad posed in practice the important question of theequivalence between heat and work.

The first experiment establishing a quantitative re­lationship between heat and work was carried out bythe well-known physicist Sir Benjamin Thompson (Countvon Humford) (1753-181'1). He worked in a factory wherecannon were manufactured. \Vhen the muzzle of a gunis bored, heat is liberated. How could it be estimated',)What should be taken as the measure of heat? I t ac­e urred to Humford to relate the work performed inboring with the heating of one or another amount ofwater by one 0[' another Humber of degrees. This in­vestigation was perhaps the first precise expression of

the idea that heat and work should have a measurein common.

The next step towards the discovery of the law ofconservation of energy was the establishment of animportant fact: a disappearance of work is accompaniedby an appearance of a proportional amount of heat;thus a common measure for heat and work was found.

The original de[llli tion of the so-called mechanicalequivalent of heat was given by the French physicistSadi Carnot (1796-1832). This outstanding person diedat the age of 36 in 1832 and left behind a manuscript,which was published only after 50 years. The discoverymade by Carnot remained unknown and did not in­fluence the development of science. Carnot calculatedin this work that the raising of 1 m3 of water to a heightof 1 m requires just as much energy as is needed for theheating of 1 kg of water by 2.7 degrees (the correct figureis 2.3 degrees) .

The Heilbronn doctor Julius Hobert von Mayer (1814­1878) published his fll'St work in 1842. Although Mayercalled physical concepts familiar to us by entirely dif­ferent names, a careful reading of his work leads never­theless to the conclusion that the essential features~)f the law of conservation of energy are presented inI t. Mayer distinguished between the internal energy("thermal"), gravitational potential energy and energyof motion of a body. He tried to infer the necessityof conservation of energy under various transformationsfrom purely theoretical considerations. In order to checkthis assertion experimentally, one must have a commonllleasure for measuring these energies. Mayer calculatedthat the heating of 1 kg of water by one degree is equiv­alent to the raising of 1 kg by 365 m.

In his second work published three years later, Mayernoted the universality of the law of conservation off3nergy-the possibility of applying it to questions of

Molecules 26 1. Building Blocks 01 the Universe 27

...

'4;bii/< -~}r' ", II" .." //~

I

I

\

III

IiHermann Helmholtz [1821-1894J-a famous German scientist. Helm­holtz worked in 11](' fields of physics, mathematics and physiologywith great snccm's. Ill' was the first (181j7) to give a mathematic'llinterpretation of Ihe ];1\V of conservation of en('r~y emphasizingf,/lf) nniver"al cllarnctr![· 01" litis Ja\\. lIelnlboltr. obt:Jinef] ollts1nnd·

chemistry, biology and cosmic phenomena. To thevarious known forms of energy JVfayer added magnetic,electric and chemical.

A lot of cred it for the discovery of the law of con­serva tion of energy goes to the remarkable Englishphysicist (a brewer from Salford in England) James Pres­cott Joule (1818-1889), working independently of Mayer.

While a certain inclination to an indeterminate phi­losophy is characteristic of Mayer, Joule's basic traitis a strict experimental approach towards the phenom­ena umler consideration. Joule posed a question be­fore nature and obtained an answer to it by means ofspecial experiments set up in an exceptionally painstak­ing manner. There is no doubt that in the entire seriesof experiments performed by Joule, he was guided by asingle idea-to find a common measure for evaluatingthermal, chemical, electrical and mechanical actions,to demonst rate that energy is conserved in all thesephenomena. Joule formulated his idea as follows: "Thedestruction of forces performing work does not occurin nature without a corresponding action."

Joule reported on his first work on January 24, 1843,ancl on August 21 of the same year, he communicatedhis I'e:-'ults OIL the establishment of a eommon measurefor he'at and work. Heating' '1 kg of water by one degreeprove(1 equivalent to rai:-'ing 1 kg by 4(iO m.

----- -----------

illli' 1'I'''ltll', ill lIl('rJI:od~I];!lllic,,; Ill' lIas tlill f'Jr"l. 10 apl'l,\' il. to th!,:'11111,\' III' ,.jll<mica] [1['0,'0""[1", HI' 111("111"; III' hi,.; lIo,'k 011 Llle \'orl.pxIllotion of liq1lids, ifelilllJOllz laid till' fOlllld,ilillll,':' of hYf]ro­d\,lla[llics :\lld ,wJ'()dynaIilic:,. ][e ca[Til'd Ollt a number of vnlttahleinwsligatiolls ill l1w I'l()lds or acolI"tics and electromagnelism,1I,'llllllOltz den,loped the physical tlieory of musil:. ITe applied[lOll (,dill :111'] Ol'igill,li l11atlll'JlI,lIic;!l Illl'thofl" iJl liis ph\'sicalf'l':';L'';1 l'C 11.

In the following years, Joule and a number of otherresearchers spent a great deal of effort in order to finda more precise value for the thermal equivalent, andalso attempted to prove the complete l111iversality ofthe equivalent. During the late forties, it became clearthat, regardless of how work is transformed into heat,the amount of heat arising will always be proportionalto the amount of work expended. In spite of the factthat Joule laid the experimental basis for the law ofconservation of energy, he did not give a clear formula­tion of this law in his works.

The credit for this. belongs to the German physicistHermann Helmholtz.'On July 23, 1847, at a meet inaof the Berlin Physical Society, Helmhol tz gave a lectur:on the principle of conservation of energy. The mechanicalbasis of the law of conservation of energy was clearlypresented for the first time in this talk. The world con­sists of atoms; atoms possess potential and kinetic ener­gies. The sum of the potential and kinetic energies ofthe particles which a body or system is made of cannotchange, if the body or system is not subjected to ex­ternal influences~ The law of conservation of energy,as we outlined it several pages above, was first formulatedby Helmholtz.

After the work of Helmholtz, it remained for otherphysicists to merely verify and apply the law of conserva­tion of energy. The success of these investigations ledto the fact that by the end of the fifties the law of con­servation of energy was universally recognized as afundamental law of natural science.

Phenomena casting doubt on the law of conserva tionof energy have already been observed in the 20th century.~owever, ~xplnnations were later found for the apparentdlscrep;mcles. The law of conservation of energy hasso f;1I' always stood the test with cmdit.

Molecules 28 2. Structure of Matter

Intramolecular Bonds

Molecules consist of atoms. Atoms are hound in mol­ecules by forces which are called chemical forces.

There exist molecules consisting of two three four'l'h I ' ,a~oms. e argest molecules, protein molecules, consist

of tens and even hundreds of thousands of atoms.The ~o~ecule kingdom is exceptionally varied. By

now, mIlllOns of substances built up out of variousmoleeules have already been isolated bv ehemists fromnatural materials and created in their faboratories.

Properties of moleeules are determined not only byIJO\: many ato~s of one or another sort partieipate int~l('[[·. const.ructlOn hut also by the order and eonflgura­tlon l~ wInch they arc bound. A molecule is not a heapof bncks, but a complieate<] arehiteetural strueture,wher: eaeh .brick has its plaee and its eornpletely de­termllled neIghbours. The atomie strueture formina aIlloleeule ean be rigid to a greater or lesser degree.'" Inany case, eaeh of the atoms earries out an oseillationabout its equilibrium position. In certain eases, someJl:1I'~S of ~ ..molecule can even revolve around other parts.~·lvlng dIl[erent and the most fantastic configurationsIt) a fn'p molecule in the process of its tlwrmal motion.

LPl us'lIlalyze the inlnaction bptwccn atoms in greaterdetail. The potential energy curve of a diatomie mole­\'ule is depicted in Figure 2.1. I t has a characteristie~orm-it first goes down, then turns up forming a "well",dn~l afterwards rises more slowly towards the horizontalaXIS on whieh the distance between the atoms is marked.

Molecules 30 2. Structure of Matter 31

Figure 2.1

We know that the state in which the poten ti al energyhas the minimum value is stable. \Vhen an atom formsa part of a molecule, it "sits" in a potentinl well, car­rying out small thermal oscillations ahoul its equi­librium position. .

The distance from the vertic a1 ax is to the bottom 01the well can be called the equilibrium llisLance. 'rheatoms would be located at this distance if the thermalmotion were to. cease. .

The potential energy curve tells about all the cletrulsof the interaction between atoms. \VheLher particlesattract or repel each other at one or another distance,whether the strength of the interaction increases ordecreases when the particles separa Le or approach allthis information can be obLained from the analysis ofthe potential energy Cllrve. Points to the left of Lhe "bot­Lorn" of the well cOJTespond Lo repulsion. On the contrary,points to the right of the bottom of the well ch;~racte~iw

attraction. 'rhe steepness of the curve also yIelds im­portant information: the steeper the curve, the greaterthe force.

\Vhen atoms arc at great distances from each other,they are attracted; this force decreases rather rapidly

with au iucrea::ie in Lhe dbLaJlce bet\\een them. As the~

approach each other, Lhe force of attraction grows andreaches its maximmn value when the atoms come veryclose to each other. l\S they come even closer, the at­traction weakens and, finally, at the equilibrium distancethe force of the interaction vanishes. 'When the atomsbecome closer than the equilibrium distance, forces ofrepulsion arise which sharply increase and quickly makea further decrease in the distance bet\veen the atomspractically impossible.

Equilibrium distances (below we shall say distancesfor the sake of brevity) between atoms are different forvarious types of atoms.

For various pairs of atoms, not only are the distancesbetween the verLical axis and the bottom of the welldifferent but so are the depths of the wells.

The depth of a well has a simple meaning: in orderto roll out of the well, an energy just equal to the depthis needed. Therefore, the depLh of a well can be calledthe binding energy of the particles.

The distances between the atoms of a molecule areso small that it is necessary to choose appropriate unitsfor their measurement; otherwise, their values wouldhave to he expressed, for example, in the following form:0.000 000 012 em. This figure is for an oxygen molecule.

Units especially convenient for describing the atomic\Vorld are called angstroms (true, the name of the Swedish1'cionList in whose honour these units were named isproperly spelt Angstrom; in order to remember this,H 1'mall circle i::i placed over the letter A);

1 "\ c= 10-8 em

i.e. one-hundred-millionth of a centimetre.The distances between the atoms of a molecule lie within

1118 limits of 1 to I[ ;\. The equilibrium distance for oxy­gen, which has been written out above, is equal to 1.2 A.

Molecules 322. Structure of Matter 33

:3- - 0:38 \l

Diagrams of the distrihtltion of t,he ,~en~res of theatoms in these molecules l1're shown -Hl F IgUles 2.2 and2.3. The linos symbolize the bonds,. •

A chemical reaction occurred; there were moleculesof one type, and then others were formed. Some bondswere broken, while others were newly created: I~ °t~)to break the bonds between atoms (rec,all Fl~U1e b'llit is necessary to perform work, just as. lI1,rolhng a ,aout of a well. On the contrary, cnergy'lshberated "',hennew bonds are formed, just as when -!'l ha}l JaIls .Into

a well. , . I' 1 '''Vng or\Vhich is greater, the wor]"lIlvolvc( lJI,)] e,l d. Iin creating bonds',' \Ve come across reactIons of bot I

types in natnre.. . l ·~r 't or more'fll" ...·xces" energT is called the thel ma eJ./ e( , 1

co V" . , . J , '( ') 'II' lcatcOIlci"elv the heat 0/ trans/ormatlOlI reactIOn. H, '

of react'i;l!l i" u"u:dly a quantilY of lite OI'<!N ol tens

..

Interatomic distanccs, as you soe, are very small.1£ we gird the Earth with a string at the equator, thonthe length of the "belt" will be as many times greaterthan the width of your palm as:thc laLter is greater thanthe distance between the atoms of a molecule.

Ordinary calories are used for measuring bindingenergies, but they are related not to one molecule, whichwould, of course, yield a negligible number, but toone mole, i.e. to the number of grams equal to the rel­ative molecular mass.

It is clear that the binding energy per mole dividedby Avogadro's number, N A = G.023 X 1023 mol-I, yieldsthe binding energy of a single molecule.

The binding energy of the atoms in a molecule, justas interatomic distances, varies within narrow limits.

For the same oxygen, the binding energy is equal to116000 cal/mol, for hydrogen 103000 cal/mol, etc.

We have already said that the atoms in a moleculeare distributed in an entirely defmite manner with re­spect to each other, forming in complicated cases ratherintricate structures.

Let us present several simple examples. In a moleculeof CO 2 (carbon dioxide), all three atoms are lined upin a row, with the carbon atom in the middle, A moleculeof H 20 (water) has an angular form, with the oxygenatom at the vertex of the angle (it is equal to105°).

In a molecule of l\lL; (ammonia), the nitrogen atomis at the vertex of a three-faced pyramid, in a moleculeof CIJ. (methane), thc carbon alom is located at thecentre of a four-faced liglu'e with equal sides, whichis called a tetrahedron.

The carbon atoms of C6 H6 (benzene) form a regularhexagon. The bonds of the carbon atoms with the hydro­gen atoms go from all the vertices of the hexagon. Allthe atoms are situated in one plane.

CARBON DIOXIDEo C 0

(j) 8 ~

o

/l'..K ./ "- "'-.,. K

WATER

Figure 2.2 Figure 2.3

Molecules 34 2. Structure of Matter 35

of thousands of calories per mole. The heat of reac­tion is often included as a summand in the formula fora reaction.

For example, the reaction whereby carbon in the formof graphite burns, Le. unites with oxygen, is writtenout as follows:

C + °2 = CO 2 + 94 250 cal

This means that when carbon combines with oxygenan energy of !:J4 250 calories is liberated.

The sum of the internal energies of a mole of car­bon and a mole of oxygen is equal to the internal energyof a mole of carbon dioxide plus 94 250 calories.

'rhus, such formulas have the transparent meaning ofalgebraic equalities written in terms of the values ofthe internal energies.

With the aid of such equations, one can find the heatsof reaction for which direct methods of measurement,as a result of one or another cause, are unsuitable. Hereis an example: if carbon (graphite) were to combinewith hydro~n, the acetylene would be formed:

2C + H 2 = C2H 2

The reaction does not proceed in this manner. Never­theless, it is possible to find its thermal effect. We writedown three known reactions:

(1) the oxidation of carbon2C + 20 2 = 2C0 2 + 188 000 cal

(2) the oxidation of hydrogen1

H2+"'2 02 = H20 + 68 000 cal

(3) the oxidation of acetylene5

C2H2+T 02 = 2COz+ H20 + 312 000 cal

All these equalities may be regarded as equations forthe binding energies of molecules. If so, we may operateon them as on algebraic equalities. Subtracting thefirst two equalities from the third, we obtain:

2C + H 2 = C2H 2 - 56 000 cal

Therefore, the reaction we are interested in is accom­panied by the consumption of 56 000 calories permole.

Physical and Chemical Molecules

Until investigators had formed a detailed conceptof the structure of matter, no such distinction was made.A molecule was simply a molecule, i.e. the smallestrepresentative of a substance. It seemed that nothingmore could be said. This is not so, however.

The molecules we have just discussed are moleculesin both senses of the word. Molecules of carbon dioxide,ammonia and benzene (mentioned above), and the mol­ecules of practically all organic substances (which werenot discussed) consist of atoms strongly bonded to oneanother. These bonds are not ruptured by dissolution,melting or evaporation. The molecule continues to be­have as a separate particle or small physical body uponany physical action or change in state.

But this is not always true. For most inorganic sub­stances, we can speak of the molecule only in the chem­ical sense. The finest particles of such well-knowninorganic substances as common salt or calcite or sodado not even exist. We do not find separate particlesof these substances in crystals (this will be discusseda few pages further on); when they are dissolved, themOlecules break down into their component atoms.

Sugar is an organic substance. Therefore, the sugardispersed in a cup of sweetened tea is in the form of3*

';1!

,,.

~~. Mmolecules. Salt is a difie-eeht matter. We find no moleculesof common salt (sodium chloride) in salty water. These"molec~.rles" (we have to use quotation marks) exist inwater In the form of atoms (actually, ions-electricallycharged atoms-that will be discussed later).

The same is true of vapours; and in melts a partoC the molecules live their own independent lives .. When we speak of the forces binding the atoms togetIierIn a physical molecule, we call thent' valence forces.Intermolecular forces are not of the valency kind. Thegeneral shape of the interaction curve, of the type il­lustrated in Figure 2.1, is the same for both kinds offorces. The difference lies in the depth of the potentialwell. For valence forces the well is hundreds of times'deeper.

Interaction of Molecules

There can be no doubt of the fact that molecules at­tract each other. If they stopped doing so for ,an instantall liquids .'and solids would decompose into molecules:

Molecules repel each other, and neither can this bedoubted, because liquids and solids would otherwisecontract with extraordinary ease. ., Forces ine exerted between molecules which resemblein many respects' the forces between atoms spoken ofabove. The potential energy curve which we have justdrawn for atoms gives a true picture of the basic featuresof molecular interaction. However, there are also es­sential differences between these interactions.

Let us compare, for example, the equilibrium distancesbetween oxygen atoms forming a molecule and oxygenatoms of two neighbouring molecules attracted in solidifiedoxygen before the equilibrium position. The differencewill be very notice~ble: the oxygen atoms forming l\l

"

2. Structure of Matter 37

molecule settle down at a' distance of1'.2 'j...,; while theoxygen atoms of. different molecules approach each otherto within 2.9 A.. .

Analogous results have also b~~n obtained for, otheratoms. Atoms of different inolecules settle down fartherfrom each other than atoms"of the same molecule. It istherefore easier to tear m.olecule.s ap'art from each otherthan atoms from a molecule; moreov;er,. the differehcein energy is much greater than thkt in distance. .wliiJethe energy necessary for breakin¥. the bonds betweenoxygen atoms forming a molecule 18. a~out 100 kcallmol,the energy neeped to pull oxyge'n molecules asunderis less than 2 kcallmol. .", . ,

Hence, on a potential energy curve for, molecules,the potential well lies farther ~w:ayfrom the, verticalaxis and, furthermore, the well ~s mUCh, shallower.

However this does not exhaust. ther diff~ence betweenthe interaction beiween atoms fonning -'a molecule andthe int.eraction of' Jllolec.ule's,: .'. .'

Cbemistshaves'hownth'at atoms are bound in a mole­cule with a _fully determined number of other. atoms.H, two hydr,ogen at()1n~ hay~ for~ed a ll?-olecule, nothird atom will join them to this end. AI). oxygen atomin water is bound to two. hydrOgen'atoms, a':£1(1' it is im-possi,ble ,to bind _~nqther atoJll to, t~~m'. ,.'. '

We do nqtfi,nd' an,ything sim~la~ ~!l ~!1t~r~<?leyularinteraction. Having attracted one neighbojir 'to itself,a molecule' d~es not rose its' "attr~ctlve force" 'to' anydegree. The approach of .'¥,~ighbours' will'-','Cd,ntinue c aslong as there. is enough room., . '.. ., What does "thimi,is 'enopgh room" mean? A-re moleculesrea}ly somethin~ like'apples. ot: '.~gi~?, <?fc(lllrse, in acertain sense such a. compa.nson _IS JustIfj cd: molecule~

ll:rt') physical bodi~s possessing definite" size' and shape.The equilibrium distance between molecules is nothingbut their size.

Molecules 88 2. structure of Matter

What Thermal Motion Looks Like

The interaction between molecules can have greateor smaller values during the "lives" of the molecules'

The three states of matter-gaseous, liquid and solid­differ from one another in the role which molecularinteraction plays in them.

The word "gas" was thought up by scientists (derivedfrom the Greek chaos meaning "disorder"~.

And as a matter of fact, the gaseous state of matteris an example of the existence in nature of complete,perfect disorder in the mutual distribution and motionof particles. There is no microscope which would permitone to see the motion of gaseous molecules, but in spiteof this, physicists can describe the life of this invisibleworld in sufficient detail.

Thereisan enormous number of molecules, approximate­ly 2.5 X 1019 molecules in a cubic centimetre of airunder standard conditions (room temperature and atmo­spheric pressure). Each molecule's share is a volume of4 X 10-20 cms, that of a small cube whose sides areapproximately 3.5 X 10-7 em = 35 A. However, themolecules are much smaller. For example, moleculesof oxygen and nitrogen-the basic components of air­have an average size of about 4 A.

Therefore, the average distance between molecules isten times as great as the size of the molecules. And thisin tu:n implie~ that the average volume of air per mole­cule IS approxImately a thousand times as great as thevolume of the molecule itself.

Imagine a plane surface on which coins have beenthrown in a random manner: where there is an average ofa hundred coins to each square metre. This means one ortW? c?ins on a page of the book which you are reading.ThIs IS roughly how sparsely gas molecules are distrib­uted.

Every molecule of a gas is in a state of continualthermal motion.

Let us follow a single molecule. Here it is swiftlymoving somewhere to the right. If it met no ohstaclesin its path, the molecule would continue its motion withthe same velocity along a straight line. But the pathof the molecule is crossed by its innumerable neighbours.Collisions are inevitable, and the molecules fly apartlike two colliding billiard balls. In which direction willour molecule gallop? Will it acquire or lose its speed?Anything is possible: for its collisions can be of the mostvarious kinds. Blows are possible from the front or frombehind, from the right or the left, which are stronf! orweak. It is clear that being subject to such irregularimpacts during these random collisions, the moleculewhich we are observing will rush about through allparts of the vessel in which gas is confined.

How far are gas molecules able to go without a col­lision?

It depends on the size of the molecules and the densityof the gas. The larger the molecules and the more mole­cules there are in a vessel, the more often will theycollide. The average distance travelled by a moleculewithout any impact-it is called the mean free path-isequal to 11 X 10-8 cm = 1100 f for hydrogen mole­cules and 5 X 10-8 cm = 500 j, for oxygen moleculesunder ordinary conditions. The distance of 5 X 10-8 em<one-twenty-thousandth of a millimetre) is very small,but it is far from small in comparison with molecularsizes. A distance of 10 m for a billiard ball correspondsin scale to a path of 5 X 10-8 em for an oxygen molecule.

The structure of a liquid differs essentially from thatof a gas whose molecules are far from each other andonly rarely collide. In a liquid, a molecule is constantlyfound in the immediate vicinity of others. The moleculesof a liquid are distributed like potatoes in a sack. True,

-,-- _.-._ .._-,._---~..._-_ .."--

Molecules 40 2. Structure of Matter 41

with one distinction: the molecules of a liquid are ina state of continual and chaotic thermal motion. Becausethey are so crowded, they cannot move around as freelYas the molecules of a gas. Each of them is aJways "marking.time" in practically one and the same place surroundedby the same neighbours, and only gradually movesthrough the volume occupied by the liquid. The moreviscous the liquid, the slower this displaclWlent. Buteven in such a "mobile" liquid as water, a moleculemoves 3 j\ oduring the time required by a gas molecule tocover 700 A.

The forces of interaction between molecules deal veryresolutely with their thermal motion in solids. In solidmatter, the molecules are almost always ina fixed posi­tion. The onlv effect of the thermal motion is that the mole­cules are coritinually vibrating about their equilibriumpositions. The lack of systematic displacements by themolecules is precisely the cause of what we call solidity.In fact, if molecules do not, ehangeneighbours, all themore will the sepa ra te parts of the body remai n in afixed bond with one, another.

.Compressibility of Bodies, I

As raindrops drum on a roof, so do gas molecules beatagainst the walls of a vessel. The number of these blowsis immense, and it is their united action that creates thepressure whic'h can move the piston 'of an engine, explodea shell or blow up a balloon. A hail of molecular blows­this is atmospheric pressure; this is the pressure thatmakes the lid of a boiling tea-kettle jump, this is theforce driving a bullet out of a rifle.

What is gas preSSlll'e related to? It is clear that thestronger the blow intlicted by a single molecule, thegreater the pressure. It is no less obvious that the pressure

•

will depend on the number of blows inflicted in a second.The more molecules in a vessel, the more frequent theblows and the greater the pressure. Hence, the pressure pDf' a 'given gas, is proportional, first of all" to itsdensity., ': If the mass of a gas '.is (:onstant., then decreasing its-yolume, we increase its, densjty by the correspondingfactor. Therefore, the pressure of a gas in a closed ,vesselwill, be inversely proportional to its volume. Or, inother words, the product of the pressure by the volumemust be constant;

. pV - const

Tbis' si~ple 'Jaw was disc~vered by the English 'physi­cist Robert. J30yle (1627-1691) arid the Frenell scientistEdme Mariot,te (c. 1620-1684). Boyle's law (also knownas Mildotte's law) is one of the first quantitative law;s inth~histou of Ilhysical science. Of course, it holds whenthetemperatpre is constant., " ., ..

As a gas is compressed, the Boyle eqvation is :;iatis,fwd,vQrseim'd "vorse. The moleqlle,s Vpproqch e,ach other:1Ild, the" interactions between them begin to influence-the, beh~vio,-!r of thega~,. ,. ., '- Bovle's law is 'valid 'in those cases wh\ln the inter-ference' ~f lhe.forces of iriter~ction ,between th~ gas mole­cures is completely i,risigJ;lifican t~ On~' therefore .speaks of'Boyle's law qS a law' of .f4eal gases.!' , .. The~djechve "ide;;l" sounds rather hi'riny" whim modi­

fying the word "gas"., Idealm~ans perfd~t, s{! that it isimpos~il~ly 'to bebet,ter.. 1",: .: .. . :,. ,:' ',' .. ' . ,

.Th!3 .sJmpler a model or dlqgram" the moreld,eal It ISfor the plwsieist. :Compu'tations ale simplified, flxplana­~iori~ of pliysical phenome!].a" bec,ome easy and elear. Theferm ''i,deal gqs" 'pertains 'to thesimplpst model ..of a gas.The behaviour 6f sufficiently rarefied gases is pra,cticallyindistinguishable from that of ideal gases.

Surface Tension

Is it possible to emerge dry from water? Of course itis, if one smears oneself with a non-wettable substance.

Rub your finger with paraffin and put it under water.When you take it out, you will find that except for twoor three drops there is no water on your finger. A slightmotion-and the drops are shaken off.

In this case we sav that water does not wet paraffin.Mercury behaves in 'such a manner towards almost allsolid bodies: it does not wet leather, glass or wood.rWater is more capricious. It adheres closely to somebodies and tries not to touch others. Water does not wetoilv surfaces, but thoroughly wets clean glass. Water wetswood, paper and wool.

If a drop of water is placed on a clean plate of glass,it will spread out and form a very shallow. small puddle.If such a drop is put on a piece of paraffin, it will justremain a drop, almost spherical in shape and slightlyflattened by gravity.

Among the substances which "stick" to almost allbodies is kerosene. Striving to flow along glass or metal,kerosene is capable of creeping out of a loosely closedvessel. A puddle of spilled kerosene can spoil one'sexistence for a long time: kerosene will seize a large sur­face, creep into cracks and penetrate one's clothes. Thatis why it is so difficult to get rid of its not very pleasantodour.

The failure to wet bodies can lead to curious phenomena.Take a needle, grease it and carefully place it flat onwater. The needle will not sink. Looking attentively,you can notice that the needle depresses the water andcalmly lies in the small hollow so formed. However, aslight pressure is enough to make the needle go to thebottom. For this it is necessary that a considerable partof it turns out to be in the water.

Molecules

. L~quids are much less compressible than ga~e~. In alIqUId, the moleculeR are already in "contact". Compres­sion consists only in improving the "packing" of themolecules, and for very high pressures, in pressing themolecules t.hemselves. The degree to which the'forces ofrepulsion hmder the compression of (liquid can be seenfrom the following figures. A rise-in pressure from oneto two atmospheres entails a decrease in the volulIle ofa gas by a factor of two, while the volume of water changesby 1120 000. and l'that of mercury by "'a total of11250000.. ~ven the enormous pressure in the depths of an oceanIS mcapable of compressing water to any noticeableextent. In fact, a pressure of one atmosphere is createdby a ten-metre column of water. The pressure under a10-km layer of water is equal to tOoo atm. The volume of,,:ather decreases by 1000/20 000, i.e. by one-twen­tiet .

r;;--T.he .compre.ssi.bility of solids differs little from thatof lIqUIds. ThIs IS understandable since in both cases themolecules are ~lready in contact, and so compressioncan only be achIeved at the expense of a fnrther drawingtogether of molecules which are already strongly repellingeach other. By means of ultrahigh pressures of 50-100 thou­sand atmospheres, we are able to compress steel bv one­thous.andth, and lead by one-seventh, of its v~lume

It .I~ clear from these examples that, under terrestriaicondItIOns, we cannot succeed in compressing solid mat­ter to any significant extent.L~ But in the Universe, there are bodies where matter iscompressed with incomparably greater strength. Astron­ome~s discovered the existence of stars in which thedenSIty of matter reaches 10e g/cm3 . Inside these starscall.ed wltite dw"arfs ("white" for the nature of their lumi~nosIty, dwarfs because of their relatively small size)there should therefore be enormous pressures. '

2. Structure of Maner 43

Molecules 44 2, Structure of Matter

Figure 2.4

This intoresting property is mado uso of Lv \yatl)r strid­PI'S running swiftly along tho surface of th~ walor with­Ollt. wetting their feet.

\Vetting is used to dress ores by merms ·of floatation.The word "floatalion" moans "surfacing". '1'110 essenco ofthe phenomenon is ns follows. Finely cru:;hod oro islonded into a V!l.t containing \Yater. A'small nmOllnt of~pecial oil is added; this oil must .wet the particles of the:nineral and not wet the particles of. the gangue (thisI.S what the valueless rock qr aggregales of miner,dsin an 01'9 are called). \Vhen mixed, the p,1l'ticies of lhemineral are cOqted with an oil v filnl.

Air is blown into the black· ;nixture of ore, waleI' ,IJHloil. There is formed n Illass of little hnbblps of air­foam. The ail' bubblps camp to the surfaep. 'rho [H'Oces:,of floatation i:-; b,ISpd on thpJart. that I!l(' p,Hlicll's (,ovl'rpdwith oil cling to tho air buhble's. ;\ Luge IJllhhll' c,uTil':;up a small particle like a bnlloon.. The mineral collects ·at the smfnce in the fonn of fo,nnand the gangue remains at thp ·bottoJll, Thp foam is ['P­moved and sent for fnrtllPT processing in OJ'del' 10 obtaina so-called "concen trn te", which con ta ins tpns of timesas little gangue.

F'mcr's of ('olll'sion bt'twepn slld,j('p:, are capable ofviolating the levelling of a liquid in communicatingvessels. [t is very easy to verify the tl'Uth of this.

If a thin glass tube (with a diameter of a fraction of amillimetre') is lowered into water, then in violation ofthe law of communiCill ing vesspls, the wnter in it willIjllickly begin rising, ilnd its lpvd will become cOllsid­I'rilbly higher than in tIl(' large vessel (Figure 2.4).

Unt what took place? What forces are supporting theweighl of the column of liquid that has risen up? Therise is accomplished by the forces of cohesion betweenthe water and the glass.

Forces of col11'sion between surfaces dearly manifestthem:;plvps only when a liquid rises in a sufficientlyI!Jin t u!Je. 'flte narrower the tube, the lligher the liquid,11\([ the more distinct the phenomenon. The nnme of thesesllI'face phenomena is related to the n<1me of the tubes.The imide d i,llneter of such ,l tube is measnl'ed in fractionsof a mill i lIletl'P, such a tube is called capillary (meaning"thin as a hair"). 'fhe phenomenon of the I'iso of liquidsin thin tubes is called capillarity.

nut to what height are capill"ry tubes capable ofraising a liquid? It turns out that water rises to a heightof 1.5 mm in a tnbo of i-mm diameter. For a diameterof 0.01 mill, tl1(' height of the rise will increase as manytimes as the diameter of the tube decreases, i.e. to1;) CIll.

Of COIlI'Sp, the elevation of a liquid is possible onlyin tlw C"St' of welting. It is not hard to gness that mercurywill not ri,.,o in glass tubes. On the eontrary, mercuryLdls in gla:;s tnhps. !\[ercllry is so "intolerant" of contactwitlt glas,., tllat it strives to redllcP the totnl :;urface to11ll' lllininllllD allo\ved by gravity.

ThpI'p p:\ist m,lnv hodip,., which are something likesystpms of vpry tll'in tube,.,. Capil1:lry phpllomena cana Iways Ill' 0 bserved in such hod ies.

Crystals and Their Shape

Many people think that crystals are beaUotiful, rarelyfound stones. They occur in various colours, are usuallytransparent and, what is most remarkable, possessa beautiful regular shape. Crystals are most often poly­hedra with ideally plane sides (faces) and strictly straightedges. They please the eye with a marvellous play ofcolours at the faces and an amazingly regular structure.

In the technology of the dyeing industry, frequentuse is also made of the ability of a fabric to draw ina liquid through the thin pores formed by the threadsof the fabric.

We have not said anything yet about the molecularmechanism of these interesting phenomena.

The differences in the surface tension of various sub­stances can be excellently explained by intermolecularinteractions.

A drop of mercury does not spread out over the surfaceof glass. The reason is that the energy of interactionbetween the mercury atoms is greater than the energy ofcohesion between the glass and mercury atoms. For the samereason, mercury does not rise in narrow capillary tubes.

It is a different matter with water. It' was found thatthe atoms of hydrogen of the water molecules readilycohere to the atoms of oxygen in the silica of whichglass mainly consists. The water-glass intermolecularforces are greater than the water-water intermolecularforces. Therefore, water spreads out into a thin film overthe surface of glass and rises in glass capillary tubes.

Surface tension or, more precisely, the energy of cohe­sion (depth of the potential well in Figure 2.1) for variouspairs of substances can be both measured and calculated.A discussion on how this can be done would lead us toofar away from our subject.

Molecules

Figure 2.5

Plants and trees have an entire system of long ductsand pores. The diameters of these ducts are less thanhundredt~s of. a mi.1limetre. Because of this, capillaryf~rce~ raIse soIl mOIsture to a considerable height anddlstnbu.te water ~hroughout the plant.Blo~tmg paper IS a very convenient thing. You spilled

some I.nk on a ~age a.nd want to turn it over. But you'renot gomg to walt untIl the blot dries up You take a sheetof b~otting. paper, dip one of its edges' in the drop, andthe mk s~Iftly runs upwards against gravity.

A typIcal capillary phenomenon has occurred. Ifyou look a~ the blotting paper through a microscope,you can see ItS structure. Such a paper consists of a sparsenetwork of paper fibres forming thin and long ducts witheach other. These ducts play the role of capillarytubes.

The same ~ind ?f sy~tem of long pores or ducts formedb~ fibres eXIsts m wIcks. Kerosene rises through the":Ick of a. lamp. A siphon can also be created with theaId of a. wICk by p~acing one of its ends in a glass partiallyfilled wIth water, m such a way that the other end hangingover the edge is lower than the first (Figure 2.5).

1. Structure of Matter 47

Molecule$ 2. Structure 01 MaHer 49

Amung them are the unassuming crysLals of rock s,dt-­natural sodium chloride, i.e. common salt. They are foundin nature in the form of rectangular parallelepipeds orcubes. Calcite crystals also have a simple form-trans­parent oblique-angled parallelepipeds. Quartz crYstals,Ire much more complicatcd. EachlitLle crystal has a g'r'ea~

many faces of different shapes, intersecting in edges ordifi'erent lengths. . ' ' --'

However, a crystal is anything but a museum-piece.:Crystals surround us everywhere. The solid bodies will\.which we build homes and make machines, the substanceswhich we use in our daily lives almost all of them ahlcrystals. But why do we not see this'? 1'lle reason is thatill nature we rarely come across bodies in the form' ofsingle indivi dual crystals (ur, as is said,' IT:lOnocrystals).Substances a re most often found in the form of firmlylinked crystalline grains of very small "size, less thana thousandth of a millimetre'. SMh a structure can onlybe seen through a' min'oscope. ...

'Bodie's con'sisting of ,tiny crystallinc' grains are calledpolyer'ysl\l1Tine (ddl'iverl' from the Greek pulys meaningH many"'. I ;.. ... ).~ . '._ .,

Of. eourse, p'o'IY~'rystallinebodiesrhust also 'be includedamong the crystals.' Jt'\dIY 'th'en turn out that almost allthe solid bodies'surroUnding- us' are erystals. Salld andgranite, copper and·it'on,' the salol sold in a drug storeand paint all these are crysfals ..'- .

There are exceptions; glass and plasties do not con­sist of small crystals. Sueh so~i(L bodies are calledamorphous. .

Thus, stuclyin'g crystals mean~~tllrlying almost· all thebodies surrounding liS. It is obvions fba-[ ~ltis is important.

Single crystals'm'e recognized at'o'nco' by the regularityof theirsbapl~g: Plane faces and st'raight edges'are char­acteristic properties of a' crystal; the' re'gularity of shapeis undoubtedly related to the reglllil]'it~"01' till" illternal

".1 t ••• I;

structure of a crystal. If a crystal has been especiallystretched in a certain direction, it means that the struc­ture of the crystal in this direct ion is also special insome way.

But imagine that a ball has been made by machine outof a large crystal. Will we succeed in figuring out that wehave a crystal in our hands, in distinguishing this ballfrom a glass ball? Since the different faces of a crystalare developed to differen t degrees, this suggests that thephysical pl'Operties of a crystal also differ in differentdirections. This, of course, refers to the strength of acrystal, its electrical conductivity and to many otherproperties. This peculiadty of a crystal is called theanisotropy of its properties. Anisotropic means differentproperties in different directions.

Crystals are anisotropic. On the contrary, amorphousbodies, liquids and gases are isotropic, i.e. possess iden­tical (derived from the Greek isos meaning "equal")properties in different directions (derived from the Greektropos meaning "turning"). The anisotropy of the proper­ties of a crystal is precisely what pOL'mits us to find outwhether or not a transparent, formless piece of matteris a crystal.

Let us visit a mineralogical museum and closely exam­ine various monocrystalline specimens of crystals ofthe same substance. I t is quite possible that specimensof both regular and irregular shapes are on display. Someof the crystals resemble fragments, others have one or twoabnormally developed faces.

From the collection of specimens we select the onesthat seem to be of perfect shape and sketch them. Thedrawing we obtain is shown in Figure 2.G. Quartz hasagain been taken as an example. As the crystals of othersubstances, quartz can develop a different number of facesof each "kind", as well as a different number of "kinds"of faces. Even if their similarity in appearance is notI, - 0389

_-----------_~=~!l!'II"'i!:_!mIl'l:_II!ClI!l_••-------------------Molecules

50'1. Structure 01 Matter 51

Figure 2.6

self-evident, ,",uch small crystals do resemble one anotheras do close relatives or, sometimes, as twins. In whatdoes their similarity consist?

Look carefully at Figure 2.6, which illustrates a numberof quartz crystals. All of them are close "relatives".They can be made truly similar by grinding off variousfaces by different amounts in such a way that they remainparallel to their initial positions. It is readily evidentfor instance, that crystal II can be made, in this way:entirely similar to crystal I. This is possible becausethe angles are equal between like faces, for example, be­tween faces A and B, Band C, etc.

It is this equality of angles that gives the crystalstheir "family" resemblance. As the faces are ground offparalIc I to thei i' ini tial positions, the shape of the crystalis changed, bill tlte angles between the faces areretained.

\~ ----•. ~~~ "'_L.__

Figure 2.7

As a crystal grows, certain of its faces, due to variousrandom factors, may be in more favourable, and others inless favourable conditions for the addition of new layersof atoms or molecules. Crystals grown under differentconditions may not appreciably resemble one another, asfar as their appearance is concerned, but the angles be­tween the corresponding faces of all crystals of the sub­stance being considered are always the same. The shape(or habit, as it is called) of a crystal is a matter of chance,but the angles between its faces (and you will understandwhy further on) depend upon its internal structure.

The flatness of its faces is not the only feature thatdistinguishes a crystal from shapeless bodies. A crystalalso possesses symmetry. The meaning of this \vard, inthe sense we are about to employ, can best be demonstrat­pd by examplps.

Illustrated in Figure 2.7 is a sta tue (reminding us of thefamous ones on Easter Island) standing in front of a largemirror. The reflection in the mirror is an exact copy(actually a mirror image) of the statue. The sculptorcould have carved two statues and arranged them in thesame manner as our statue and its reflection in the mirror.

This double sculptme is a symmetrical ftgme; it consistsof two equal parts, one being a mirror image of the otherand equal in the sense that one's left hand is equivalentto one's right.

Let us assume that a flat mirror is placed as in Fig­ure 2.7. Then the right-hand part of the sculptureexactl v coincides with the reflection of its left-hand part.Such ,; symmetrical arrangement would have a verticalplane of mirror symmetry passing through the centrebetween the two parts. The plane of symmetry is animaginary one, but we sense it di~inctly in examininga symmetrical body.

The bodies of animals have planes of symmetry; a ver­tical plane of external symmetry can be passed througha man as well. Symmetry is only approximate in theanimal world, alld, in ,general, there is no ideal sym­metry in the world around us. All a1'chitecto1' can designa house consisting of two ideally symmetrical halves.But when the house is built, no matter how well, youcan always find some difference in the two symmetricalparts: one may have a crack at some place where theother has not.

The most exact symmetry is found in the world ofcrystals, though it is hardly ideal here either. Fissuresinvisible to the naked eye, scratches and other flawsalways make equal faces differ slightly from one another.

The child's toy called a pinwheel is illustrated inFigure 2.8. I t is also symmetrical, but you cannot passa plane of symmetry through it in any way. \Vhat thenis symmetr'ical about this toy? First, let us considerits symmetrical parts. How many are there? Obviously,fo1ll'. What makes their mutual alTangement regular?This, too, is simple enongh. Let us turn the pinwheelconnterclockwise through a right angle, i.e. one-fourthof a revolution. Now vano 1 has turned to the previousposition of vane 2,2 to that of vane 3, :J to that of vane 4,

and 4 to that of vane 1. The new position of the pinwheelcannot be distinguished from the previous one. We saythat such a fwure has an axis of symmetry or, more pre­cisely, an axi~ of four-fold symmetry because it coincideswith itself after turning through one-fourth of a revolu-tion. .

Thus, an axis of rotation symmetry is an imaglllarystraight line about which a body, when turned througha unit fraction of a revolution, is in a position that can­not he distinguished from its initial position. ~he .orderof rotation symmetry (four-fold in our case) lllchcatesthat such coincidence occurs after turning the body one­fourth of a revolution. Consequently, after four suchturning motions we return to th? init.ial position. .

Do we lind svmmetry of all kmds III the crystal klllg­dom'l) Investig,;tions ]~ave shown that we do not. Infaet, the only axes of rotation symmetry found f~ ?ry.stalsare tho two-, three-, four- anel six-fold ones. Iht:' 1:'3 ~o

more chance. Crystallographers have shown that It ISrelaled with the j'nteI'nnl structure of crystals. Therefore,the nlllnber of dj1'I'ol'pnI kinds or, as we say, elnsses oferystal symmetry is rdatively small---only 32.

Molecules 522. Structure of Matter

Figure 2.8

53

Strudure of Crystals

Why is the regular shape of a crystal so beautiful?Its faces, so smooth and shining, look as though they werepolished by a skilled lapidary. Different parts of thecrystal repeat one another, forming a handsome sym­metrical figure. This exceptional regularity of crystalshas been known since time immemorial. But the con­cepts of the ancient scholars on the nature of crystalsdiffered only slightly from the tales and legends madeup by poets whose imagination was fascinated by thebeauty and elegance of crystals. It was believed thatrock crystal is formed of ice, and diamond.f rock crystal.Many wonderous properties were attributed to crystals;they were thought to heal diseases, protect one againstpoisons, have an influence on one's fate and manyothers.

The first scientific views on the nature of crystalsappeared only in the 17th and 18th centuries. An ideaof these conceptions is given by Figure 2.9, which wasrepr.oduced from Traite de Mineralogie, written by theFrench abbe, Rene Just Hafty, near the end of the 18thcentury. According to its author, crystals are made upof tiny "building blocks", fitting tightly to one another.This conclusion is one that is naturally arrived at. Letus break up a crystal of Iceland spar (calcite, or calciumcarbonate) with a sharp blow of a hammer. It flies apartinto pieces of various sizes. Examining them carefully,we find that the pieces are of regular shape, quite similarto that of the large crystal, their "parent". Evidently,reasoned Hafty, if we continue to break up the piecesinto smaller and smaller ones, we will finally reach thesmallest building block, invisible to the naked eye,which is a crystal of the given substance. These ultimateblocks are so small that steps formed by them, comprisingthe faces of the large crystal, seem to us to be immaculate-

55

Figure 2.9

ly smooth. Well, and what is this final building blocklike? The scientists of that day could not answer thisquestion.

The "building block" theory of crystal structure was ofgreat benefit to science. It explained the origin of thestraight edges and flat faces of a crystal. As a crystalgrows, new building blocks attach themselves to thoseof the crystal, and a face grows like the wall of a housebuilt by the hands of bricklayers.

Hence, the question concerning the reason for theregularity and beauty of the shapes of crystals has beenanswered a long time ago. The reason for this phenom­enon is their internal regularity. This regularity con­sists in the endless repetition of one and the same ele­mentary parts.

Imagine a park fence made of rods of different lengthsand distributed helter-skelter. An ugly scene. A goodfence is constructed from identical rods distributed ina regular sequence at equal distances from each other.We find such a self-repeating pattern on wallpaper. Herean element of a drawing, say, a girl playing with a ball,is repeated not only in one direction, as in a park fence,but so that it fills a plane.

:t Structure of Matter54Molecules

Molecules 56 2. Structure of Matter 57

Figure 2.10

But what relationship do a park fence and wallpaperh.ave to a. crystal? A. most direct one. A park fence con­SIStS of lmks repeatmg along a line, wallpaper of pat­terns repeating in a plane, and a crystal of groups ofatoms repeating in space. One therefore says that theatoms of a crystal form a space (or crystal) lattice.

There are certain details concerning a three-dimension­al, or space, lattice that we must now discuss. To sim­plify . the w?rk of t.110 illustrator, we shall explain allthat IS r~qlllre.d, USIng wallpaper as an example, ratherthan askmg hIm to construct complicated drawings ofthree-dimensional figures.

Separated out in Fig-nre 2.10 is the smallest piece thatcan be transfermd or shifted to aMain the whole pat!p/"IIof the w:lllpap.er. To separate out such a piece, we~draw

~wo stralg-ht hnes from any point of the drawing, forlUstance, the centre of the heach ball, connecting it to

the same points of the two adjacent balls. As can beseen in our drawing, we can use these two lines to con­struct a parallelogram. By shifting or transferring thisparallelogram along the directions of the two initialbasic lines over distances equal to its sides, we canobtain the whole wallpaper pattern. This smallest repeat­ed portion, commonly called a unit cell, can be selectedin various ways. It is evident from Figure 2.10 thatseveral differen t parallelograms could be selected, eachcontaining one drawing. We emphasize that in the givencase it makes no difference to us whether the pictureinside the cell is a whole one or it is divided up by thelines bounding the cell.

It would be a mistake to suppose that after drawingthe repeating picture, called the motif, for the wallpaperthe artist can always consider his job to be finished. Thiswould be true if the only way to make a pattern forwallpaper was to add to the given portion another identi­cal portion, shifted parallel to the first along the twoinitial basic lines.

In addition to this simplest method, however, thereare sixteen more ways to fill in the wallpaper designwith an orderly repeated drawing, or motif, i.e. seventeentypes of mutual arrangement of drawings on a plane.They are illustrated in Figure 2.11. Here a simplerrepeating motif has been selected, but, like the one inFigure 2.10, it has no symmetry in itself. The patternscomposed of this motif are symmetrical and tIw differencein the patterns is due to the difference in Lhe symmetricalarrangements of the motifs.

We see, for example, that in the first three cases thepattern has no plane of mirror symmetry: you cannotposition a vertical mirl'Or so that one part of tIll' pattprnis a mirror reflection of another part. Cases 4 and ;),on the contrary, have planes of symmetry. Two mutuallyperpendiclliar :mirrors can be set up in cases Sand !J.

59

cco

'"'":iu..o<J'lUJz::l LINES OF MIRROR SYMMETRY

2. Structure of Matter

LINES OF MIRROR SYMMETRY

Figure 2.12

axis (or plane) of symmetry, you can readily find theadjacent point and consecutive points, all spaced at thesame distance from one another, through which the samekind of axes (or planes) of symmetry can be passed.

The seventeen types of symmetry of a plane patterndo not, of course, exhaust all of the great variety of pat­terns that can be composed from a single motif. Theartist must specify one more condition: how the motifis to be positioned with respect to the boundary linesof the cell. Figure 2.12 illustrates two wallpaper pat­terns with the same initial motif which is differentlypositioned in each pattern with respect to the lines rep­resenting the vertical mirrors (in a plane pattern theyare simply lines of symmetry). Both patterns come undercase 8 in Figure 2.11.

Each body, including crystals, consists of atoms.Simple substances, or elements, consist of atoms ofa single kind. Complex substances, or compounds, con­sist of atoms of two or several kinds. Assume that insome superpowerful microscope we could examine thesurface of a crystal of common salt and see the centresof the atoms. Figure 2.13 shows how we would see the

58

~~~~~~~~~ ~§§-m~~§

~~,. '-.r~,.-.r,..~ §§§:.....~§~~~,.~~...,.~~~~,.~ ....~.,.~4t" ~§§§§"" .....§~...~..:~~ ........ -....,. §§§§?~~~F-,:.-....,.. .......,.....£ ~~~~~~~~~~ 11 ~§~§§~~;"j 2 '.ist;I$';t';~IJ,IJ~ 3

.4 • ~~

5 ~6

- ~~~~~ 1119mmmn: 7J &

~.,.IVV' ,

.~,,. ,,. "'

}O~' /SZSf~ ~ ?ood .! } \/\/\~~»~~ 'ooor )()<$2)- J\J~7~~~~~dJ W_O.S-!. 1. X,(; }12 "... '\ ...~ '- "";1\ 'jJ

...-{oo..;--{oo~

Its':(J".,t#f'-.~

~~.~#' ~ 'It.,..

- I........... ~ . .....~ '-"L+---.bo... .. ..i............'" ....

~lSlS1'l+~~*~ 14 #'J"~'''~''lt.~i,.. ...~........ 16

Figure 2.11

Molecules

Case 10 has an axis of four-fold rotation symmetry, per­pendicular' to the plane of the drawing. In case 11, theaxis is of three-fold symmetry, and in cases 13 and 15,of six-fold rotation symmetry.

Planes and axes of symmetry are found on our drawingsin parallel series or families, rather than singly. If youhave found one point through which you can pass an

Molecules 60

• • • • •0 c~ t- ('"..

--.:2':0.;.='"-.' -" -i:;V

OJ

tit • Ii

~•r;I,

r~i

0 ,~~~.

'DL .r::/' (9'1, '''~~

:1ii • • 6) • •'~j,,!I'

! Figure 2.13

atoms) of two kinds: sodium and chlorine. More complexcrystals may be made up of molecules which, in theirturn, consist of atoms of many kinds.

We can always single out in a crystal the smallestrepeating groups of atoms (or single atoms, in the simplestcase). Such a group is called the unit cell.

The dimensions of unit cells may vary in wide ranges.The shortest distances are found between the adjacentlattice points (corners of the unit cell) of the simplestcrystals, consisting of atoms of a single kind. Themaximum distances are found in complex protein crystals.These distances, called lattice constants, range from2 or 3 angstroms to several hundred angstroms (hund­redths of one-millionth of a centimetre).

There are a great variety of crystal lattices. The prop­erties common to all crystals are due to their latticestructure. To begin with, we can readily understandthat the ideally flat faces of a crystal are the planes passingthrough the lattice points at which atoms are located.But any number of lattice planes can be passed in themost diverse directions. Which of these planes boundthe grown crystals, i.e. become its faces?

Let us turn our attention, first of all, to the followingcircumstance: various lattice planes and lines are notfilled equally densely with lattice points. Experimentsshow that a crystal is faceted by the planes that aremost densely studded with lattice points and that theseplanes intersect at edges which are also most denselyoccupied by lattice points.

A view of a crystal lattice perpendicular to a face isshown in Figure 2.14. Shown also are the traces of latticeplanes perpendicular to the drawing. It should be clearfrom what has been said that faces on the crystal parallelto planes I and III are apt to develop. No faces will bedeveloped parallel to plane I I which is so sparsely dottedwith lattice points (and atoms).

6l2. Structure of Matter

atoms arranged along a face of the crystal like a wallpaperpattern. You can readily understand now how a crystalis constructed. We could say that a crystal resembles"three-dimensional wallpaper". A crystal is composed byjoining three-dimensional unit cells tightly togetherinstead of plane cells as for wallpaper patterns. Thesethree-dimensional unit cells are the building blocks ofwhich the crystals are built.

How many different ways exist for building up three­dimensional wallpaper patterns from these elementarypieces? This complex mathematical problem was solvedat the turn of the century by the founder of structuralcrystallography Yevgraf Stepanovich Fedorov (1853­1919). This eminent Russian scientist established thefact that there can be only 230 different ways to buildup a crystal or, as they say today, 230 Fedorov groups.

All that is known today about the internal structure ofcrystals was obtained by X-ray structure analysis, whichwe shall describe in some detail in the fourth book.

Simple crystals exist, made up of atoms of a singlekind. Diamond, for example, is of pure carbon. Crystalsof common salt consist of ions (electrically charged

~I !

63

(C)(b)

2. Structure of MaHer

(a)

Figure 2.ts

tribution of the atoms in this lattice, let us take somebilliard balls and start packing them as closely as pos­sible. First of all, let us form a close layer-it looks likebilliard balls which have been gathered in by a rackbefore the beginning of a game (Figure 2.16). Note thatthe ball in the middle of the triangle is in contact withsix neighbours, and these six neighbours form a hexagon.We continue the packing by laying one layer upon ano­ther. If we place the balls of the second layer directlyabove the balls of the first, such a packing would notbe close. Trying to distribute the greatest number ofballs in a definite volume, we should place the balls ofthe second layer in the holes formed by the first, theIalls of the third layer in the holes of the second, etc.In a close-packed hexagonal lattice, the balls of thethird layer are placed in such a way that their centreslie directly above those of the balls of the first.

The centres of the atoms in a close-packed hexagonallattice are distributed just like those of the balls whichare closely packed in the manner described.

A great many elements crystallize in the lattices of thethree types described above:

62Molecules

Figure 2.U

The structures of many hundreds of crystals are knowntoday. We shall discuss the structures of the simplest crys­tals, first of all, those built up of atoms of a single kind.

Three types of lattices are most common. They areshown in Figure 2.15. The centres of the atoms are repre­sented by points; the lines joining the points do r1&t haveany actual meaning. They have been drawn merely tomake the nature of the spatial distribution of the atomsclearer to the reader

Figures 2.15a and b depict cubic lattices. In order tovisualize these lattices more clearly, imagine that youhave arranged building blocks in the simplest manner­edge to edge and face to face. If you now conceptuallyplace points at the vertices and centres of the cubes,the cubic lattice depicted in Figure 2.15a appears. Sucha structure is called body-centred cubic. If points areplaced at the vertices of the cubes and at the centres oftheir faces, the cubic lattice depicted in Figure 2.15bappears. It is called face-centred cubic.

The third lattice (Figure 2.15c) is called close-packedhexagonal (Le. having six angles). In order to understandthe origin of this term and more clearly visualize the dis-

65

Figure 2.t8Figure 2.t7

2. Structure of MaHer

The presence of weakly bound atomic layers makes iteas"J f?r graphite. crystals. to break up along these layers.ThIs IS why solId graphIte can serve as a lubricant int~ose cases when it is impossible to apply lubricatingOIl-for example, at very low or very high temperatures.Graphite is a solid lubricant.

Friction between two bodies reduces, roughly speaking~o the fact that microscopic protuberances of one body faliIII hollo.ws of t~e other.. The force required for breakingup a mICroscopIC graphIte crystal is much smaller thanthe !rictional forces; therefore, a graphite lubricationconsIderably facilitates the sliding of one body alonganother.

There is an endless variety in the structures of crystalsof chemical compounds. The structures of rock salt andcarbon dioxide depicted in Figures 2.19 and 2.20 canserve as extreme examples.5-0389

64

Figure 2.t6

Molecules

Close-packed hexagonal lattice Be, Co, HI, Ti, Zn, ZrFace-centred cubic . . . . . AI, Cu, Co, Fe, Au, Ge, Ni, TiBody-centred cubic .•... Cr, Fe, Li, Mo, Ta, Ti, D, V

We shall mention only a few of the other structures.The structure of diamond is depicted in Figure 2.17.What is characteristic of this structure is that a carbonatom in diamond has four immediate neighbours. Let uscompare this number with the corresponding numbersfor the three most common structures just described.As is evident from the figures, each atom has 12 immediateneighbours in a close-packed hexagonal lattice, theatoms forming a face-centred cubic lattice have justas many neighbours, and each atom has eight neighbours ..in a body-centred lattice.

We shall say a few words about graphite, whose struc­ture is shown in Figure 2.18. This structure has a strikingpeculiarity. Graphite consists of layers of atoms, whereatoms of a single layer are more firmly bound to eachother than atoms of neighbouring layers. This is relatedto the values of the interatomic distances: the distancebetween neighbours in a single layer is 2.5 times as smallas the shortest distance between layers.

fl:

5*

indicates that the substance is constructed from the samenumber of atoms of sodium and chlorine.

The question of the existence of molecules in a sub­stance is decided by its structure. If no groups of closeatoms can be distinguished in it, there are no!molecules.

A crystal of carbon dioxide, CO 2 (the dry ice which liesin cartons of ice cream), is an example of a molecularcrystal (Figure 2.20).

The centres of the oxygen and carbon atoms of a mole­cule of CO 2 are situated along a straight lin,!'l (see Fig­ure 2.2). The distance C-O is equal to 1.3 A, and thedistance between oxygen atoms of neighbouring moleculesis about 3 A. It is obvious that under such conditionswe immediately "recognize" a molecule in a crystal.

Molecular crystals are close packings of molecules. Inorder to see this, we must outline the contours of themolecules. This is precisely what has been done inFigure 2.20.

All organic substances consist of molecular crystals.Organic molecules frequently consist of many tens andeven hundreds of atoms (those made up of tens of thousands

i. Structure of MaHer

Figure 2.21

66

Figure 2.20Figure 2.19

Molecules

•

C.rystals of rock salt (Figure 2.19) consist of atoms ofsodIUm ~small dark balls) and chlorine (large light balls)alter~atmg ~l?ng the a~es of a cube. Each sodium atomhas s~x eqUIdIstant neIghbours of the other kind. Thesame I.S also tru~ of chlorine. But where are the moleculesof s?dIUm chlOrIde? There are none; not only are groupsof smgle atoms of sodium and single atoms of chlorineabsent from salt crystals, but in general, no group ofatoms ~hatso~v~r can be distinguished from the othersby theIr prOXImIty.

The chemical formula of NaCI does not give us anygrounds for saying that "this substance is built up outof molecules of NaCI". The chemioal formula merely

and what is happening at their boundaries. For thisreason, physicists have devoted much research to thestudy of polycrystalline substances. By means of X:raystructure analysis, about which we have alrell:dy.promlsedto tell our readers, they found that each gram IS a smallcrystal. We repeat our promise. .

All treatment of a metal affects its grams. Suppose wehave a piece of cast metal. Its grains are .in disorder ~nd

their size is quite large. Then we make WIre of the pIeceof metal by drawing it through a ~ie. W~at happen~ tothe crystalline grains in this plastic workmg. operatIOn?Investigations have shown that the change I~ shape. ofa solid in drawing wire or in some other plastIC workmgtechnique breaks up the crysta~line grains .. At the sametime, a certain element of order IS produced m the arra~ge­

ment of these grains by the action of the mechamcalforces. But what possible kind of order can there be here?The fragments of the grains are absolutely shapeless.

This is true: the fragments may be of any random shape,but a fragment of a crystal is still a crystal a~d the atomsin its lattice are packed as regularly as many well­faceted crystal. Hence, we can ind~cate in eac.h frag~enthQw it~ uJ;lit cells ar~ Qril;lnt~d. PrIOr to plastic workmg!

69

Figure 2.22

2. Structure of MlIIIer

•

Moleculet 6S

of atoms are to be discussed in a separate chapter). It isimpossible to properly depict their packing arrangementsgraphically. For this reason, you may find drawingssimilar to Figure 2.21 in books in this field. The mole­cules of this organic substance are built up of carbonatoms. The bars between the atoms represent the valencebonds. The molecules seem to be suspended in air. Butdo not believe what you see here. They have been drawnthus to give you an idea of how the molecules·are arrangedin a crystal. For the sake of simplicity, the illustratordid not show the hydrogen atoms joined to the outeratoms of carbon (as a matter of fact, chemists frequentlymake their similar omissions). He did not even considerit necessary to outline the molecule, imparting a definiteshape to it. If he had, we would see that the principle ofmolecular packing, with the "key fitting the lock", isjust as valid in this case as in other similar ones.

Polycrystalline Substances

We have already mentioned the fact that amorphousbodies are rare in the world of solids. Most of the objectssurrounding us consist of tiny crystalline grains, abouta thousandth of a millimetre in size.

Investigators discovered the granular structure of met­als in the 19th century with the aid of ordinary opticalmicroscopes. All that was required was an arrangementenabling the specimens to be observed in reflected insteadof transmitted light. Up-to-date metallurgical micro­scopes operate on this same principle.

The image seen in such a microscope may be like thatshown in Figure 2.22. The grain boundaries are usuallyquite distinct. As a rule, impurities accumulate at theboundaries.

The properties of materials depend to an exceptionallylarge degree on the size of the grains, their orientation

71Molecules

Figure 2.23

the cells are strictly ordered only within each separategrain; there is, as a rule, no total order. After the metalis worked, the grains align themselves so that a certaingeneral order becomes evident in their cells. This iscalled texture. For example, the diagonals of the cells inall the grains become oriented approximately parallelto the direction in which the metal is worked.

Texture is illustrated in Figure 2.23 (at the right) bythe example of certain definite, indicated planes in thegrains. These planes are most densely filled with atomsand are shown by rows of dots. The illustration atthe left shows the grains of (the metal before beingworked.

The various kinds of plastic working operations (roll­ing, forging, and wire-drawing) produce textures of differ­ent types. In some operati~s, the grains turn so that thediagonals of their unit cells are aligned along the direc­tion of working, while in others the edges of the cells arealigned, etc. The more advanced the rolling or wire­drawing technique, the more perfect the texture of thecrystalline grains of metal. Texture has a striking effectOn thlil mlilc4anical propertilils of metal articles. A study

2. Structure of Malter

of the arrangement and size of the crystalline grains .intals led to an understanding of the fundamental pn.n­

:~les involved in the plastic working of met~ls. ThIs,in turn, led to the improvements of the techmques em-ployed. 1

Another important kind of metal treatment, annea-. 's also associated with the rearrangement of the met­~~gg:ains. If rolled or drawn metal is heated to a su~fi­ciently high temperature, new crystals are f~rmed, WhIChgrow at the expense of the old ones. Annealmg gradual.lydestroys the texture; the new crystals are arranged dI~­orderly. As the temperature is raised (or the metal. ISheld longer at the annealing temperatu~e), new gramsgrow and the old ones disappear. The graI~s can gr?w toa size visible to the naked eye. Annealmg drastIca~lyalters the properties of a metal. It bec?mes more ductIleand softer. This occurs because the grams become coarserand the texture disappears.

3. Temperature

Thermometer

If two differently heated bodies are brought into con­tact, the warmer one will cool off and the colder one will

hwarm up. It is said that two such bodies exchangeeat.As we have already said, heat exchange is a kind of

~ne~gYh transfer; the body which gives off energy is saido e otter. We feel that a body is hot if it makes our

hand ~arm, transfers energy to it. On the contrary if abody IS felt to be cold, this means that it is taking'e~eaway from our body. _ rgy

Concerning a body which is giving off heat (i e "off ene:gy ?y m~ans of heat exchange), we say: it~ ie~;~:

taht1!re

hIs hIgher than that of the body which is taking in

IS eat.i Ob~rVing wh?ther an object of interest to us is cooI-ng 0 or warmmg up in the presence of one or another

body, we. find :'its place" in a row of heated bodies. Tem­p~~ature I~ a kmd of mark indicating for which bodies theo Je~t of mterest to us will be a giver and for wh' hreceIver, of heat. ,Ie a

Temperature is measured by thermometers!?e ~tructure of thermorAters can be bas~d on the

~tIhzatlOn of various properties of bodies sensitive tot~mperatur~. The most frequently utilized property is, e expansIon of bodies durin¥ a rise in temperaturl'l,

3. Temperature 73

If the body of a thermometer changes its volume whenin contact with various bodies, this implies that thesebodies have different temperatures. When the volume ofthe body of a thermometer is greater, the temperature ishigher, and when the volume is smaller, the temperatureis lower.

There are various substances"that can~serve as thermom­eters: liquids (such as mercury or alcohol), solids (suchas metals), and gases. But different substances expanddifferently, and so mercurial, alcoholic, gaseous and other"degrees" will not coincide. Of course, two basic points,the melting point of ice and the boiling point of water,can always be marked on all thermometers. Therefore,all thermometers will indicate 0 and 100 degrees centi­grade identically. But bodies will not expand identicallybetween 0 and 100 degrees. One body expands rapidlybetween 0 and 50 degrees on a mercury thermometer, andslowly in the second part of this interval, but another,vice versa.

Having made thermometers with differently expand­ing bodies, we will discover noticeable discrepancies intheir readings, in spite of the fact that their readings willcoincide for the basic points. Moreover, a water thermom­eter would lead us to the following discovery: if a bodycooled to zero is placed on an electric stove, its "watertemperature" would first fall and then rise. This happensbecause water at first decreases its volume when heated,and only later behaves "normally", i.e. increases itsvolume when heated.

We see that a rash choice of material for a thermometercan bring us to an impasse.

But then what should we be guided by in choosing 8.

"true" thermometer? Which body would be ideal for thispurpose?

We have already spoken about such ideal bodies. Theyare ideal gases, There are no interactions between the

Molecules

particles of an ideal gas and, studying the expansion ofan ideal gas, we study how the motion of its moleculeschanges. This is precisely the reason why an ideal gas isan ideal body for a thermometer.

And it really is a striking fact that while water doesnot expand like alcohol (nor alcohol like glass, nor glasslike iron), then hydrogen, oxygen, nitrogen and any othergas in a sufficiently rarefied state to deserve being calledideal expand in exactly the same fashion when heated.

Therefore, the changes in volume undergone by a def­inite amount of ideal gas serve as the basis for definingtemperature in physics. Of course, in view of the fact thatgases are highly compressible, one must be especiallycareful in seeing to it that the gas is under constant pres­sure.

In order to graduate a gas thermometer, we shouldaccurately measure the volume of the gas we have chosenat 0° and at 100°C. We shall divide the difference be­tween the volumes V100 and Vo into 100 equal parts. Inother words, the change in the volume of the gas by(V100 - Vo)/100 corresponds to one degree centigrade(1°C).

Let us now suppose that our thermometer shows avolume V. What temperature t °C corresponds to thisvolume? It is not difficult to comprehend that

tOC= v-vo 100, i.e. tOe = v-voV100 - Vo 100 V100 - Vo

By means of this equality, we assign each volume V toa temperature t and obtain the temperature scale* whichphysicists use. ..

*The centigrade scale at which 0 °c is taken as the melting pointof ice, and 100°C as the boiling point of water (both at thestandard pressure of 760 mm Hg) is '\;ery convenient. In spiteof this, the British and the Americans have so far been usinga temperatur~ :;;cal«;J w4ich s\leIlls very strange to U;3. How, for

3. Temperature

With a rise in temperature, the volume ?f a ~as. in­creases without bound-there is no theoretIcal lImIt tothe growth in temperature. On the contrary, low (nega­tive on the centigrade scale) temperatures have alimit. . I d?

What will happen when the temperat~re IS 0v:ere .A real gas will eventually turn into ~ liqUl?, .and WIth aneven greater fall in temperature, WIll solIdIfy. The g:,-smolecules will gather in a small volume. But wha.t WIllthis volume be equal to for a thermometer filled WIth anideal gas? Its molecules do not interact with each otherand do not have any volume of their own. Hence, a de­crease in temperature brings an ideal gas to a zer~ vo.l­ume. It is quite possible to come as clo~e ~s we wI~h mpractice to a behaviour that is characterIstIc of an Idealgas for example, to a zero volume. For this it is necessaryto fill up the gas thermometer with more and mo~e rare­fied gas. Therefore, we won't go wrong by assummg theminimum volume of the gas equal to zero ..

According to our formula, the lowest possIble temper~­ture corresponds to a zero volume. This temperature IScalled the absolute zero of temperature.

example will you react to the following sentence taken froman English novel: "The summer wasn't hot, the. tempera~urewas 60-70 degrees." A misprint? No, the Fahrenhelt

oscale ( F).

The temperature in England rarely falls belo~ -20 C. ~ahrer­heit selected a mixture of ice and salt havIllg app.roxlmatelsuch a temperature and took this temperature for hIS zero. nthe words of the inventor, the standard temperature !>f a ~um~nbody was taken for 100° on this scale. However, III or er 0determine this point, Fahrenheit probably made use. of theservices of a slightly feverish person. On the Fr:den~ett98c~~,the average standard temperature ot

Fa hudmbai ~ ~12s0F Th~

On this scale, water freezes at +32 an 01 sa.conversion formula will be:

I'

Molecules76

In ord.er to determine the position of absolute zero on~:e c(~tIgrade ~cale, we must substitute zero for the vol-C -10) ID the temperature formula just derivedtoonse{Ouoevnt/y(V' the temperature of absolute zero is equai

a 100 - Va).t It turns out that this remarkable point corresponds to ae~perature of about -273 °C (more precisely -273 150C)

fo~T~~s, therrdare no temperatures below ~bsolut~ zero:ltd ey,~ou k correspond to negative volumes of a gas'i . oesn .ma e ~ense to speak of lower temperatures. IiIS tJust as ImpossIble to.obtain temperatures below abso-u I~ z.er~ as to ~balke a WIre with a diameter less than zero

IS ImpoSSI e to cool a body at absolute zero . .one. cannot take energy away from it. In other ' I.e.bodI~s and the particles they are made of have th:~~~:ik?ss~~le energy at absolute zero. This implies that the

IDe IC. energy equals zero and the potential enersu:es ItS least possible value at absolute zero gy as-

jUIDce absolute zero is the lowest temperat~re 't'~nlr natural that the absolute scale in which re'aJin 1:thgID a} ~bsolute zero be used in physics, especially In. ose 0 ItS branches where low temperatures laImportant role. It is clear that Tabs = (t + ~7 Y oanRoom temperature will be about 300°C on th b 3)1 C.~cale. The absolute scale is also called the Ke~v~ns~::z~eI~ honour of the well-known 19th century English . 'tISt, and the notation T K is employed in place of s~Ien-I f {ormula for a gas thermometer determining the aba:;~u e emperature T can be written down in the form

T=100 v-vo +273 •v100 - Vo

Using the eq.uality. 100VO/(VIOO - Vo) = 273at the followmg sImple result: ' we arrive

T v273 = v;;-

3. temperaturA 1tTherefore, the absolute temperature is simply proportion­al to the volume of an ideal gas.

Exact measurements of temperature require all kinds ofcontrivances on the part of the physicist. Mercury, al­cohol (for Arctic regions) and other thermometers aregraduated by comparison with a gas thermometer overa rather wide temperature interval. However, it too isalso unsuitable for temperatures very close to absolutezero (below 0.7 K), when all gases liquefy, and also fortemperatures above 600°C, when gases penetrate glass.Other principles of temperature measurement are usedfor high and very low temperatures.

lAs for practical methods of measuring temperature,they are manifold. Instruments based on electrical phe­nomena are of great significance. It is now important toremember only one thing-during all measurements oftemperature, we should be convinced that the readingobtained completely coincides with what a measurementof the expansion of a rarefied gas would give.

High temperatures arise in ovens, furnaces and burners.Temperatures of 220-280 °C are attained in baking ovens.Higher temperatures are applied in metallurgy-harden­ing furnaces yield 900-1000 °C, forges yield 1400-1500 °C.Temperatures of 2000°C are attained in steel furnaces.

The records for high temperature in a furnace are ob­tained with the aid of electric arcs (about 5000 0C). Thearc flame makes it possible "to deal" with the most re­fractory metals.

And what is the temperature of the flame of a gas burn­er? The temperature in the inner bluish cone of flame isonly 300 °C. The temperature in the outer cone attains1800°C.

Incomparably higher temperatures arise during theexplosion of an atomic bomb. Judging by indirect esti­mates, the temperature at the centre of the explosionattains several million degrees.

Molecules 18Attempts have recently been made to obtain such ul­

trahigh temperatures in special laboratory installationsbuilt in the Soviet Union and in other countries. It hasproved possible to attain temperatures up to several mil­lion degrees for very brief moments.

Ultrahigh temperatures also exist in nature, not onthe Earth, but on other bodies in the Universe. In thecentres of stars, the Sun in particular, the temperatureattains tens of millions of degrees. But the surfaces ofstars have a considerably lower temperature, not exceed­ing 20 000 °e. The surface of the Sun gets heated up to6000 oe.

Ideal Gas Theory

The properties of an ideal gas, giving us the definition oftemperature, are very simple. The Boyle law is valid forconstant temperatures: during changes in volume or pres­sure, the product p V remains constant. For a constantpressure, the quotient VIT is conserved, no matterhQwthe volume or temperature changes. It is easy to unitethese two laws. It is clear that the expression pVIT re­mains the same as for a constant temperature but withchanging V and p; the same is true for a constant pressurebut with changing V and T. The expression p VI T re­mains constant during a change not only in any pair, butalso simultaneously in all three of the quantities p, Vand T. The law pVI T = const is known as the equationof state of an ideal gas.

An ideal gas is chosen for a thermometer because itsproperties depend only on the motion (but not on the in­teraction) of its molecules.

What is the nature of the relationship between tp.EAno­tion of molecules and temperature? To answer this ques­tion, it is necessary to find the relationship between thepressure of a gas and the motion of the molecules in it.

). temperatur.

Figure 3.1

In a spherical vessel of radius R, N molecules of a gasare contained (Figure 3.1). Let us follow an arbitrarymolecule, for example, one which is moving at a givenmoment from left to right along a chord of length l. Weshall not pay attention to molecular collisions: suchimpacts do not affect the pressure. Having flown to theboundary of the vessel, the molecule will strike againstthe wall and fly off in some other direction with the samespeed (the collision is elastic). Ideally, such a journeythrough the vessel might continue eternally. If v is themolecular velocity, each impact will occur after lIvseconds, i.e. each molecule will strike the wall vIZ timesa second. The continual hail of impacts by the N mole­cules unites into a single force of pressure.

According to Newton's law, the force is equal to thechange in momentum during a unit of time. Let us de­note the change in momentum at each impact by d. This

Molecules 80

c~ang~ occurs vIZ times a second. Consequently, the con­tr~butlOn to the force on the part of a single moleculeWIll be I1vf l.

The momentum vectors befo"e and after an impact anda~so the momentum transfer Ll have been construct~d inFI~~re 3:1. It follows from the similarity of the trianglesar~sIll~ III the construction that I1fl = mvfR. The con­tr~butlOn to the force on the part of a single moleculewIll take the following form:

*Of Swiss origin D. Bernoulli worked and lived in Russia;he was a membe~ of the St. Petersburg Academy of Sciences:No less well known is the activity of Johann (Jean) BernoullIand Jakob (Jacques) Bernoulli. All three were brothers, notnamesakes.

6-0389

3. Temperature 81

This equation was first derived by Daniel Bernoulli in1738*.

I t follows from the equation of state of an ideal gasthat pV = const· T; we see from the equation just derivedthat pV is. proportional to zrav. Hence,

T 0:: V~Vt or Vav 0:: Yiii.e. the average velocity of an ideal gas molecule is pro­portional to the square root of the absolute temperature.

Avogadro's Law

Assume that a substance is a mixture of different mole­cules. Isn't there a physical quantity characterizing amotion which would be identical for all these molecules,say, for hydrogen and oxygen, provided their tempera­tures are identical?

Mechanics yields an answer to this question. It can beproved that the average kinetic energy mzravf2 of the trans­latory motion will be identical for all molecules.

This implies that for a given temperature, the averagesquare of the molecular velocities is inversely proportion­al to the mass of the particles:

2 1 1vav 0:: -. or vav 0:: ,,-

m r m

Let us return to the equation pV = (1/3) Nmv~v· Sincethe quantities mzrav are identical for all gases at a giventemperature, the number N of molecules contained in agiven volume V at a definite pressure p and temperatureT is identical for all gases. This remarkable law wasfirst formulated by Amedeo Avogadro (1776-1856).

Nmv~v

3V

Nmv~v

p= 4nRlIR

If"

Since t~e .length of the chord does not occur in the for­mula, It IS clear .that .molecules moving along arbitrarychords make an Id~ntlCal contribution to the force. Ofcou~se, ~he change III momentum will be smaller for anoblIque Impact, but then the impacts in this case will bemore frequent. Calculations show that these two effectsexactly compensate for each other

Since .there are N molecules in the·sphere, the resultantforce wIll be equal to

:Nm'1vR

where Vav is the average molecular velocity.The pressure p of the gas equal to the force divided by

the sur~ace area of the sphere, 4nR2, is1"3 Nmv:v

..i nR83

where V is the volume of the sphere.Therefore,

pV= ~ Nmv~v

-Molecules

But how many molecules are th . 1 3 8!out that there are 2.7 X 1019 1 erellD. em ? It turnsand 760 mm Hg Th" mo ecu es lD 1 cm3 at 0 °C. IS IS an enorm byou can feel just how great it· 1 ous n,um er. So thatSuppose that gas is flowing o~i ~t uSllve an example.such a speed that a millio 1 0 1 a -cm

3. vessel with

It isn't hard to calculate ~h~~ ~cu ~s leave each second.million yea~s to get rid of the ;:s~lll take the vessel a

Avogadro s law shows that u d .and temperature, the ratio of th n er ~ defimte pressurethe volume in which the a e n?m er of molecules totity. that is identical fo~ aW contalDed, NIV, is a quan-

SlDce the density of a as gasetdensities of gases is equal t~ th~tf thm!V, the ratio of the

o eIr molecular masses'PI m1 •

p;-=m.The relative masses f 1 1

mined by simply w~ ::0 ecu es can therefore be deter-measurements once la :dI:g gaseous. substances. Suchof chemistry. It al~o lollow~r~atrole lD the d~velopmentfor a mole of any substance' ~hm Avogadro s law thatp V = kNAT, where k is ~n e state of an ideal gas,the famous German PhY:i~~~vrs~l ~onsBtant (named afterto 1.38 X 10-16erg/K Th ~ wIg oltzmann) equalthe un~versal gas con~tant~ pro uct R = kNA is called

The Ideal gas law is often written aspV = ILRT

where IL is the amount of s b tThis equation is frequently ~pspla?cde .express~d in moles.

Ie lD practICe.

Molecular Velocities

Theory shows that for a constant tage ~inetic energy of molecules 2empe~at.ure, ~he aver­cordmg to our definition of ,mvav/2, IS Ide~tICal. Ac­

temperature, Jbls average

3. Temperature 83

kinetic energy of the translatory motion of the moleculesof a gas is proportional to the absolute temperature.

Combining the ideal gas equation with Bernoulli'sequation we obtain

( mv· ) = .!. kT2 av 3

Temperature measurements with a thermometer filledwith an ideal gas add a meaning of rare simplicity to thismeasure. The temperature is proportional to the averagevalue of the energy of translatory motion of the mole­cules. Since we live in three-dimensional space, we cansay that a point moving at random has three degrees offreedom. Consequently, there is kT12 energy per degr~e

of freedom of a moving particle.Let us determine the average speed of oxygen mole­

cules at room temperature, which we take to be 27°C ­= 300 K in round numbers. The molecular mass of oxy­gen is 32, so the mass of one molecule equals 32/(6 X1023

) g.A simple computation yields Vav = 4.8 X 104 cmls, Le.about 500 m/s. Molecules of hydrogen move considerablyfaster. Their masses are 16 times as small, and theirspeeds are V16 = 4 times as great, Le. are about 2 kmlsat room temperature. Let us estimate the thermal speedof a small particle which is visible through a microscope.An ordinary microscope permits us to see a dust part:cleof 1ILm (10-4 em) in diameter. The mass of such a particlewith a density close to unity will be in the neighbourhoodof 5 X 10-13 g. We obtain about 0.5 cmls for its speed.It is not surprising that such motion is quite noticeable.

The speed of the Brownian movement of a particle witha mass of 0.1 g will be only 10-6 cmls in all. It is nowonder that we do not see the Brownian movement ofsuch particles.

We have spoken of the average speed of a molecule.But not all molecules move with the same speed; a cer-6*

of 11 km/s is equal to 32 X 10-12 erg. The average energyof a molecule at room temperature is equal to only6 X 10-14 erg. Therefore, the energy of an "eleven-kilo­metre molecule" is at least 500 times as great as the ener­gy of a molecule with the average speed. It is not surpris­ing that the fraction of the molecules with speeds higherthan 11 km/s is equal to an unimaginably small number­of the order of 10-800 •

But why are we intrigued with the speed of 11 km/s?In the first book we spoke of the fact that only bodieshaving this speed can escape from the Earth. Hence,molecules which have risen to a great height can losetheir ties to the Earth and take off in a distant interplane­tary trip, but for this it is necessary to have a speed of11 km/s. The fraction of such fast molecules, as we haveseen, is so negligible that there is no danger of the Earth'slosing its atmosphere even in the course of a thousandmillion years.

The rate of leaving the atmosphere depends to an ex­traordinarily great degree on the gravitational energyGMm/r. If the average kinetic energy of a molecule ismany t~mes less than the gravitational energy, the es­cape of the molecules from the Earth is practically impos­sible. The gravitational energy on the surface of the Moonis 20 times as small, which gives an oxygen molecule a"runaway" energy of 1.5 X 10-12 erg. This value exceedsthat of the average kinetic energy of a molecule by a factorof only 20-25. The fraction of the molecules capable ofbreaking away from the Moon is equal to 10-17 • This isalready entirely differentHhan 10-800 , and computationsshow that the air would leave the Moon quickly enoughfor interplanetary space. It is not surprising that there isno atmosphere on the Moon.

Molecules

1/00

Figure 3.2

tain fraction of the molecules move faster bmove slower. It turns out th t thO l' ut othersWe shall only present the r~sult~~ can a 1 be calculated.

At a temperature of about 15°C for exampl thage speed of "t ' e, e aver­59 % of the IDl rfgen mole~ules is equal to 500 m/s;700 m/s. On~; ~c6%s ::;0;; WIt~ speeds between 300 andspeeds-from 0 t~ 100 mls e Tmho ecules move with small

1 1 · . ere are only 5 4% f f tmo ecu es wIth speeds greater than 1000 mls Fi 0 0 as

Each column IS constructed with't b (gu:e 3.2).velocity range it refers to and th I S fase covermg theproportional to th e area 0 each column isity l~es within thi~ ~:~~~~tage of molecules whose veloc-

CUf:SI~~~~Ot1~s:~;:gtyO :t~~ul.atet theldistribution of mole-The number of eIr rans atory motion.

double the average~sol:~~l:: whos: energy is more thanmore "energetic" molecules fa~~ 1~~. The fraction of stillenergy increases. Thus the nu 0 b ast~r and faster as theenergy is at least four' t' m1er 0 molecules whoseonly 0 7% ei ht· Imes as arge as the average isX 10-4% '16 f. tImes as large as the average 0.06 X

Th ' Imes as large as the average 2 X 10-8%e energy of an oxygen molecule moving with a spe:d

3. Temperature 85

873. Temperature

The thermal expansion of solids is considerably lessthan that of liquids. It is hundreds and thousands oftimes les'l than the expansion of gases. .

Thermal expansion is an annoying hindrance In manycases. Thus, a change in the sizes of the moving parts ofa clockwork with a change in temp~rature ",:ould lead .toa change in the speed of the clock, If a speCIal alloy,,,~n­var (invariant means unchanging, whence the name In­var"), were not used for these delica~e c~mponents. !nvar,steel with a large nickel content, IS WIdely used In theinstrument manufacture. An invar rod is lengthened byonly one-millionth when ~its temperature increases by

1°C. . f I'dAn apparently negligible thermal expanSIOn 0 a so 1

body can lead to serious consequences. The reason f~r

this is that the low compressibility of solids makes Ithard to hinder their thermal expansion.

When heated by 1°C, a steel rod will increase in lengthby only one-hundred-thousandth, Le. by an amount un­noticeable to the unaided eye. However, in order to pre­vent the expansion and compress the rod one-hundred­thousandth a force of 20 kgf on 1 cm2 is needed. And thisis merely f~r cancelling the effect of a rise in temperatureby only 1 DC. -

The forces arising from thermal expansion can lea~ tobreakages and catastrophes if they are not reckoned WIth.Thus in order to avoid the action of these forces, therails 'of a railroad-bed are laid with clearances. One has.to remember these forces when handling glassware whichis easily cracked by non-uniform heating. It is thereforethe practice in laboratories to use vessels made of quartzglass (fused quartz, silicon dioxide, exists in an amorphousstate), which lack this drawback. ~or one and the samerise in temperature, a copper bar wIll be lengthened by. amillimetre while the same sized bar of quartz glass wIllchange its' length by the unnoticeable amount of 30-

86Molecules

Thermal Expansion

.If a body i~ heate~, the moti~n of its atoms (molecules)WIll be more IntensIve. They wIll start pushing each otheraway and will occupy more space. The following well­known fact is explained by this: solids, liquids and gasesexpand when heated.

We don't have to say much about the thermal expansionof gases: in fact, the proportionality of gas temperature tovolume was made the basis of our temperature scale.

We see from the formula V = VoT/273 that the volumeof a gas under a constant pressure grows by 1/273 (Le. by0.0037) of its value at 0 °c for each 1 °c increase in tem­perature (this situation is sometimes called Gay-Lussac'Blaw).

Under ordinary conditions, Le. at room temperatureand standard atmospheric pressure, most liquids expand.one-third to one-half as much as gases.

We have already spoken more than once of the anoma­lous expansion of water. The volume of water decreasesas it is heated up from 0 to 4°C. This anomaly in the ex­pansion of water plays a colossal role in organic life onthe Earth. In autumn, the upper layers of water becomedenser and sink to the bottom as they cool off. Warmerwater, rising from below, takes their place. But such amixing takes place only until the temperature of the waterfalls to ~ DC. With a further fall in temperature, the upperlayers WIll no longer contract, will therefore not becomeheavier and will not sink to the bottom. Starting withthis temperature, the upper layer, gradually cooling off,reaches zero degrees and freezes.

It is only this peculiarity of water that prevents riversfrom freezing down to their beds. If water were to sud­denly lose its remarkable anomaly, the disastrous con­sequences of this can be easily pictured even by a personwithout a rich imagination.

Molecules 88

40 !-tm. The expansion of quartz is so insignificant thata quartz vessel can be heated by several hundred degreesand then thrown into water without any fear.

Heat Capacity

The internal energy of a body depends, of course onits temperature. The more a body must be heated,' thegreater is the energy required. In order to raise the tem­perature of a body from Tt to T 2' it is required to supplyan energy

Q = C (T 2 - T t )

to it in the form of heat. Here C is the proportionalityfactor, which is called the heat capacity of the body. Thedefinition of the concept of heat capacity follows from theformula: C is the amount of heat necessary for raising thetemperature by 1°C. The heat capacity also derends onthe temperature: rises in temperature from 0 to 1 °C andfrom 100 to 101°C require somewhat different amounts ofheat.

The quantity C is usually referred to a unit 'mass andcalled the specific heat. It is then denoted by the smallletter c.

The amount of heat which goes to heat up a body ofmass m is given by the following formula:

Q = mc (T2 - T t )

In what follows we shall make use of the concept ofspecific heat capacity, but shall speak of the heat capac­ity of a body for the sake of conciseness. An additionalguide will always be the dimension of the quantity.

The value of heat capacity varies within a'rather widerange. Of course, the heat capacity of water in calorie'per degree is equal to unity by definition

3. Temperature 89

Most bodies have a heat capacity less than that ofwater. Thus, most oils, alcohols and other liquids haveheat capacities close to 0.5 cal!g·K. Quartz, glass andsand have a heat capacity of the order of 0.2 cal/g·K.The heat capacity of iron and copper is about 0.1 callg·K.And here are examples of the heat capacities of somegases: hydrogen, 3.4 callg·K; air, 0.24 cal/g·K.

The heat capacities of all bodies decrease, as a rule,with a fall in temperature, and assume negligible valuesfor most bodies at temperatures close to absolute zero.Thus, the heat capacity of copper is equal to only 0.0035at 20 K; this is twenty-four times less than at room tem­perature.

A knowledge of heat capacities may prove useful forsolving various problems on the distribution of heatamong bodies. .

The difference between the heat capacities of water andsoil is one of fhe causes determining the distinction be­tween maritime and continental climates. Possessingapproximately five times as great a heat capacity as soil,water warms up slowly and cools off just as slowly..

In summer in maritime regions, the water, havmgwarmed up more slowly than the land, cools the air, butin winter the warm sea gradually cools off, yielding heatto the air' and making the frost less severe. It is not diffi­cult to calculate that 1 mS of sea water, cooling off by1 °C warms up 3000 mS of air by 1°C. Consequently, inmaritime regions the variations in temperature and thedifference between winter and summer temperatures areless substantial than in continental regions.

Thermal Conductivity

Each object can serve as a "bridge" along which he.atpasses from a warmer body to a cooler o~e. For exaI?ple,a tea spoon placed in a glass of hot tea IS such a brIdge.,

Molecules 90

Metallic objects conduct heat very well. The top of thespoon placed in the glass will become warm in the courseof a second., If it is necessary to stir a hot mixture the handle of

the stirrer must be made of wood or plasu'c. These solidsconduct heat a thousand times worse than metals. Wesay "conduct heat'" but could just as well have said "con­duct cold". Of course, the properties of a body do notchange as a result of'the direction in which a heat flow ispassing through it. In freezing weather we are carefulnot to touch metals with our bare hands outdoors butgrasp wooden handles without fear. '

Among the poor heat conductors, they are also calledheat insulators, are wood, brick, glass and plastic. Thewalls of houses, ovens and refrigerators are made of thesematerials.

Among the good conductors are all the metals. Thebest conductors are copper and silver-they conduct heattwice as well as iron.

Of course, not only solids can serve as "bridges" forthe transfer of heat. Liquids also conduct heat, but muchworse than metals. The thermal conductivity of'metals ishundreds of times greater than that of solid and liquidnon-metallic bodies.. In order to demonstrate the poor thermal conductiv­Ity of water, the following experiment is performed. Apiece of iee is fastened to the bottom of a test tube filledwith water, while the top of the test tube is heated overa .gas burner; the water begins boiling, but the ice isstIll not meltIng. If the test tube were without water andmade of metal, then the piece of ice would begin meltingalmost immediately. Water conducts heat 'about 'two 'hun­dred times worse than copper.

Gases conduct heat tens of times worse than condensednon-metallic bodies. The thermal conductivity of air istwenty thousand times smaller than that of copper.

3. Tempemure

The poor thermal conductivit~ of gases permits us ~ohold in our hands a piece of dry Ice whose temperature IS-78 °e, and to even hold on our palms a drop of liquidnitrogen having a temperature of -196 °e. If we do notsqueeze these cold objects with our fingers, there will beno "burn". The reason consists in the fact that when thedrop of liquid or the piece of solid is boiling very energet­ically it is covered by a "vapour jacket", and the layerof gas'so formed serves as a.he~t ins1!l8:tor.

The spheroidal state of a lIqUId, thIS IS what .one callsthe state in which drops are covered by vapour, IS formedwhenever water gets into a very hot frying-pan. Dropsof boiling water, having fallen on one's palm, severelyburn one's hand, although the difference. In temperaturebetween boiling water and a human body IS less than thatbetween a hand and liquid air. Since one's hand is colder,than the drops of boiling water, heat leaves the drops, the.boiling ceases and no vapour jacket is formed.

It isn't hard to understand that the best heat insulatoris a vacuum-emptiness. There are no carriers of heat ina vacuum, and so the thermal conductivity will be at Il

minimum. . 'dTherefore if we want to create a thermal shIeld, hI e

something ~arm from something cold or vice versa" th&best thing to do is to erect a casing with double walls ~nd,pump the air out of the space between them. When dOIng·this we come across the following curious phenomenon.If ~e keep track of the change in thermal conductivity:of the gas as it is being rarefied, we shall observe thatright until the moment when the pressure reaches sev­eral millimetres of mercury column the thermal conduc­tivity remains practically constant., Our expectations areonly justified when, with the passage to a hIgher vacuum,the thermal conductivity sharply falls off.

But what is the cause of this?

atI

Molecules

I~ ord~r to understand this phenomenon, we must tryto visualIze the process of heat transfer in a gas.

Heat transfer from a warm place to a cold one takesplace by means.of the ~ransmission of energy from onemolecule to a neighbourIng one. It is clear that collisionsbetwee~ fast and slow molecules usually lead to an ac­celeratIOn of the slow molecules and a deceleration of thefast ones. ~nd this means that the hot place will becomecolder, whIle. the cold place will warm up.

. But how wIll a .decrease in pressure affect heat transfer?SInce a de~r~ase In pressure lowers the density, the num­ber. of collIslOn~ b~tween fast and slow molecules, duringwh~ch a transmiSSIOn of energy occurs, will also decrease.ThiS would .decrease the thermal conductivity. However,~ decrea~e In pressure leads, on the other hand, to anIncrease In the mean free path of the molecules whichthe.refore transfer heat by greater distances, and this tendsto Increase the thermal conductivity. Computations showth~t. these effects compensate for each other, and so theabllIt.y ~o tr~nsfer heat does not change for some time asthe all' IS beIng pumped ou t.. This will be the case until the vacuum becomes so con.

sl.derable that the mean free path is comparable to thedistance ~etween the walls of the vessel. Now a furtherdecrease In the pressure can no longer change the meanfree path of the molecules, which are "hopping around"from wall to wall: the fall in density is not compensatedfor, !,-nd so the thermal conductivity rapidly falls in pro­P?rtlOn to th~ pressure, reaching negligible values as ahigh v~cuum IS attained. The construction of a vacuumbottle IS based on the use of the properties of a vacuumVacuum bottles are very widespread: they are applied noi?nly.for the conservation of hot and cold foods but alsoIn sCience and technology. In such a case they are calledD.ewar. fJ.as~s (vessels), in honour of their inventor. Liq­UId all', Dltrogen and oxygen are transported in such

3. Temperatur6 Mvessels. Later we shall tell how these gases are obtainedin a liquid state*.

ConvectionBut if water is such a poor heat conductor, how does

it warm up in a tea-kettle? Air conducts heat even worse;then it isn't clear why the same temperature is establishedin all parts of a room.

Water in a tea-kettle quickly boils because of gravity.The lower layers of water, having warmed up, expand,become lighter and rise to the top, with cold water tak­ing their place. A rapid heating occurs thanks only toconvection (derived from the Latin convectus meaning"bring together"). It wouldn't be so easy to heat up waterin a tea-kettle located in an interplanetary rocket.

Somewhat earlier explaining why rivers do not freezedown to the bottom, we spoke of another case of convec­tion currents of water without using this word.

Why are the radiators of a central heating placed nearthe floor? Why is the ventilation window made in theupper part of a window? It might be more convenient toopen a ventilation window if it were lower down, and itmight not be a bad idea to place radiators under the ceil­ing, so that they did not get in ~the way. If we wereto follow such advice, we would soon discoverthat the room is not being warmed up by the radia-

,II tor and not being aired when the ventilation window isopen.

*Everyone who has seen cylinders of vacuum bottles noticedthat their walls are always silver-plated. But why? The factis that thermal conductivity is not the only means of trans­ferring heat. There exists yet another way to transfer heat,which we shall speak of in another book, so-called radiation.Under ordinary conditions, it is much weaker than thermalconductivity, but is nevertheless quite noticeable. The wallsof vacuum bottles are covered with a coating of silver preciselyin order to weaken the radiation.

The same thing takes place with the air in a room aswith the water in a tea-kettle. When the radiator is turnedon, the air in the lower layers of the room begins warm­ing up. It expands, becomes lighter and rises towards the~eiling. Heavier layers of cold air arrive in its place. Andthey, having warmed up, leave for the ceiling. A contin­uous air current thus arises in the room, with warm airmoving up from below and cold air moving down fromabove. Opening a ventilation window in winter, we admita stream of cold air into the room. It is heavier than theair in the room, and so goes down, forcing out the warmair, which rises towards the top of the room and leavesthrough the ventilation window.

A kerosene lamp flames up well only when it is coveredwith a tall piece of glass. One should not think that theglass is needed only in order to shield the flame from thewind. Even in the calmest weather, the brightness ofthe flame immediately increases when the glass is put onthe lamp. The role of the glass consists in intensifyingthe stream of air approaching the flame-in creating adraught. This occurs because the air inside the glass,deprived of the oxygen that was used for the burning, quick­ly warms up and rises, while pure cold air moves intoits place through the holes made in the burner of the lamp.

The taller the glass, the better will the lamp burn. Infact, the speed with which cold air rushes into the burnerof the lamp depends on the difference in weight betweenthe heated column of air in the lamp and the cold airoutside it. The higher the column of air, the greater thisdifference in weight, and so the faster the movement.

Factory chimneys are also made high for this reason.An especially rapid influx of air, a good draught, is need­ed for a factory furnace. It is achieved as a result of ahigh chimney.

The lack of convection in a rocket devoid of weightmakes it impossible to use matches, lamps or gas burners:

~. temperaturo 95

the products of combustion would smother the flame.Air is a poor conductor; we can conserve heat with its

aid, but only under one condition: if we avoid convection,the mixing of warm and cold air, which will bring tonaught the thermal-insulation properties of air.

The elimination of convection is achieved by applyingvarious kinds of porous and fibrous bodies. It is difficultfor air to move inside such bodies. All bodies of this kindare good heat insulators, thanks only to their ability toretain a layer of air. But the thermal conductivity of thesubstance itself of which the fibres or the walls of thepores consist can be not very small.

A good fur coat is made of a dense fur containing asmany fibres as possible; eiderdown can be used to makewarm sleeping bags weighing less than half a kilogram,due to the exceptional thinness of its fibres. Half a kilo­gram of this down can "detain" as much air as tens ofkilograms of sheet wadding.

Storm windows are made in order to reduce the con­vection. The air between the panes does not participatein the mixing of layers of air which takes place withinthe room.

On the contrary, every movement of the air intensifiesthe mixing and increases the transfer of heat. This isprecisely why we fan ourselves or turn on the ventilatorwhen we want the heat to go away faster. This)s also whatmakes it colder in the wind. But if the air temperature ishigher than one's body temperature, the mixing has theopposite effect, and a wind feels like a hot breath of air.

The problem involved in a steam boiler consists inobtaining steam heated to the required temperature asquickly as possible. The natural convection in a gravita­tional field is quite insufficient for this. Therefore, thecreation of an intensive circulation of water and steam,leading to the mixing of warm and cold layers, is' one ofthe basic problems in the construction of steam boilers.

94Molecules

4. States of Matter

Iron Vapour and Solid Air

A strange combination of words, isn't it? However,this is by no means nonsense: iron vapour and solid airexist in nature, only not under ordinary conditions.

But what conditions are we talking about? The stateof a substance is determined by two circumstances: tem­perature and pressure.

Our lives proceed under conditions which change rela­tively little. The air pressure varies by several per centabout a value of one atmospherej the temperature of theair, say, near Moscow, lies in the interval from -30to +30 °Cj in the absolute scale, in which the lowest pos­sible temperature (-273°C) is taken as zero, this intervalwill look less impressive: 240-300 K, which is also only±10% of the average value.

It is quite natural that we have become accustomed tothese ordinary conditions, and so when speaking simple'truths such as "iron is a solid, air is a gas", we forget toadd "under standard conditions".

If iron is heated, it will first melt and then vaporize.If air is cooled, it will first liquefy and then solidify.

Even if the reader has never come across iron vapour orsolid air, he or she will probably believe without difficultythat by means of a change in temperature it is possibleto obtain any substance in a solid, liquid and gaseous stateor, as is also said, in a solid, liquid or gaseous phase.

4. States of Matter 97

I t is easy to believe this because everyone has observeda substance without which life on the Earth would beimpossible, and that substance is in the form of a gas,a liquid, or a solid. We are speaking, of course, o.f water.

But under what conditions does a transformatIOn of asubstance from one state to another occur?

Boiling

If we lower a thermometer into water which has beenpoured into a tea-kettle, turn on the electric stove andwatch the mercury in the thermometer, we shall see thefollowing: the level of the mercury will inch upwardsalmost immediately. Now it is already 90, 95 and finally100°C. The water begins boiling and simultaneously themercury stops rising. The water has already been boilingfor many minutes, but the level of the mercury does notchange. The temperature will not change until all thewater has boiled away (Figure 4.1).

But what is the heat used for if the temperature of thewater does not change? The answer is obvious. The processof transforming water into steam requires energy.

Let us compare the energy of a gram of water and a gramof the steam created out of it. The molecules of the steamare distributed farther from each other than the watermolecules. It is obvious that because of this the potentialeI4rgy of the water will differ from that of the steam.

The potential energy of attracting particles decreasesas they approach. The energy of the steam is thereforegreater than that of the water, and so the transformationof water into steam requires energy. This excess energyis imparted by the electric stove to the water boiling inthe tea-kettle.

The energy needed for transforming water into steamis called its heat of vaporization. In order to transform 1 gof water into steam, 539 calories are required (this figure7-0389

Dependence of Boiling Pointon Pressure

The boiling point of water is equal to 100°C; one mightthink that this is an inherent property of water, thatwater will always boil at 100°C, no matter where andunder what conditions it may be.

4. $tates of Mattet 09

But this is not so, and peopie who iive high up in themountains are perfectly well aware of this.

There is a tourist cabin and a scientific station nearthe top of Mt. Elbrus. Novices are sometimes amazedat "how hard it is to boil an egg in boiling water" orwonder "why boiling water doesn't scald". In such cases.it is pointed out to them that water is already boilingat 82°C on the top of Mt. Elbrus.

But what causes this? What physical factor interfereswith boiling? And does the height above sea level haveany significance?

This physical factor is the pressure acting on the surfaceof theliquid . It isn't necessary to climb to the top of amountain in order to check the validity of what we havesaid.

If we place a bell glass over water that is being heatedand pump air into or out of it, we can. convinc~ ourselvesthat the boiling point is raised by an mcrease m pressureand lowered by a decrease in pressure.

Water boils at 100°C only at a definite pressure-760 mm Hg (or 1 atm). . . .

The curve showing the dependence of the bOIlmg pomton the pressure is depicted in Figure 4.2. The pressure isequal to 0.5 atm on the top of Mt. Elbrus, and a boilingpoint of 82°C corresponds to this pressur~.

But it is even possible to refresh oneself m hot weatherwith water boiling at 10-15 mm Hg. At such pressures,the boiling point will fall to 10-15 DC.

One can even obtain "boiling water" having a temper­ature of freezing water. One has to lower the pressure to4.6 mm Hg for this. .

It is possible to observe an interesting scene by placmgan uncovered vessel with water under a bell glass andpumping the air out. This will make the water boil, butboiling requires heat. But since there is no air, the waterhas to give up its own energy. The temperature of the

7*

9

t, min'

BOILING

Molec..l..

Figure 4.t

is for a temperature of 100 DC). If 539 calories are usedfor 1 g, then 18 X 539 ~ 9700 calories will be suppliedto 1 mole of water. This amount of heat must be consumedin breaking the intermolecular bonds. One can compare thisfigure with the amount of work necessary for breaking theintramolecular bonds. In order to split one mole of steaminto atoms, about 220000 calories are required, i.e.25 times as much energy. This directly proves the weak­ness of the forces binding molecules to each other, ascompared to the forces finding atoms together in a mole­cule. I

t!

,i

Moleculet 1004. States of Matter 101

b?iling water starts falling, but since the pumping con­tinues, the pressure also falls. Therefore, the boiling willnot cease, the water will continue cooling off and willfinally freeze.

Such a boiling of cold water occurs not only as a resultof air being pumped out. For example, when a marinescrew propeller rotates, the pressure in the layer of watermoving rapidly about the metallic surface will fall sharp­ly, and so the water in this layer will boil, Le. manybubbles filled with steam will appear in it. This phe­nomenon is called cavitation (from the Latin cavus mean­ing "hollow").

Decreasing the pressure, we lower the boiling point.And raising it? A graph analogous to ours answers thisquestion. A pressure of 15 atm can so delay boiling ofwater that it will begin only at 200°C, while a pressureof 80 atm will make water boil only at 300°C.

Thus, a definite boiling point corresponcll to a definiteexternal pressure. But we may also turn this assertion"around" by saying that "a definite pressure correspondsto each boiling point of water". This pressure is calledthe vapour pressure.

The curve depicting the dependence of the boilingpoint on the pressure is simultaneously the curve of thevapour pressure as a function of the temperature.

The numbers plotted on the boiling point graph (or onthe vapour pressure graph) show that the vapour pressurechanges very sharply with a change in the temperature.At 0 °e (Le. 273 K) the vapour pressure is equal to 4.6 mmHg, at 100°C (373 K) it equals 760 mm Hg, Le. has in­creased by a factor of 165. With a doubling of the tem­perature from 0 °e (Le. 273 K) to 273°C (Le. 546 K),the vapour pressure grows from 4.6 mm Hg to almost60 atm, Le. by a factor of about 10000.

Therefore, the boiling point, on the contrary, changesrather slowly with a change in the pressure. When thepressure is doubled, from 0.5 to 1 atm, the boiling pointgrows from 82°C (Le. 355 K) to 100°C (Le.373 K),and with a doubling from 1 to 2 atm, from 100°C (Le.373 K) to 120°C (Le. 393 K).

The same curve that we are now considering also con­trols the condensation of steam into water.

Steam can be transformed into water by either compress­ing or cooling it.

During the course of condensation, just as for boiling,the point will not move off the curve until the transfor­mation of steam into water or water into steam is com""pletely finished. This can also be formulated as follows:the coexistence of the liquid and the vapour phase is pos­sible under the conditions of our curve and only underthese conditions. If, moreover, no heat is supplied orremoved, the amount of vapour and liquid in a closedvessel will remain constant. We say that such a vapourand liquid are in equilibrium, and a vapour in equilibriumwith its liquid is called saturated.

The boiling and condensation curve has, as we see, yetanother meaning-it is the curve of the equilibrium ofliquid lind vapour. The equilibrium curve divides the

IOQ

so411

~o

20~-+-_..--+---Il!I~r...,.H-Pl'f

~.;:J,'~~~!. "- ~ ..- ...

Figure 4.2

Evaporation

Boiling is a rapid process, and not even a trace of boil­ing water remains after a short time-it is transformedinto steam.

But there is also another phenomenon wtereby wateror some other liquid is transformed into a vapour-evap­oration. Evaporation takes place at any temperatureand regardless of the pressure, which is always close to760 mm Hg under ordinary conditions. Evaporation,unlike boiling, is a very slow process. A bottle of eau-de­colo~ne which we for~ot to clOSe will turn ou.t to Pe emp-

plane of the diagram into two parts. To the left and abovethe curve (towards higher temperatures and lower pres­sures) is the stable-vapour region. To the right and belowthe curve is the stable-liquid region.

The vapour-liquid equilibrium curve, i.e. the curve ofthe dependence of the boiling point on the pressure, or,what is the same thing, of the vapour pressure on thetemperature, is approximately identical for all liquids.In some cases, the change may be somewhat sharper, inothers somewhat slower, but the vapour pressure alwaysgrows rapidly with rise in temperature.

We have already used the words "gas" and "vapour" manytimes. These two words are more or less synonyms. Onemay say: water gas is the vapour of water, oxygen gasis the vapour of liquid oxygen. Nevertheless, a certainhabit has been formed regarding the usage of these twowords. Since we are accustomed to a definite, rather smallrange of temperatures, we usually apply the word "gas"to those substances whose vapour pressure is higher thanatmospheric pressure at standard temperatures. On thecontrary, we speak of a vapour when a substance is morestable in the form of a liquid at room temperature andatmospheric pressure.

1034. States of Matter

ty after several days; water will r~main i~ a saucer fora longer time, but sooner or later It too WIll turn out tobe dry. .

Air plays a big role in the process of evapora~IOn. Itdoes not by itself, prevent water from evaporatIng. Assoon as ~e uncover the surface of a liquid, water mo~e­cules will begin moving into the neares~ laye: of a.Jr.The density of the vapour in this layer WIll qUIckly I~­crease; after a short time, the pressure of the vapour WIllbecome equal to the vapour pressure at the temperatu.reof the surroundings. Moreover, the vapour pressure WIllbe exactly the same as in the absence of air.

The passage of vapour into the air does not, of c?urse,mean an increase in pressure. The total p~essure In. t~espace on top of the wat.er surface doe~ no! Increase; It ISonly the fraction of thIS pressure W~lCh IS born~ by thevapour that increases, and the fractIOn of the aIr corre­spondingly decreases as it is displaced by the vap0';lr.rrhere is vapour mixed with air over the ~at.er; ~Igherup are layers of air without vapour. They wIl~ InevI~ablymix. Water vapour will continually move Into hIgherlayers, and its place in the lower layer will be taken byair which does not contain any water molecules. There­fore room will always be made for new water moleculesin the layer closest to the water. Water will continuallyevaporate, maintaining the pressure of the water vapourat the surface equal to the vapour pressure, and the pro­cess will continue until the water has completely evapo­rated.

We began with examples involving eau-de-co.logne. andwater. It is well known that they evaporate WIth dIffer­ent speeds. Ether flies away with e.xceptional rapidity,alcohol is rather quick and water IS much slower. Weshall immediately understand why this is so if we findthe values of the vapour pressure for these liquids in ahandpook l saYI at room temperatllr~. Here are the figures;

102Molecules

Critical Temperature

How can we turn a gas into a liquid? The boiling pointgraph answers this question. A gas can be turn~d. into aliquid by either lowering the temperature or raISIng thepressure.

In the 19th century, the problem of raising pressuresseemed to be easier than that of lowering temperatures.At the beginning of that century, the great English phys­icist Michael Faraday (1791-1867) succeeded in compres~­ing gases to the value of their vapour press'!res and Inthis manner transforming many gases (chlorIne, carbondioxide, etc.) into liquids. .

However, certain gases, such as hydrogen, llltrogen,oxygen, simply-could' notbe liquefied. ~o matter ~owmuch the pressure was incre~sed, they dId not turn UIto

4. States of Matter 105

thannone-hundred-thousandth of a gram per cubic centi­metre.

Although this is a small number, it leads to an impres­sive amount of water in a room. It is not difficult tocalculate that in an average sized room of 12-m2 areaand 3-m height, "there will be room" for about a kilogramof water in the form of saturated vapour.

, Consequently, if we place an open barrel of water i,na room sealed up tight, then, regardless of the barrel svolume a litre of water will evaporate .

.. It is 'interesting to compare this result for water withthe corresponding figures for mercury. At the same tem­perature of 20°C, the density of saturated mercury vapouris 10-8 g/cm3 • There will be room for at least 1 g of mercuryvapour in a room of the size we have just considered.

Incidentally, mercury vapour is very poisonous, andone gram of it can seriously injure any person's healt~.When working with mercury, it is necessary to see to Itthat not even the smallest drop is spilt.

104alcohol-44.S mm and water-17.S mm

Molecules

ether-437 mm,Hg.

.The gr.eater the ~apour pressure, the more vapour therewIll be In the adjacent layer of air and the faster theliquid will evaporate. We know that vapour pressure in­creases with temperature. It is clear why the rate of evap­oration increases with heating.

It is also possible to influence the rate of evaporationby other means. If we want to aid the evaporation, we~ust take the vapo~r !1way from the liquid more rapidly,I.e. speed up the mIXIng WIth air. This is precisely whye~aporation is greatly. speeded up by blowing on the liq­UId. W~ter~ although It has a. relat~vely low vapour pres­sure, wIll dIsappear rather qUIckly If the saucer is placedin the wind.

It is therefore clear why a swimmer, having come outof the.~ater, fee.ls c~ld in the wind. T~e wind speeds upthe mIXIng of aIr WIth vapour and so increases the rateof evaporation, but the swimmer's body is forced to giveup heat for the evaporation.

The way a person feels depends on how much watervapour there is in the air. Both dry and moist air areunpleasant. The humidity is regarded as standard whenit is equal to 60%. This means that the dllnsity of watervapour is 60% of the density of saturated water vapourat the same temperature.

If moist air is cooled, the pressure of the water vapourin it will eventually equal the vapour pressure at thistemperature. The vapour will become saturated and willbegin condensing into water with a further fall in tem­perature. The morning dew moistening the grass and theleaves appears precisely as a result of this phenome~non.

At 20°C the density of saturated water vapour is about0.00002 g/cm3

• We shall feel fine if the amount of watervapour in the air is 60% of this figure, only a bit more

liquids. One might have thought that oxygen and othergases cannot be liquid. They were regarded as true, orconstant, gases.

But as a matter of fact, the failures were caused by alack of understanding of one important circumstance.

Let us consider a liquid and a vapour which are in equi­librium, and think of what happens to them with an in­~rease in .the boiling point and, of course, a correspondingIncrease ~n the press~r~. In other WOrtts, let us imaginethat a POInt on the ~oIIIng point graph is moving upwardsalong, th~ curve. It IS clear that as the temperature rises,the lIqUId expands and its density falls. But as for thevap?ur, an ,increase in the boiling point is, of course, con­ducIve to ItS expansion, but, as we have already said,the pressure of the saturated vapour grows considerablyfaster than the boiling point. Therefore, the density oft?e va~our do~s not fa~l, but, on the contrary, rapidlynse~ WIth an Increase In the boiling point.

SInce the density of a liquid falls, and the density of avapour rises, moving "upwards" along the boiling pointcurve, we shall inevitably arrive at the point for whichthE: densities of the liquid !\nq the vapour are equal(FI~lIre 4.3), .

At this remarkable point, called critical, the boilingpoint curve breaks off. Since all distinctions between agas and a liquid are related to a difference in density,the properties of the liquid and the gas become identicalat the critical point. Each substance has its own criticaltemperature and critical pressure. Thus, the critical pointfor water corresponds to a temperature of 374°C and apressure of 218.5 atm.

If we compress a gas whose temperature is below criti­cal, the process of its compression can be depicted by anarrow intersecting the boiling point curve (Figure 4.4).This implies that at the moment of attaining a pressureequal to the vapour pressure (the point of intersection ofthe arrow with the boiling point curve), the gas startscondensing into a liquid. If our vessel were transparent,then at this moment we would see a layer of liquid form­ing at the bottom of the vessel. If the pressure does notchange, the layer of liquid will grow until all of the gashas finally been transformed into the liquid. A furthercompression will now require an increase in pressure..

The situation is entirely different for the compreSSIOnof a gas whose temperature is above critical. The processof compression can again be still depicted in the form ofi'l,n arrow ~oin~ upwards: But nQW this !\ffOW ~Q~s pot

f07

P

Pc

Figure 4.4

4. States of Matterfoo

'lCr 7'

LIQUID

D .:;) CRITICAl~r POINT

p~1lISATURATE"D VA I

IJ

Molecules

Figure 4.3

Molecules

"

fOBinter~ect the boiling P?int curve. Hence, during the com­p!eSSlOn the vapour wIll not condense, but will only con­tmually become denser.

The existence of a gas-liquid interface is impossible ata tempera~u!e above. critical. When compressed to arbi­trary densIties, a unIform substance will be found underthe piston, and it is hard to say when one may call ita gas and when a liquid.

The presence of the critical point shows that there isno difference in principle ~etween the liquid and gaseousstates. At first sIght, It mIght seem that there is no suchdifference in principle only in case we are dealing withtemper~tures above critical. This, however, is not so.T~~ eXIstence of t?e crit~cal point indicates the possi~bIlIty of transformIng a lIquid, a genuine liquid, whichcan be poured into a glass, into the gaseous state avoid­ing the boiling process.

The path of such a transformation is shown in Figure 4.. 4.<\n unquestionable liquid is marked by a cross. If we~owe.r the pre.ssur~ somewhat (arrow pointing downwards),It WIll boll; It WIll also boil in case we raise the tempera­ture somewhat (arrow pointing to the right). But let uschoose a ~ifferent way. Let us compress the liquid sohard. that ItS pressure exceeds critical. The point repre­sentIng the state of the Ptquid will rise vertically. Thenlet us heat the liquid-this process will be depicted bya horizontal line. Now, after we have found ourselves tothe right of the critical temperature, let us lower thepressure to its initial value.' If we now decrease the tem­perature, we can ob.tain a most genuine vapour whichcould have been obtaIned from this liquid along a simplerand shorter path.

Therefore, by changing the pressure and temperatureso that ~he critical point is avoided, it is always possibleto obtam a vapour by means of a continuous transition

4; 'States of Matter f09

from a liquid and vice versa. Such a continuous transitiondoes not require boiling or condensation.

Early attempts to liquefy such gases as oxygen, nitro­gen and hydrogen were unsuccessful because the existenceof the critical temperature was unknown. These gaseshave very low critical temperatures: -119°C for oxygen,-147°C for nitrogen and -240 °C or 33 K for hydrogen.The record holder is helium, whose critical temperatureis equal to 4.3 K. It is possible to transform these gasesto the liquid phase in only one way: we must lower theirtemperatures below the indicated values.

Obtaining Low Temperatures

A significant decrease in temperature can be attainedby various means. But the idea involved in all these meth­ods is one and the same: we must force the body we wantto cool to expend its internal energy.

But how can this be done? One of the ways is to make theliquid boil, not bringing in any heat from without.For this, as we know, it is necessary to decrease the pres­sure-to reduce it to the value of the vapour pressure.The heat expended on the boiling will be taken from theliquid, and hence the temperature of the liquid and va­pour will fall; therefore, so will the vapour pressure.Consequently, in order that the boiling not cease butproceed more rapidly, it is necessary to continually pumpair out of the vessel with the liquid.

However, a limit is reached to the fall in temperatureduring this process: the vapour pressure will eventuallybecome completely negligible, and so even the most pow­erful pumps will be unable to create the required pres­sure.

In order to continue the lowering of temperature, wecan, by cooling a gas with the aid of the liquid alreadyobtained, convert it into a liquid with a lower boiling

turbine, it will cool off so abruptly that it liquefies.If carbon dioxide is let out of a cylinder with great speed,it will cool off so abruptly that it is converted into "ice"in the air.

Liquid gases have found wide application in technology.Liquid oxygen is used for explosives and as a componentof the fuel mixture in jet engines.

The liquefaction of air is used in technology for sepa­rating the gases constituting air.

The temperature of liquid air is widely used in variousbranches of technology. But this temperature isn't lowenough for many physical investigations. In fact, if weconvert the relevant temperatures expressed on the cen­tigrade scale to their values on the Kelvin scale, we shallsee that the temperature of liquid air is approximatelyone-third of room temperature. Much more interestingfor physics are "hydrogen" temperatures, Le. tempera­tures of the order of 14-20 K, and especially "helium" tem­peratures. The lowest temperature obtained by pumpingout liquid helium is 0.7 K.

Physicists have succeeded in coming much closer toabsolute zero. At the present time, temperatures havebeen obtained which are only several thousandths of adegree above absolute zero. However, these extremelylow temperatures are obtained by methods which do notresemble those described above.

In recent year.s, cryogenics, the physics of low tem­peratures, has created the need leading to the foundingof a new branch of industry. This branch is engaged inthe manufacture of equipment and apparatus enablinglarge volumes and long conductors to be held at tempera­tures close to absolute zero.

Molecules HO

p~int. It is now possible to repeat the pumping processwIth the second substance and in this way to obtain lowertemperatures. In case of necessity, such a cascade methodof obtaining low temperatures can be extended.

.Precisely in such a manner was this problem dealtwIth at the end of the past century, the liquefaction ofgases was carried out in stages: ethylene, oxygen, ni­trogen and hydrogen, substances with boiling points of-103, -183, -196 and -253°C, were successively con­verted into liquids. Having liquid hydrogen availableone can also obtain the lowest boiling liquid -heliuo:.(-269°C). The neighbour "to the left" helped obtainthe neighbour "to the right".

The cascade method of cooling is about a hundred yearsold. Liquid air was obtained by this method in 1877.Liquid hydrogen was first obtained in 1884-1885. Finally,the last stronghold was taken after another twenty years:helium, the substance with the lowest critical tempera­ture, was converted into a liquid in 1908 by Heike Kamer­lingh Onnes (1853-1926) in Leiden, Holland. The 70thanniversary of this important scientific achievement waswidely celebrated.

For many years, the Leiden laboratory was the only"low-temperature" laboratory. But now there exist tensof such laboratories in many countries, not to mention thef~ctories producing liquid air, nitrogen, oxygen and he­hum for technical1'urposes.

The cascade method of obtaining low temperatures isnow rarely applie~. In technical installations for loweringtemperatures, a dIfferent means of decreasing the internale:r~.ergy of a gas is applied: the gas is forced to expand ra­pIdly and perform work at the expense of its internalenergy.

If, for example, air is compressed up to several atmo­spheres and let into an expander, then when it performsthe work involved in displacing a piston or rotating a

4. States of Matte: Hi

Supercooled Vapours andSuperheated Liquids

In passing through the boiling points, a vapour oughtto condense, be transformed into a liquid. However, itturns out that if a vapour is very pure and does not comein contact with a liquid, we are able to obtain it in theform of a supercooled or supersaturated vapour-a vapourwhich should have already become a liquid a long timeago.

A supersaturated vapour is very unstable. Sometimesit is sufficient to shake the vessel containing the vapouror throw a couple of grains into it for the delayed con­densation to immediately begin.

Experience shows that the condensation of steam mole­cules is greatly eased by the introduction of small alienparticles. The supersaturation of water vapour does notoccur in dusty air. It is possible to bring about conden­sation with puffs of smoke, since smoke consists of tinysolid particles. When the particles enter the steam,they gather molecules around themselves and becomecentres of condensation.

Thus, even though unstable, a vapour can exist in theregion of temperatures fit for liquid "life".

But can a liquid "live" in the region of a vapour underthose same conditions? 1P other words, is it possible tosuperheat a liquid?

It turns out to be possible. For this it is necessary toprevent the molecules of the liquid from breaking awayfrom its surface. A radical means of achieving this is liq­uidating the free surface, i.e. placing the liquid in avessel where it would be compressed on all sides by solidwalls. Liquids have been successfully superheated inthis manner by several degrees, i.e. one is able to displacea point depicting the state of a liquid to the right of itsboi~ing point curve (see Figure 4.4).

Molecules 112 4. States of Malter 113

Superheating is the displacement of a liquid into theregion of a vapour; therefore, the superheating of a liquidcan be achieved by lowering the pressure, as well as bysupplying heat.

The former method can be used to obtain a surprisingresult. Water or some other liquid thoroughly freed ofdissolved gases, which is not easy to do, is placed in avessel with a piston reaching the surface of the liquid.The vessel and the piston should be wet by the liquid.If we now draw the piston towards ourselves, the watercohering to the bottom of the piston will move along withit. But the layer of water cohering to the piston will pullthe next layer of water after it, this layer will pull theone lying below it. As a reSUlt, the liquid will stretch.

The column of water will finally break (it is preciselythe column that will break, but the water will not breakaway from the piston), but this will occur when the forceon a unit of area attains tens of kilograms. In other words,a negative pressure of tens of atmospheres is created inthe liquid.

The vapour phase of a substance is stable even forsmall positive pressures. And a liquid can be made tohave a negative pressure. You couldn't think of a morestriking example of "superheating".

Melting

There is no solid body which would withstand a con­tinual rise in temperature. Sooner or later a solid pieceis transformed into a liquid; true, in certain cases wewill not succeed in reaching the melting point-a chem­ical decomposition may take place.

Molecules move more intensively as the temperatureincreases. Finally, the moment arrives when the pres­ervation of order among the wildly "swinging" mole­cules becomes impossible. The solid body melts. Tungsten

8-0389

has the highest melting point: 3380 DC. Iron melts at1539 DC, gold at 1063 DC. Incidentally, there are alsoeasily melted metals. Mercury, as is well known, willeven melt at a temperature of -39 ec. Organic substancesdo not have high melting points. Naphthalene meltsat 80 ec, toluene at -94.5 ec.

It is not at all difficult to measure the melting pointof a body, especially if it melts within the interval oftemperatures which can be measured by an ordinary ther­mometer. It is completely unnecessary to keep one'seye on the melting body. It is sufficient.. to look at themercury column of the thermometer. Tne temperatureof the body increases until the melting begins (Figure 4.5).As soon as the melting begins, the rise in temperatureceases, and the temperature remains constant until theprocess of melting is completely finished.

Just as in the conversion of a liquid into a vapour, theconversion of a solid into a liquid requires heat. The.amount of heat needed for this is called the latent heat offusion. For example. the melting of one kilogram of icerequires 80 kcal.

_ Ice is one of the bodies possessing a large heat of fusion._The melting of ice requires, for example, more than tentimes as much energy as the melting of the same mass

"

Molecules

Figure 4.5

114 4. States of Matter 115

of lead. Of course, we are talking about the melting prop­el" here we are not dealing with the fact that beforele~d begins to melt, it must be heated up to +327 ec.The thawing of snow is delayed as a result of the largeheat of fusion of ice. Imagine that its heat of fusion wereten times smaller. Then the spring thaws would lead eachyear to inconceivable disaste~s. . . .

Thus the heat of fusion of Ice IS great, but It IS alsosmall if we compare it with the heat of vaporization­540 kcallkg (about seven times as small). This difference,by the way, is completely natural. Converting a liquidinto a vapour, we must tear its molecules away from eachother but in melting a solid, we only have to destroythe o~der in the distribution of the molecules, leavingthem almost at the same distances from each other.It is clear that less work is required in the latter case.

The possession of a definite melting point is an importantfeature of crystalline substances. It is on the basis ofprecisely this property that they are easily: distinguishedfrom other solids, called amorphous bodIes or glasses.Glasses are found among organic substances as well asamong inorganic ones. Window glass is usually made outof silicates of sodium and calcium; an organic glass (itis also called Plexiglas) is often placed on a desk.

In contrast to crystals, amorphous substances do nothave a definite melting point. Glass does not melt, butsoftens. When heated, a piece of hard glass becomessoft, so that it can be easily bent or stretched; it beginsto change its form at a higher temperature under theinfluence of its own weight. As it is further heated, thethick viscous mass of glass assumes the form of the vesselin which it is lying. This mass is at first as thick as honey,then as sour cream a!d, finally, becomes almost like aliquid with a small viscosity, such as water. ~ith thebest will in the world, here we are unable to srngle outa definite temperature at which the solid entered the

8*

Molecules

liquid phase. The reasons for this lie in the radical differ­ence between the structure of glass and that of crY,stal­line bodies. As has been said above, the atoms in amor­phous bodies are distributed disorderly. The structureof glass resembles that of a liquid. The molecules inhard glass are already distributed disorderly. Hence,a rise in the temperature of glass merely increases theamplitude of the oscillations of its molecules, graduallygiving them more and more freedom of movement. There­fore, glass softens gradually and does not display thesharp transition from "solid" to "liquid", which is char­acteristic of a transition from the distribution of mole­cules in a strict order to their disordered distribution.

When discussing a boiling point curve, we said that aliquid and a vapour can exist, although in an unstablestate, in alien regions-a vapour can be supercooled andmoved to the left of the boiling point curve, while a liq­uid can be superheated and drawn off to the right of theboiling point curve.

Are the analogous phenomena with respect to a crystaland a liquid possible? It turns out that the analogy hereis incomplete.

If a crystal is heated, it will begin melting at its melt­ing point. We shall not succeed in superheating a cry.tal.On the contrary, in cooling a liquid, we can, if we fakecertain steps, "slip past" the melting point with compa­rative ease. We are able to achieve considerable supercool­ings of certain liquids. There are even such liquids whichare easy to supercool, but hard to make them crystallize.As such a liquid is cooled, it becomes more and more vis­cous and finally hardens without having crystallized.Glass is like that.

It is also possible to supercooJ.'water. Drops of mistcan fail to freeze even during the most severe frosts. Ifa crystal of a substance (priming) is thrown into a super­cooled liquid, then crystallization will immediately begin.

4. States of Matter 117

Finally, in many cases a delayed crystallization ca.nbegin as a result of shaking or other random events. It ISknown, for example, that crystalline gly~erine was fi.rstobtained while being transported by tram. After lymgaround for a long time, glass can begin to crystallize(devitrify) .

How to Grow a Crystal

Under definite known conditions, crystals can be grownfrom almost any substance. Crystals can be obtained froma solution or from a melt of the given substance, as wellas from its vapour (black rhombic crystals of iodine,for instance, are readily deposited from iodine vapourat standard pressure without an intermediate conversioninto the liquid state).

Start by dissolving common salt or sugar in water.At room temperature (20 0q, you can dissolve up to 70 gof salt in a thick glass (holding about 200 g of water).If you keep adding salt, it will not di~solve, but ,;illsettle to the bottom in the form of a reSIdue. A solutIOnthat can dissolve no more solute (as it is called) is saidto be saturated with the given substance. If we changethe temperature, the solubility of the solute in the solve?tis changed as well. Everyone knows that hot water dIS­solves most substances more readily than cold water.

Imagine now that you have prepared a saturated solu­tion, say, of sugar, at a temperature of 30°C and co?lit to 20°C. At 30 °C, you can dissolve 223 g of sugar m100 g of water and at 20°C, only 205 g. Then, in coolingfrom 30 to 20 6C, 18 g of sugar turn out to be "redund~nt"and, as they say, are precipitated fro~ the solu~IOn.

Hence, one of the possible ways to obtam crystals IS tocool a saturated solution.

We can accomplish the same in a diffe~ent way., Pre­pare a saturated solution of salt and let It stand III an

It has been found that the growth of a nucleus or seedcrystal consists in the seemingly outward motion of itsfaces in a direction perpendicular to each face So thatthey remain parallel to their initial positions (the crys­tal seems to expand). Naturally, the angles between thefaces remain constant (we already know that the con-.sta~y of these angles is one of the vital features of crys:­tals and is due to their lattice structure).

The successive outlines during the growth of three differ­ent crystals of the same substance are shown in Figure 4.6.Similar pictures can be observed in a microscope. Forthe crystal shown at the left the number of faces remainsconstant during the growth. The middle crystal is anexample of how a new face appears during the growth(at the upper right) and subsequently disappears.

It is important to note that the rate of growth of thefaces, i.e. the velocity at which the faces seem to movewhile remaining parallel to their initial positions,isnot the same for various faces. We also find that the facesthat disappear are those that grow fastest, for instance,the lower left face in the middle crystal. The slowest­growing faces, on the contrary, become the v:idest:ot,

Molecules 118

open glass. After some tIme has passed you will find crys­tals of salt at the bottom. Why did they form? Look atthe glass again, more carefully, and you can see thatanother change occurred together with the appearanceof crystals: there is less water. The water evaporated andleft "redundant" matter in the solution. Thus, still an­other way to form crystals is to evaporate the solution.

How are crystals formed from a solution?We mentioned that the crystals precipitate from the

solution. Does this mean that no crystal was observed fora whole week and then, in a flash, it appears as if bymagic? No, this is not so; crystals grow. You cannot,of course, observe the initial instant of growth withthe naked eye. First, a few of the randomly moving mole­cules or atoms of the solute assemble by chance in approx­imately' the c same order required to form the crystallattice. Such a group of atoms or molecules is called anucleus.

Experiments show that nuclei are more frequentlyformed when the solution contains extremely fine par­ticles, dust specks, of some foreign substance. Crystalli­zation proceeds at the highest rate and most easily if atiny seed crystal is put into the saturated solution. Thenthe precipitation of the solid substance from the solutionconsists in the growth of the seed crystal rather thanin the formation of new small crystals, i.e. in nucleation.

The growth of a nucleus does not, of course, differ inany way from the growth of a seed crystal. The advantageof using a seed crystal is that it "draws" to itself the sub­stance separating out of the solution, hindering, in thisway, the formation of a large number of nuclei. If a greatmany nuclei are formed at once, they impede each otherin growing and we cannot obtain large crystals.

How do new portions of atoms or molecules, separatingout of the solution, arrange themselves on the surfacesof the nucleus?

4. States of Ma"er

Figure 4.6

119

as we say, best-developed, ones. It is a principle thata grown crystal is always bounded by its slowest-growingfaces.

This is especially clear from the illustration at theright in Figure 4.6. Here a formless fragment acquiresthe same shape as the other crystals precisely owing tothe anisotropy of growth rates. Quite definite faces devel­op at the expense of others more and more resolutely,and impart the shape, or habit, to the crystal that isinherent in all specimens of this substance.

Very handsome intermediate shapes are observed ifa sphere is taken as a seed crystal and the solution isalternately cooled slightly and then heated slightly.Upon being heated the solution becomes undersaturatedand the seed crystal is partly dissolved. Cooling leadsto oversaturation of the solution and growth of the seedcrystal. But the molecules attach themselves differentlythan they were before being dissolved, seeming to give

121

preference to certain locations. The substance is thustransferred from certain parts of the sphere to other parts.

First small circular faces appear on the surface of thesphere. These circles gradually increase ~n size and, con­tacting one another, merge along straIght edg~s. Thesphere is converted into a polyhedron. Then certam facesov.rtake others in their growth, a part of the faces becomesmaller and smaller and disappear. Finally, the crystalreaches its characteristic shape, or habit (Figure 4.7).

In observing the growth of a crystal, one of i~s featu~esmay astound us-the seemingly parallel mot~on of Itsfaces. This implies that the substance separatmg out ofthe solution is deposited on the faces in layers, and thata succeeding layer is not started until the preceding oneis completed.

Illustrated in Figure 4.8 is an "incompleted" packingarrangement of atoms. In which of the positions indicatedby letters is a new atom, attaching itself to the ~rysta~,most likely to remain secured? Without .doubt,. m p.OSI­tion K because here it is subject to attractIOn by Its neIgh­bours from three sides. In position L the new atom isattracted from two sides, and in position M from only

Figure 4.8

4. States of Matter120Molecules

Figure 4.7

123Molecules 122

one. This is' why a row is first completed, then a wholelayer and finally a new layer is started.

In many cases, crystals are formed from a molten massor melt. This takes place on an immense scale in nature:basalts, granites and other kinds of igneous rocks wer~formed of fiery magma.

Let us heat some kind of crystalline substance, forexample, rock salt. Up to a temperature of 804°C, thesalt crystals change very slightly; they only expandnegligibly, but the substance remains a solid. A thermom­eter placed in the vessel with the substance indicatesa continuous increase in temperature during heating.At 804°C, we discover two new interrelated phenomena:the substance begins to melt and the temperature ceasesto increase any further. Until all of the substance is con­verted into a liquid, the temperature remains unchanged.A further increase in temperature indicates that we areheating the liquid. All crystalline substances have a def­inite melting point. Ice melts at 0 DC, iron at 1527 °Cand mercury at 39°C below zero, etc.

As we already know, in each small crystal the atoms ormolecules of the substance form an ordered packing arrange­~ent and have sm8:11 vibrations about their mean posi­tIons. As the body IS heated, the velocity of the vibratingparticles increases together with the amplitude. Thisincrease in the velocity of motion of particles as the tem­perature is raised is one of the basic laws of nature andconcerns matter in any state-solid, liquid or gaseous.

When the temperature of the crystal has reached adefinite, sufficiently high value, the vibration of itsparticles becomes so violent that an accurate arrangementof the particles is no longer possible; the crystal melts.As melting begins, the heat supplied no longer increasesthe velocity of motion of the particles, it is employed tobreak up the crystal lattice. This is why the temperatureremains constant until all of the substance melts. Sub-

4. States of Matter

sequent heating increases the velocity of the particles ofliquid.

In our case of crystallization from a melt, called freez-ing, all the phenomena described above occu.r in !hereverse order: as the liquid is cooled, the chaotIc motIonof its particles slows down. When a definite, sufficientlylow temperature has been reached, the. veloci!y of theparticles is so low that some of them begm to alIgn !hem­selves with others, forming a crystal nucleus. UntIl thewhole substance has frozen, or solidified, the temperatureremains constant. As a rule, this temperature is the sameas the melting point. ., .

Unless special measures are taken, freezmg begms mthe melt simultaneously at many places. The smallcrystals begin to grow as regular polyhedrons character­istic of the given substance in exactly the same way aswe described above. Such free growth soon ends, however,because the growing small crystals run into one another,with further growth ceasing at the points of contact.This imparts a granular structure to the solidified body.Each grain is a small crystal that was unable to assumeits regular shape. . .

Depending on many factors, and pnmanly on the c?ol-.. ing rate, a solid can consist of coarser or fine~ grams.

The lower the cooling rate, the coarser the grams. Thegrain size of crystalline bodi~s .ranges from a mill~o~thof a centimetre to several mlllImetres. In the maJontyof cases, the granular crystalline st~ucture can be ob­served under a microscope. Most solIds have such a fine-crystalline structure. . .

The freezing of metals is a process of ~ltal mte.rest toengineering. Physicists have invest.igated m exceptIOnallygreat detail the phenomena occurrmg when .~olten metalis poured into a mould in a foundry and solIdlfi?s.

Mostly small tree-like crystals, called dendntes, growin a molten metal when it is <woled. Sometimes the den-

drites are oriented at random, and sometimes they areparallel to one another.. Th~ stages of growth of a single dendrite are illustratedm FIgure 4.9. Under such circumstances, the dendrite~an become overgrown, occupying all the space betweenIts branches, before encountering another dendrite in~he melt. Then we find no dendrites in the solidified cast­mg. But even~s can develop differently: dendrites maymeet and grow mto one another (with the branches of onefilling the spaces between the branches of another) whilethey are still in an early stage of growth .

. Hence, we may obtain castings whose grains (shown in~Igure 2.22) have very different structures. The proper­ties of metals depend essentially upon this structure.We can control the behaviour of a metal in solidifyingby varying the cooling rate and the system of heat dis­posal from the mould.

If we wish to grow a large single crystal, we must takesteps to have the crystal grow in one place. But if severalcrystals have already started growing, then in any casewe must take steps to make the conditions for growthfavourable for only one of them.

Here, for example, is what one does in growing crystalsof fusible metals. The metal is melted in a glass test tubewith a drawn-out bottom. The test tube, suspended on athrel).d inside a vertical cylindrical oven, is slowly low­ered. The drawn-out bottom gradually leaves the ovenand cools off. Crystallization begins. At first, severaltiny crystals are formed, but those which grow sidewayscome up against the test tube wall and their growth isslowed down. Only the crystal which grows along theaxis of the test tube, i.e. into the heart of the melt, provesto be in favourable conditions. As the test tube is lowered,new portions of the melt coming into a low-temperatureregion will "feed" this unique crystal. Therefore, of allthe tiny crystals, it alone will survive; as the test tubeis lowered, it continues to grow along its axis. At last,all of the melted metal hardens in the form of a singlecrystal.

The same idea underlies the growing of refractory crys­tals of ruby. Fine ruby powder is poured in a streamthrough a flame. As a result, the particles of powder melt;tiny droplets fall on a refractory support of very smallarea forming a mass of crystals. During the further fallof droplets on the support, all the tiny crystals will grow,but once again only the one which is in the most advanta­feous position for "receiving" the falling drops will de­velop.

But what are large crystals needed for?Industry and science are often in need of large single

crystals. Of great significance for technology are crystalsof Seignette salt and quartz possessing the remarkableproperty of transforming mechanical action (for example,pressure) into voltage.

The optical industry needs large crystals of calcite,rock salt, fluorite, etc.

Crystals of ruby, sapphire and certain other preciousstones are needed for the watchmaking. The reason for

i~54. States of Matter124Molecules

Figure 4.9

this is that individual movable parts of ordinary watchesperform up to 20000 oscillations per hour. This makesunusually heavy demands on the quality of the tips ofthe axes and the baarings. The wear will be least whenruby or sapphire serves as the bearing for the tip of anaxis of 0.07-0.15 mm in diameter. Artificial crystals ofthese substances are very durable and are worn out verylittle by steel. It is remarkable that artificial stones proveto be better for this purpose than the same natural stones.

Of greatest significance in industry, however, is thegrowing of semiconductor (silicon) monocrystals. Thecommunications-electronics of today is inconceivablewithout these crystals.

Influence of Pressure on Melting Point

If the pressure is changed, the melting point will alsochange. We came across such a regularity when dealingwith boiling. The greater the pressure, the higher theboiling point. As a rule, this is also true for melting.However, there are a small number of substances whichbehave anomalously: their melting points decrease withan increase in pressure.

The fact of the matter is that the vast majority of solidsare denser than their liquids. The exceptions to this ruleare precisely those substances whose melting pointschange somewhat unusually with a change in pressure, forexample, water. Ice is lighter than water, and the meltingpoint of ice is lowered by an increase in pressure.

Compression facilitates the formation of the denserstate. If the solid is denser than the liquid, then compres­sion aids solidification and hinders melting. But if melt­ing is obstructed by compression, this means that thesubstance remains solid, whereas previously it wouldhave melted at this temperature, Le. the melting pointrises with an increase in pressure. In an anomalous case,

Evaporation of Solids

When we say that a substance is evaporating, it isusually implied that a liquid is evaporating. But solidscan also evaporate. The evaporation of solids is sometimesca~d sublimation.

Naphthalene, for example, is an evaporating solid.Naphthalene melts at 80°C, but evaporates at room tem­perature. It is precisely this property of naphthalenewhich enables it to be used for the extermination ofmoths. A fur coat powdered with naphthalene will becomesaturated with naphthalene vapours, creating an atmo­sphere which moths cannot bear. Every solid with anodour sublimes to a significant degree. In fact, the odouris created by the molecules which have broken away fromthe substance and reached our nose. However, the cases

f27A. States of Matter

the liquid is denser than the solid, and so the pressurehelps to form the liquid, Le. lowers the melting point.

The influence of the pressure on the melting point ismuch less than the analogous effect for boiling. An in­crease in pressure of more than 100 kgf/cm2 lowers themelting point of ice by 1°C.

Why is it that skates glide on ice but do not slide on ahighly polished parquet floor that is just as smooth as ice?The only feasible explanation, evidently, is the formationof water which lubricates the runners of the skates. But thepressure exerted by the runner does not in any case exceed100 kgf/cm2 , and this would not seem to lower the melt­ing point of the ice sufficiently so that it melts under therunners. All I can say to eliminate this contradiction isthe following: skates with blunt runners slide very poorlyon ice. The runners must be ground so that they cut theice. Then only the sharp edges of the runners contact theice, exerting a pressure of tens of thousands of atmospheresand the ice does melt.

126Molecules

Triple Point

Thus, there are conditions under which a vapour, aliquid and a crystal can exist pairwise in equilibrium.

Can all three states be in equilibrium? Such a pointexists on the pressure-temperature diagram, it is calledthe triple point. Where is it located?

If ice floating on water is placed in a closed vessel ata temperature of zero degrees, water (and "ice") vapourwill start entering the free space. At a pressure of 4.6 mmHg, evaporation will cease and saturation will set in.Now the three phases, ice, water and vapour, will bein a state of equilibrium. This is precisely the triple point.

The relationships between the various states are graph­ically and clearly shown by the diagram for water depict­edin Figure 4.11. Such a diagram can be constructedfor any substance.

We are acquainted with the curves in the diagram-theyare the equilibrium curves for ice and water vapour, iceand water, and water and water vapour. As is customary,the pressure is plotted along the vertical axis, and thetemperature along the horizontal.

The three curves intersect in the triple point and dividethe diagram into three regions-the living spaces for ice,water and water vapour.

A ,hase diagram is a concise handbook. Its aim is toanswer questions as to what state of a substance is stableat a given pressure and a given temperature.

If water or water vapour is placed under the conditionsof the "left-hand region", it will turn into ice. If wateror ice is introduced into the "lower region", water vapourwill be obtained. In the "right-hand region", water va­pour will condense and ice will melt.

The phase diagram permits one to immediately saywhat will happen to a substance when it is heated or com­.pressed. Heating at a constant pressure will be depicted9-0389

128

Figure 4.10

where a substance sublimes to an insignificant degree,sometimes to a degree that cannot be detected by eventhe most careful investigation, are more frequent. In prin­ciple, any solid (yes, any, even iron or copper) evaporates.If we do not detect sublimation, this merely means thatthe saturated vapour density is very negligible.

The saturated vapour density in equilibrium withthe solid grows rapidly with temperature (Figure 4.10).It is possible to convince oneself that a number of sub­stances having sharp odours at room temperature losethem at low temperatures.

In most cases it is impossible to considerably increasethe saturated vapour density of a solid for a simple rea­son-the substance will melt beforehand.

Ice also evaporates. This is well known to housewiveswho hang up their wash to dry during a frost. At first thewater will freeze, but then the ice evaporates and thewash turns out to be dry.

4. States of Matter 129

p,mm "g1000

500--- ---- --- ---- -_ul --+ -- --~

200 1\ }100

, WATER

50-y

ICE \20\10 ,

5 -- - ---2 VAPOUR

I ....0 +0.01 t,OC

on the diagram by a horizontal line. The point depictingthe phase of the substance will move from left to rightalong this line.

Two such lines are depicted in the figure, one of whichis heating under standard pressure. This line lies abovethe triple point. Therefore, it will first intersect the melt­ing point· curve, and then, beyond the bounds of thediagram, the evaporation curve too. Under standardpressure, ice melts at a temperature of 0 DC, while thewater so formed will boil at 100°C.

IThe situation will be different for ice heated under avery low pressure, say, a bit less than 5 mm Hg. Theheating process is depicted by a line passing beneath thetriple point. The melting and boiling point curves do notintersect this line. Under such a negligible pressure,heating leads to a direct transition of ice to water vapour.

In Figure 4.12, this same diagram shows what an inter­esting phenomenon will occur when water vapour in

i31

P WATER

Pmelt

Pcryst

TFigure 4.U

•• States of Matter

9*

the state marked on the figure by a cross is compressed .The vapour will first be converted into ice, which willthen melt. The diagram enables us to say at once at whatpressure the growth of crystals will begin and when themelting takes place.

The phase diagrams for all substances resemble each. other. Significant, from the everyday point of view, differ­

ences arise because the location of the triple point onthe diagram can be most diverse for various substances.

After all, we exist under conditions which are close to"standard", i.e. in the first place, under a pressure closeto Qfle atmosphere. The way in which the triple point of asubstance is located with respect to the standard pressureline is very important for us.

H the pressure at the triple point is less than atmospher­ic, then for us, living under "standard" conditions, thesubstance will be one of those that melt. As the temper­ature rises, it will first be transformed into a liquid,which will then boil.

In the opposite case, when the pressure at the triplepoint is higher than atmospheric, we will not see anyliquid when the substance is heated, since the solid willbe converted directly into a vapour. This is how "dryice"behaves, which is very convenient for ice cream vendors.

t30Molecules

Figure 4.U

are determined by the mutual distribution of their atoms.Fireproof crucibles, withstanding temperatures up' totwo or three thousand degrees, are made of graphite, butdiamond burns at temperatures above 700 ac; the spe­cific gravity of diamond is 3.5, of graphite 2.3; graphiteconducts' electricity, but diamond does not, etc.

This property of forming various crystals is possessednot only by carbon. Almost every chemical element,and not-only element but also any chemical substance,can exist in several varieties. Six varieties of ice, ninevarieties of sulphur and four varieties of iron are known.

In discussing a phase diagram, we did not speak of thevarious· types of crystals, but drew a single region for thesolid. But for a great many substances, this region isdivided up into sections each of which corresponds to adefinite "sort" of solid body or, as one says, a definitesolid phase (a definite crystal modification).

Each crystal phase has its own region of stable statebounded by definite intervals of pressure and temperature.The laws of transformation of one crystal variety intoanother are the same as the laws of melting and evapora­tion.

Given any pressure, one can find a temperature at whichbotlwtypes of crystals will peacefully coexist. If we raisethe temperature, a crystal of one form will be convertedinto a crystal of the second. If we lower the temperature,the reverse transformation will take place.

In order to convert red sulphur into yellow under stan­dard pressure, a temperature below 110 ac is needed.Above this temperature, right up to the melting point,the order of the distribution of atoms characteristic ofred sulphur is stable. When the temperature falls, theoscillations of the atoms decrease, and, beginning with110 ac, nature finds'a more convenient order for the dis­tribution of the atoms. A transformation of one crystalinto another occurs.

Molecules .' f32

Th~y can distri~ute pieces of ."dry ice" among the portionsof Ice cream wIthout worrymg that the ice cream willbecome wet as a result. "Dry ice" is solid carbon dioxideCO 2 , Its triple point lies at 73 atm. Therefore when solidCO 2 i~ heated, .the 'point de~icting its state will'move alonga horIZontal lme mtersectmg only the evaporation curveof the solid (just as for ordinary ice at a pressure of about5 mm Hg).

We have already explained how one degree of temper­ature on the Kelvin scale, or kelvin (as we are now requiredto call.it by. the SI system), is determined. The pointof.t~e dIscussIOn there was the principle involved in deter­mmm.g temperatures. But not all metrological instituteshave Ideal gas thermometers at their disposal. Therefore,a temperature scale is usually established with the aidof points of equilibrium, fixed by nature between var-ious states of substances. '

?f especial significance in this matter is the triplepomt of water. A kelvin is determined today as 1/273.16of the thermodynamic temperature of the triple point ofwater. The triple point of oxygen is taken equal to 54.361 K.The freezing point of gold is equal to 1337.58 K; Usingthese datum points we can readily graduate any ther­mometer.

The Same Atoms but Different Crystals .

The dull, soft graphite with which we write and thebright, transparent, hard, cutting glass, diamond are madeof one and the ~ame atoms- atoms of carbon. Why thenare there such dIfferences between the properties of thesetwo substances identical in composition?

Recall the lattice of flaky graphite each of whose atomshas three neatest neighbours, and the lattice of diamondw~ose atoms h~ve four nearest neighbours. It is clearlyeVIdent from thIS example how the properties of crystals

4. States of Matter f33

Molecules

96°C, the yellow sulphur will be eaten u~ by th~ red,but the yellow will swallow the red at 95 C. Unhke a"crystal-liquid" transition, a "crystal-crystal" .trans!or­mation is ordinarily delayed during superheatmg, Justas during supercooling.

In certain cases, we come across states of a substancewhich are supposed to exist at entirely different temper-atures.

White tin must turn into grey when the temperaturefalls to +13 DC. We ordinarily use things made of whitetin and know that nothing will happen to them in winter.White tin withstands supercooling of 20-30 degrees per­fectly well. However, under conditions of a severe winter,white tin is transformed into grey. The lack of knowledgeof this fact was one of the circumstances destroying Scott'sexpedition to the South Pole (1912). The liquid fuel takenalong by the expedition was kept in vessels solderedwith tin. During severe frosts, the white tin was trans­formed into a grey powder, the vessels came unsolderedand the fuel was spilt. Not without reason is the appear­ance of grey spots on white tin called tin plague.

Just as in the case of sulphur, white tin can be convert­ed into grey at a temperature a bit lower than 13°C,if a tiny grain of the grey variety falls on a tin object.

The e,aistence of several varieties of one and the samesubstance and the delays in their mutual transformationshave great significance for technology.

At room temperature, iron atoms form a body-centredcubic lattice, occupying the vertices and centre of eachcube. Every atom has 8 neighbours. At a high tempera­ture iron atoms form a denser "packing"-each atomhas 12 neighbours. Iron with 8 neighbours per atom is soft;iron with 12 neighbours per atom is hard. It turns outthat it is possible to obtain iron of the latter type at roomtemperature. The method whereby this is done, hardening,is widely applied in metallurgy.

134

~o~ody has thought of names for the six different ices.ThIs IS what one .says: ice one, ice two, ... , ice seven.How.come seven If there are only six varieties? The rea­~on IS that repeated experiments have failed to detectIce four.

If water is compressed at a temperature of about zeroice ~ve ~ill be formed at a pressure of about 2000 atm:and Ice SIX at a pressure of about 6000 atm.

Ice two and ice three are stable at temperatures belowzero degrees centigrade.. Ice seven is a hot ice; it appears when hot water is sub­Jected to a pressure of about 20000 atm.

All ice~, except the ordinary one, are heavier than water.Ice obtamed under standard conditions behaves anoma­l~usl~; on the contrary, ice obtained under conditionsdlffermg from the norm behaves normally.

We say that each crystal modification is characterized~y a defin~te region of ~xisten~e. But if so, how can graph­I~e and dIamond pOSSIbly eXIst under identical condi­tIOns?

Such "lawlessness" is found very often in the world ofcrystals. The ability to live under "alien" conditions isal~ost.a rule for crystals. While one must turn to varioustrI~ks m or~er to transfer a vapour or a liquid to alienregIOns of eXIstence, a crystal, on the contrary, can almost~ever be forced to stay within the frontiers marked off forIt by nature.~he superheati~g and supercooling of crystals are ex­

plamed by the dIfficulty in transforming one order intoanother under conditions of extreme overcrowding. Yel­low.sulphur should be transformed into red at 95.5°C.Durmg a m~)fe or less rapid heating, we "slip past" thistransformatIOn point and drive it up to 113° C.

The ~rue transformation point is most easily detectedwhen dlff?rent crystals are in contact. If we put one upclose agamst the other and maintain a temperature of

4. States of Metfer

/

135

Figure 4.13

right-hand corner of the diagram) melted carb~n. Cooli;'lgit at a high pressure, we should enter the regIOn of dIa-mond. d'

The practical possibility of such a process ,:as pro~e In

1955, and at the present time the :problem. IS consIderedto be solved from technological POInt of VIew.

An Amazing Liquid

If we lower the temperature of a body, sooner or laterit will solidify and acquire a crystal structure. ~oreover,it makes no difference at what pressure the coolIng takesplace. Tly.s cir.cumstance seems perfectly natural ~ndunderstat1Hable from the point of view of the ph?,SIcallaws with which we have already become acquaInted.In fact, by lowering the temperature, we decrea~e theintensity of the thermal motion. When the motIOn ofmolecules becomes so weak that it has already ceased tointerfere with the forces of interaction between th.em, themolecules will line up in an accurate order-wIll formcrystals. A further cooling will take away from the mole­cules all the energy of their motion, and at absolute zero,a substance should exist in the form of molecules at restdistributed in a regular lattice.

137

VAPOUR

o 1000 3000 :;000 1; K

J.O

4. States of MaHerMolecules 136

~ar~ening is accomplished quite simply-the metallicobject IS made red-hot and then thrown into water or oil.Cooling occurs so rapidly that there is no time for thetransformation of the structure stable at a high temper­ature to take place. Consequently, the high-temperaturestructure will exist indefinitely under conditions unnat­ural to it; recrystallization into a stable structure OCcursso slowly that it is practically unnoticeable.

In speaking of the hardening of iron, we were not quiteexact. Steel, Le. iron containing a fraction of a per centof carbon, is hardened. The presence of very small admix­tures of carbon delays the transformation of hard iron intosoft and permits the hardening to be carried out. As frocompletely pure iron, one cannot succeed in hardening it­there is time for the conversion of its structure to takeplace even during the most rapid cooling.

Depending on the form of the phase diagram, one oranother transformation is achieved by changing the pres­sure or the temperature.

Many transformations of a crystal into a crystal areobserved during a change of only the pressure. Blackphosphorus was obtained in this way.

Graphite was able to be converted into diamond only by,simultaneou~lyusing both high temperature and pressure.The phase dIagram for carbon is depicted in Figure 4~13.At pressures below ten thousand atmospheres and at tem­peratures less than 4000 K, graphite is the stable modi­fication. Therefore, diamond exists under "alien" con­diti~ns, and .so it can be transformed into graphite withoutp.artI~ular dIfficulty. But the inverse problem is of prac­tical Interest. We cannot succeed in carrying out a trans-

- formation of graphite into diamond by only raising thepressure. A phase transformation in the solid state:appar­ently goes too slowly. The form of the phase diagramsuggests the correct solution: increase the pressure andheat the graphite simultaneously. We then obtain (upper

Experiments show that all substances behave in thismanner. All but one unique substance: this "freak" ishelium. We have already informed the reader of certainfacts concerning helium. I t holds the record for the value ofits critical temperature. Not a single substance has its crit­ical temperature lower than 4.3 K. However, this recordby itself does not imply anything amazing. Somethingelse is startling: cooling helium below the critical tem­perature and practically reaching absolute zero, we willnot obtain solid helium. Helium remains liquid even atabsolute zero.

The behaviour of helium is completely unexplainablefrom the point of view of the laws of motion presented byus, and is one of the signs of the limited validity of thelaws of nature which seemed universal.

If a substance is liquid, its atoms are in motion. Butin cooling it down to absolute zero, we have taken allthe energy of motion away from it. We have to admit thathelium has an energy of motion which cannot be taken-away. This conclusion is incompatible with the mechan­ics which we have been studying so far. According to themechanics we learned, the motion of a body can alwaysbe slowed down to a complete halt by taking away allits kinetic energy; the motion of molecules can also bestopped in exactly the same way by taking energy awayfrom them during collisions with the walls of the vesselbeing cooled. Such a mechanics will obviously not dofor helium.

The "strange" behaviour of helium is an indication ofa fact of enormous importance. This is the first time thatwe have come up against the impossibility of applying inthe world of atoms the basic laws of mechanics establishedby means of a direct investigation of the motion of visiblebodies-laws which seemed to constitute a firm founda­tion for physics.

The fact that helium "refuses" to crystallize at absolute

zero is by no means possible to reconcile with the mechan­ics which we have been studying until now. The contra­diction which we have come across for the first time,that atoms do not obey the laws of mechanics, is merelythe first link in a chain of sharper and more glaring con­tradictions in physics.

These contradictions have led to the necessity of revis­ing the foundations of the mechanics of atomic motio?This revision is very profound and has led to a change III

our entire understanding of nature.The necessity for a radical revision of the mechanics of

atQfllic motion does not imply that we must give up thelaws of mechanics we have studied as a bad job. It wouldhave been unfair to make the reader study unnecessarythings. The old mechanics is completely valid in theworld of large bodies. This is already enough for us toregard the corresponding chapters of physics with completerespect. However, it is also important that a number oflaws of the "old" mechanics pass over without change intothe "new" mechanics. Among them, in particular, is thelaw of conservation of energy.

The presence of "inalienable" energy at absolute zero isnot a special property of helium. It turns out that allsubstances have "zero" energy. Only for helium does

139

o 1 .2 3 4. r S ,

•%0 \ LIQUID HELIUM I

LIQUID ~JO HELlUM~

I

4. States of Matter

Figure 4.14

'138Molecules

Molecules t

fthis,~nergy prove sufficient to prevent the atoms from"ormmg a regular crystal lattice. ., O.ne must not ~hink that helium cannot occur in a cr s-'

tallme sta.te. It IS merely necessary to raise the press:re~o ~J?proxIm~telY25 atm in order to crystallize helium i

oo.mg carrI?d out ab?ve this pressure leads to the for:"matIon .of sohd ~rystallmehelium with perfectly ordinarypropertIes. He.hum forms a face-centred cubic lattice.I Th.e phase dIagram for helium is shown in Figure 4.14.t .dIffe~s greatly from the diagrams of all other sub­s~es m ~he absence of a triple point. The melting andbOIlmg pomt curves do not intersect.

5. Solutions

What a Solution Is

If broth is salted and stirred with a spoon, no tracesof salt will remain. It should not be thought that the grainsof salt are simply not visible to the unaided eye. One willnot succeed in detecting the tiny salt crystals in any way,because they have dissolved. If pepper is added to thebroth, no solution will be obtained. One can even stir thebroth for days on end, but the tiny black grains will notdisappear.

But what do we mean when we say that a substance hasdissolved? For the atoms or molecules composing it cannotvanish without a trace, can they? Of course not, and theydo not vanish. When dissolving, only the grains of a sub­stance, crystals, accumulations of molecules of a singlesort disappear. A dissolution consists in stirring the par­ticles.,af a mixture in such a manner that the moleculesof one substance are distributed between the moleculesof another. A solution is the mixture of molecules or atomsof different substances.

A solution can contain various amounts of a solute.The composition of a solution is characterized by itsconcentration, for example, the ratio of the number ofgrams of a solute to the number of litres of a solution.

As we add a solute, the concentration of a solutiongrows but not without bound. Sooner ~ later the solutionwill become saturated and will cease "taking in" the solute.The concentration of a saturated solution, Le. the "limit;.ing" concentration of the solution, is called the solubility. "

143Molecules 142

It is possible to dissolve a surprising amount of sugarin hot water. At a temperature of 80 oe, a full glass ofwater will accept 720 g of sugar with no residue. Thissaturated solution will be thick and viscous, and is calleda sugar syrup by cooks. The figure we just gave for sugaris for a cup which holds 0.21. Hence, the concentrationof sugar in water at 80 °e is equal to 3600 gil (which isread "grams per Iitre").

The solubility of certain substances is highly dependenton the temperature. At room temperature (20 0q, thesolubility of sugar in water falls to 2000 gil. On the con­trary, the solubility of salt changes quite insignificantlywith a change in temperature.

Sugar and salt dissolve well in water. But naphthalene ispractically insoluble in water. Different substances dis­solve quite differently in different solvents.

Solutions are used for growing monocrystals. If a smallcrystal of a solute is suspended in a saturated solution,then as the solvent evaporates, the solute will settle outon the surface of this crystal. Moreover, the moleculeswill preserve a strict order, and, as a result, the smallcrystal will be transformed into a large one, remaining amonocrystal.

Solutions of Liquids and Gases

Is it possible to dissolve a liquid in a liquid? Of course,it is. For example, vodka is a solution of alcohol in water(or, if you wish, of water in alcohol-depending on whatthere is more of). Vodka is a real solution; the moleculesof water and alcohol are completely mixed up in it.

However, such a result is not always obtained when twoliquids are -mixed.

Try adding kerosene to water. No matter how you stir,you will not succeed in obtaining a uniform solution; thisis just as hopeless as trying "to dissolve pepper in soup;

5. Solutions

As soon as you stop stirring, the liquids arra)lge them~elvesin layers: the heavier water at the bottom, the lIghterkerosene on the top. Kerosene with .water and. alco~olwith water are systems having OppOSIte propertIes WIthrespect to their solubility.. .

However there are also intermedIate cases. If we mIXether with 'water, we shall clearly see two layers in thevessel. At first glance, it might appear that on the topis ether and at the bottom is water. But as a matter offact the lower and upper layers are solutions: at the bot­tom'is water in which part of the ether is dissolved (theconcentration being 25 grams of ether to a litre of water),while on the top is ether in which there is a noticeableamount of water (60 gil).

Let us now turn our attention to solutions of gases.It is clear that an unlimited amount of an arbitrary gaswill dissolve in any other gas. Two gases always inter­mingle in such a way that the molecules of one penetratebetween the molecules of the other. For gas moleculesinteract but slightly with each other, and ~o each g~sbehaves in the presence of another gas by, In a certaInsense, not paying any "att~nti~n" .to its cohabitant. .

Gases can also dissolve In lIqUIds. However, not Inarbitrary amounts but in limited one~, not differing !n.is respect from solids. Moreover, dIfferent gases dIS­solve differently, and these differences can be very great.Immense amounts of ammonia can dissolve in water(about 100 grams to half a glass of cold water), and largeamounts of hydrogen sulphide and carb?n dioxide. O~y.genand nitrogen are soluble in water In only neglIgIbleamounts (0.07 and 0.03 grams per litre of cold water).Therefore, there is a total of only about one-hundredthof a gram of air in a litre of cold water. However, eventhis small amount plays a large role in life on the Earth,for fish breathe the oxygen of the air dissolved in water.

The greater the pressure of a gas, the more of it will

It is possible to melt cadmium and bismuth ina singlecrucible. After cooling, we can see a mixture of cadmiumand bismuth crystals through a microscope. Bismuth andcadmium do not form solid solutions either.

A necessary although not a sufficient condition for theemergence of solid solutions is the affinity of the moleculesor atoms of the substances being mixed in form and dimen­sions. In this case, crystals of one sort are formed whenthe mixture freezes. The lattice points of each crystal areusually occupied randomly by atoms (molecules) of differ­ent sorts.

Alloys of metals of great technological value are fre­quently solid solutions. The dissolution of a small amountof an admixture can radically change the properties of ametal. A striking illustration of this is the obtaining ofone of the most widespread materials in technology­steel, which is a solid solution of small amounts of carbon(of the order of 0.5 of weight per cent, that is one atomof carbon to 40 atoms of iron) in iron, where the atomsof carbon are rAtdomly distributed between the atomsof iron.

Only a small number of carbon atoms dissolve in iron.However, some solid solutions are formed by mixingsubstances in arbitrary proportions. Alloys of gold andcopper can serve as an example. Crystals of gold and copperhave lattices of the same type-face-centred cubic. Analloy of copper with gold has the same lattice. An ideaof the structure of alloys with an increasing percentageof copper can be obtained by conceptually deleting atomsof gold from the lattice and replacing them by atoms ofcopper. Moreover, the replacement occurs disorderly,with the copper atoms generally being distributed random­ly among the lattice points.

Alloys of copper with gold may be called solutions ofreplacement, whereas steel is a solution of a different type,a solution of introduction~

10-0389

Molecules 144

~e dissolved in a liquid. If the amount of a dissolved gasIS not very great, there is a direct proportionality betweenit and the pressure of the gas above the surface of the liquid.

Who has not enjoyed a glass of cold soda, so good forquenching one's thirst. It is possible to obtain soda becauseof the dependence of the amount of a dissolved gas onthe pressure. Carbon dioxide is driven into water underpressure (from the cylinders which are in every store wheresoda is sold). When soda is poured into a glass, the pres­sure falls to atmospheric and the water gives off the "super­fluous" gas in the form of bubbles.

Taking such effects into account, deep-sea divers should 'not rise rapidly from under the water to the surface. Addi­tional amounts of air dissolve in a diver's blood under thehigh pressure of deep water. When rising, the pressurefalls and air begins separating out in the form of bubbles"Which can stop up the blood vessels.

Solid Solutions

. The word "solution" is applied to liquids in everydayhfe. However, there also exist solid mixtures whose atomsand molecules are uniformly distributed. But how canone obtain solid solutions? You won't get them with theaid of mortar and pestle. Therefore, we must first turnthe substances to be mixed into liquids, that is melt themthen mix the liquids and allow the mixture to solidify:One can also act otherwise, dissolving the two substances~hich we want to mix in some liquid and,then evaporat­mg the solvent. Solid solutions might be obtained bysuch means. They might be, but they cannot ordinarilybe so obtained. Solid solutions are rarities. If a piece ofsugar is thrown into salty water, it will dissolve excel­lently. Evaporate the water; the minutest crystals ofsalt and sugar will be found at the bottom of the cup. Saltand sugar do not yield solid solutions.

5. Solutions 145

10*

that it consists of tiny crystals of ice and tiny crystalsof salt.

Consequently, the freezing of a solution is unlike thatof a simple liquid. The process of freezing is spread outover a wide temperature interval.

What will happen if we pour salt on some frozen surface?The answer to this question is well known to yard men­as soon as salt comes in contact with ice, the ice beginsmelting. In order that this phenomenon take place, it isnecessary, of course, that the freezing point of a saturatedsalt solution be lower than the temperature of the air.If this condition is met, the mixture of ice and salt is in analien phase region, namely, in the region of stable exi­stence. Therefore, the mixture of ice with salt will turninto a solution, Le. the ice will melt and the salt willdissolve in the water so formed. Finally, either all theice will melt or else a solution of such a concentration willbe formed that its freezing point will be equal to the tem­perature of the surroundings.

Suppose that a yard whose area is 100 m2 is covered bya 1-cm layer of i~-this is quite a ~it of ice, about o~e

ton. Let us calculate how much salt IS needed to melt thISice if the temperature is -3 ac. For such a freezing (melt­ing) point the concentration of a salt solution must be45 gil. Approximately one litre of water corresponds toone kilogram of ice. Hence, in order to melt one ton of iceat -3 ac, 45 kg of salt are needed. A much smaller amountis used in practice since one does not seek the completemelting of all the ice.

Ice melts when mixed with salt, and salt dissolves inwater. But heat is required for melting, and the ice takesit from its surroundings. Therefore, the addition of saltto ice leads to a decrease in temperature.

We are now in the habit of buying ice cream, but itwas previously made at home. In this connection, amixture of ice and salt played the role of a coolant.

Molecules

In the vast majority of cases, solid solutions do notarise, and, as has been said above, after cooling we cansee in the microscope that the substance consists of amixture of tiny crystals of both substances.

How Solutions Freeze

If a solution of any salt in water is cooled, it will bediscovered that the freezing point has been lowered.Zero mark is past, but solidification does not occur. Onlyat a temperature of several degrees below zero do tinycrystals appear in the liquid. They are tiny crystals ofpure ice: salt does not dissolve in solid ice.

The freezing point depends on the concentration of asolution. Increasing the concentration of a solution, weshall lower its freezing point. A saturated solution hasthe lowest freezing point. The decrease in the freezing pointof a solution is not at all small: thus, the saturated solu­tion of sodium chloride in water freezes at -21 ac. Withthe aid of other salts, we can achieve an even greater fallin temperature; calcium chloride, for example, enables usto bring the freezing point of a solution down to -55 ac.

Let us now consider how the process of freezing takesplace. After the first tiny crystals of ice have fallen outof a solution, the concentration of the solution increases.Now the relative number of alien molecules grows, theobstacles to the process of the crystallization of wateralso increase and the freezing point falls. If we do notfurther lower the temperature, then crystallization willcease. With a further lowering of temperature, tiny crys­tals of water (the solvent) continue separating out.Finally, the solution becomes saturated. A further enrich­ment of the solution· by the solute becox"nes impossible,and the solution solidifies at once; moreover, if we lookat the frozen mixture through a microscope, we can see

5. Solutions f47

How Liquids Are Freed of Admixtures

Distillation is one of the most important means offreeing liquids of admixtures. The liquid is boiled andthe vapour is sent into a refrigerator. When cooled, thevapour turns into a liquid once more, but this liquid willbe purer than the initial one.

It is easy to get rid of solids dissolved in a liquid withthe aid of distillation. Molecules of such substances arepractically absent in water vapour. Distilled water isobtained in this way-completely tasteless pure waterdevoid of mineral admixtures.

of a saturated solution of sodium chloride at the sametemperature is 13.2 mm Hg.

Vapour with a pressure of 15 mm Hg, unsaturated forwater, will be supersaturated for a saturated salt solu­tion. In the presence of such a solution, the vapour startscondensing and uniting with the solution. Of course, notonly a salt solution but also powdered salt will ,take wa­ter vapour out of the air. In fact, the very first drop ofwater falling on the salt will dissolve it and create asaturated solution.

The absorption of water vapour from the air by saltleads to the salt's becoming moist. This is well known tohosts and affords them grief. But this decrease in vapourpressure over a solution can also be of benefit: it is usedfor drying air in laboratory practice. Air is passed throughcalcium chloride, which is the record holder in takingmoisture out of the air. While a saturated solution ofsodium chloride has a vapour pressure of 13.2 mm Hg,that of calcium chloride is 5.6 mm Hg. The vapour pres­sure will fall to this value when it is passed through asufficient amount of calcium chloride (1 kg of which"has room for" aJlproximately 1 kg of water). This is anegligible humidity, and such air may be regarded as dry.

Molecules

Boiling of Solutions .

_Boiling of solutions has a lot in common with theirfreezing.

The presence of a solute hinders crystallization. Forthe very same reasons, a solute also hinders boiling. Inboth cases it is as though alien molecules were fightingto retain a solution which is as diluted as possible. Inother words, the alien molecules stabilize the state of thebasic substance (i.e. facilitate its existence) which candissolve them. '

Therefore, alien molecules impede the crystallizationof a liquid, and so lower the freezing point. In exactlythe same way, alien molecules impede the boiling of aliquid, and so raise its boiling point.

It is curious that within certain limits of concentration(for not very concentrated solutions), neither the lower­ing of the freezing point of a solution nor the raising ofits boiling point depends in the least on the propertiesof the solute but is determined only by the number of itsmolecules. This interesting circumstance is used for thedetermination of the molecular mass of soluble substances.This is done with the aid of a remarkable formula (wecannot give it here), which relates the change in the freez­ing or boiling point to the number of molecules in a unitvolume of a solution (and to the heat of fusion or boiling).

The boiling point of water is raised one-third as muchas its freezing point is lowered. Thus, sea water contain­ing approximately 3.5% of salt has a boiling point of100.6 °c, while its freezing point is lowered by 2°C.

If oneliquid boils at a higher temperature than another,then (at the same temperature) its vapour pressure islower. Hence, the vapour pressure of a solution is lessthan that of the pure solvent. One can judge the differ­ence on the basis of the following values: the vapour pres­sure at 20°C equals 17.5 mm Hg and the vapour pressure

5. Solutions 149

However, using evaporation, we can also get rid ofliquid admixtures and separate a mixture consisting oftwo or more liquids. In this connection, we make use ofthe fact that two liquids forming a mixture do not boilwith the same "ease".

Let us see how a mixture of two liquids, for example,of water and ethyl alcohol, taken in equal amounts(100 proof vodka), will behave when boiled.

Under standard pressure, water boils at 100°C and al·cohol at 78°C. The mixture we are dealing with will boilat the intermediate temperature of 81.2 DC. Alcohol boilsmore easily, so its vapour pressure is greater and for aninitial fifty-per cent composition of the mixt~re the firstportion of vaP12ur will contain 80 % alcohol. '

We c:m draw off the portion of vapour so obtained intoa r.efrigerator and obtain a liquid enriched by alcohol.ThIS process can be further repeated. However, it is clearthat such a method is not practical, for with each suc­cessive distillation less and less substance will be obtained.In order that there be no such loss, so-called fraction­ating (i.e. purifying) columns are applied for the pur­pose of cleaning a liquid.

The idea behind the structure of this interesting appa­ratus consists in the following. Imagine a vertical columnwhose lower part contains a liquid mixture. Heat issupplied below the column and cooling is carried out aboveit. The vapour formed by boiling {'ises to the top andcondenses; the resulting liquid flows down. For a fixed heatsupply from below and heat removal from above, thecountercurrents of vapour, going up, and liquid, flowingdown. are established in the closed column.

Let us fix our attention on some horizontal cross­section of the column. The liquid passes downwards, andthe vapour upwards, through this cross-section; moreover,none of the substances composing the liquid mixturestays behind. If we are dealing with a column contain-

ing a mixture of alcohol and water, the amount of alcoholpassing upwards and downwards, just as the amount ofwater passing upwards and downwards, will be· equal.Since the liquid is going down and the vapour is comingup, this means that the compositions of the liquid andof the vapour are identical at any height in the column.

As has been just explained, the equilibrium of theliquid and the vapour of a mixture of two substances re­'quires; on the contrary. a difference between the composi­tions of these phases. Therefore, the conversion of liquidinto vapour, and of vapour into liquid, takes place at anyheight in the column. Moreover, the high-boiling compo­nent condenses and the low-boiling component passes fromthe liquid to the vapour.

Consequently, it is as though the rising vapour weretaking away the low-boiling component and the down­flowing liquid were continually being enriched by thehigh-boiling component. The composition of the mixturewill become different at each height: the higher the mix­ture, the greater the percentage of the low-boiling com­ponent. In the ideal case, a pure layer of the low-boilingcomponent will~ on the top and a pure layer of the high­boiling component will be at the bottom.

The substances should now be drawn off, only as slowlyas possible, in order that the ideal picture just outlinednot be disturbed, the low-boiling one from the top, andthe high-boiling one from the bottom.

In order to carry out the separation, or rectification,in practice, we must make it possible for the countercur­rents of liquid and vapour to thoroughly mix. To;this end,the currents of liquid and vapour are delayed with the aidof plates distributed one above the other and connectedby means of overflow pipes. The liquid can flow down toalower level from an overfilled plate. The rising vapour(0.3-1 m/s) forces its way through a thin layer of liquid.The diagram of a column is shown in Figure 5.1.

Molecules 150 5. Solutions 151

One does not always succeed in purifying a liquid com­pletely. Certain mixtures possess an "unpleasant" prop­erty: for a definite composition, the ratio of the compo­nents of the evaporating molecules is the same as that inthe liquid mixture. In this case, of course, a further pu­rification by means of the method described becomes im­possible. Such is the mixture containing 96% alcohol and4 % water: it yields a vapour of the same composition.Consequently, 96% alcohol is the best that can be ob­tained by the evaporation method.

Rectification (or distillation) of liquids is an importantprocess in chemical technology. For example, gasoline isobtained from oil with the aid of rectification.

It is a curious fact that rectification is the cheapestmethod of obtaining oxygen. For this, ofcourse, one mustchange air to the liquid phase beforehand, after which onecan separate it into almost pure nitrogen and oxygen bymeans of rectification.

1535. Solufions

Purification of Solids

On a phial containing a chemical substance alongsideof the chemical name one can see, as a rule, the followingnotation: "pure", "pure for analysis" or "spectroscopicallypure". They are used to mark the degree of purity of asubstance: "pure" means a rather slight degree of purity­there may possibly be an admixture of the order of 1%in the substance; "pure for analysis" means that admixturesconstitute not more than several tenths of a per cent ofthe substance; "spectroscopically pure" is not easy toobtain, since spectral analysis detects an admixture ofseveral thousandths of a per cent. The inscription "spec­troscopically pure" permits us to be confident that thepurity of the substance is characterized by at least "fournines", Le. that the content of the basic substance is notless than 99.99 %. ,.

The need for pure solids is quite great. Admixtures ofseveral thousandths of a per cent can change many phys­ical properties, and in one special problem of exception­al interest to modern technology, namely the problem ofobtaining semiconductors, technologists require a purityof seven nines. This means that one unnecessary atomamong ten million necessary ones prevents us from solv­ing engineering problems! We take recourse in specialmethods for obtaining such ultrapure materials.

Ultrapure germanium and silicon (these are the mainrepresentatives of semiconductors) can be obtained byslowly drawing out a growing crystal from a melt. A rodon whose tip a seed crystal is attached is brought to the

152

.--_....... - -_ ...- - - """-.-'

~~~-~-- -~--' -------- "-._-_.,.

Molecules

Figure 5.1

Adsorption

Gases rarely dissolve in solids, i.e. rarely penetratecrystals. But there exists a different kind of absorptionof gases by solids. Molecules 'Of a gas accumulate on thesurface of a solid body-this peculiar adherence is called

surface of melted silicon (or germanium). We then beginslowly raising the rod; the crystal coming out of themelt is m.ade up of the atoms of the basic substance, whilethe atoms of the admixture remain in the melt.

The method of so-called zone refining has been appliedmore widely. A rod of arbitrary length and several mil­limetres in diameter is made out of the material beingpurified. A small cylindrical oven is placed alongside therod enveloping it. The temperature in the oven is highenough for melting, and the part of the metal which isinside the oven melts. Therefore, a small zone of meltedmetal moves along the rod.

The atoms of an admixture usually dissolve much moreeasily in a liquid than in a solid. Consequently, at theboundary of the melted zone, the atoms of admixture passfrom solid parts to the melted zone and do not pass back.It is as though the moving melted zone were draggingoff the atoms of the admixture together with the melt.The oven is turned off during the reverse motion, and theoperation of pulling the melted zone past the rod of met­al is repeated many times. After a sufficient number ofcycles, it only remains to saw off the polluted end of therod. Ultrapure materials are obtained in a vacuum or inan atmosphere of inert gas.

When there is a large fraction of alien atoms, purifi­cation is carried out by other methods, the zone meltingand pulling a crystal out of a melt being applied only forthe final purification of the material.

1555. Solutions

*Adsorption should not be confused with absorption, whichsimply means taking in.

adsorption*. Thus, adsorption occurs when moleculescannot penetrate within a body, but successfully stick toits surface.

To be adsorbed means to be absorbed by a surface. Butcan such a phenomenon really playa significant role? Infact, a layer one molecule thick, even on a very largeobject, will weigh a negligible fraction of a gram.

Let us perform the appropria!e calculations. The areaof a small molecule is about 10 A 2

, i.e. 10-15 cm2• Hence,

1015 molecules will fit into a square centimetre. Thatmany molecules, say, of water, do not weigh much:3 X 10-8 g. Even in a square metre, there will be roomfor only 0.0003 g of water.

More noticeable amounts of a substance will form onsurface areas of hundreds of square metres. There is asmuch as 0.03 g of water (1021 molecules) per 100 m2

•

But do we come across such sizable surface areas in ourlaboratory practice? However, it is not hard to grasp thatsometimes very small bodies, fitting onto the tip of ateaspoon, have enormous areas of hundreds of squaremetres.

A cube whose edges are 1 cm long nas a surface area of6 cm2 • Let us cut ~such a cube into 8 equal cubes with0.5-cm edges. The small cubes have faces with an area of0.25 cm2 • There are 6 X 8 = 48 such faces in all. Theirtotal area is equal to 12 cm2 • The surface area has beendoubled.

Thus, every splitting of a body increases its surface.Let us now break up a cube with a 1-cm side into smallchips one micrometre long: 1 f-tm = 10-4 cm, so the largecube will be divided into 1012 pieces. Each piece (let usassume for the sake of simplicity that it, too, is cubic)has a surface area of 6 f-tm 2 , i.e. 6 X 10-8 cm2

• The total

154Molecules

Osmosis

Among the living tissues, there are peculiar membraneswhich have the ability of letting water molecules

surface area of the chips iii equal to 6 X 104 cm2 , i.e.6 m2

• And such a splitting is by no means the limit.It is quite understandable that the specific surface area

(i.e. the surface area of one gram of a substance) can beimmense. It grows rapidly with the crushing of a sub­stance-for the surface area of a grain decreases in pro­portion to the square of its linear dimension, and thenumber of grains in a unit of volume increases in pro­portion to the cube of this dimension. A gram of waterpoured onto the bottom of a glass has a surface area ofseveral square centimetres. The same gram of water inthe form of raindrops has a surface area of about tensof square centimetres. But one gram of droplets of foghas a surface area of several hundred square metres.

If you break up a piece of coal (the finer the better),it will be capable of adsorbing ammonia, carbon dioxideand many toxic gases. This last property has assured forcoal an application in gas-masks. Coal breaks up partic­ularly well, and the linear dimensions of its particlescan be reduced to ten angstroms. Therefore, one gram ofsRecial coal has a surface area of several hundred squaremetres. A gas-mask with coal is capable of absorbing tensof litres of gas.

Adsorption is widely employed in the chemical indus­try, Molecules of different gases adsorbed on a surfacecome in close contact with each other and participate inchemical reactions more easily. In order to speed upchemical processes, one frequently makes use of finelysplit-up metals (nickel, copper and others) as well as ofcoal. Substances increasing the rate of a chemical reac­tion are called catalysts.

5. Solutions t57

pass through them, remaining impermeable to moleculesof substances dissolved in water. The properties of thesemembranes are the causes of physical phenomena bearingthe name "osmotic" (or simply "osmosis").

Imagine that such a semipermeable partition dividesa pipe made in the form of the letter U into two parts. Asolution is poured into one of the elbows of the pipe, andwater or some other solvent into the other elbow. Havingpoured the same amount of liquid into both elbows, weshall discover with surprise that there is no equilibriumwhen the levels are equal. Mter a short time, the liquidssettle down at different levels. Moreover, the level israised in the elbow containing the solution. The waterseparated from the solution by the semipermeable par­tition tends to dilute the solution. It is just this phenom­enon that bears the name osmosis, and the difference inheight is called the osmotic pressure.

But what is the cause giving rise to osmotic pressure?In the right-hand elbow of the vessel (Figure 5.2), pres­sure is exerted only by water. In the left-hand elbow, thetotal pressure is the sum of the pressure of the water andthat of the solute. But the door is open only for the water,and equilibrium in the presence of the semipermeablepartition is established not when the pressure from theleft equals the total pressure from the right, but when thepressure of the pure water is equal to the "water" portionof the pressure of the solution. The arising differencebetween the total pressures 'rs equal to the pressure ofthe solute.

This excess of pressure is precisely the osmotic pres­sure. As experiments and computations show, the osmoticpressure is equal to the pressure exerted by a gas composedof the solute and occupying the same volume. It istherefore not surprising that osmotic pressure is measuredin impressive numbers. The osmotic pressure arising inone litre of water when it dissolves 20 grams of sugar -

t56Molecules

Molecules

BEVERAGES

F.O-:;:°l1·-

OSMOTICPRESSURE:

Figure 5.2

would balance a column of water 14 m in height.Running the risk of arousing unpleasant memories in

the reader, we shall now examine how the laxative ac­tion of certain salt solutions is related to osmotic pres­sure. Tile walls of the intestines are semipermeable to anumber of solutions. If a salt does not pass through thewalls of the intestines (this is true of Glauber's salt),then in the intestines there arises an osmotic pressure andthis pressure "sucks" water through the tissues from theorganism into the intestines.

Why does very salty water fail to quench one's thirst?It turns out that osmotic pressure is guilty of this, too.The kidneys cannot eliminate urine with the osmoticpressure greater than the pressure in the tissues of theorganism. Consequently, an organism which has acquiredsalty sea water does not only fail to give it to the tis­sue liquids, but, on the contrary, eliminates water takenaway from the tissues along with urine.

6. Molecular Mechanics

Frictional Forces

This isn't the first time that we are speaking of fric­tion. And as a matter of fact, in telling about motion, howcould we have managed without mentioning friction?Almost anv motion of the bodies surrounding us is ac­companied"by friction. A car whose driver cut the motorcomes to a halt; a pendulum comes to rest after manyoscillations; a small metal ball thrown into a jar of sun­flower oil slowly sinks. What makes bodies moving alonga surface come to a halt? What is the cause of the slowfalling of a ball in oil? We answer: the frictional forcesarising during the motion of some bodies along the sur­faces of others.

But frictional forces arise not only during motion.You probably had to move furniture in a room. You

know how hard it is to begin moving a heavy bookcase.The force counteracting this effort is called static friction.

Frictional forces also arise when we slide and rollobjects. These are two somewhat different physical phe­nomena. We therefore distinguish between sliding fric­tion and rolling friction. RolliJlg friction is tens of timesless than sliding friction.

Of course, in certain cases sliding also proceeds withgreat ease. A sled slides easily along snow, and the slid­ing of skates along ice is even easier.

But what causes do frictional forces depend on? Africtional force between solid bodies depends little on thespeed of the motion and is proportional to the weight of

Molecules f60

a body. If the weight of a body doubles, it will be twiceas, hard to set it in motion and to keep pulling it. Wehaven't expressed ourselves with complete precision:what is important ~s not so much the weight as the forcethat presses the body to the surface. If a body is light,but we press down hard on it with our hand, this will ofcourse affect the force of friction. If we denote the forcethat presses a body to a surface (this is its weight in mostcases) by P, the following simple formula will be validfor the frictional force Ffr :

Ffr=kP

But how are the properties of surfaces taken into ac­count? For it is well known that one and the same sledwill slide completely differently on the very same runners,depending on whether or not the runners are bound withiron. These properties are taken into account by the pro­portionality factor k, which is called the friction coeffi­cient.

The friction coefficient of metal on wood is approximate­ly equal to one-half. Only with a force of"1 kgf will onesucceed in setting in motion a metallic slab with a massof 2 kg which is lying on a smooth wooden table. But thefriction coefficient of steel on ice is equal to only 0.027.That same slab can be moved on ice with a force of only0.054 kgf.

One of the earliest attempts to lower the coefficient ofsliding friction is depicted in a fragment of a mural froman Egyptian tomb dated approximately 1650 B.C. (Fig­ure 6.1). We see a slave pouring oil under the runners ofa sledge carrying a heavy statue.

The surface area does not occur in the above formula:the force of friction does not depend on the area of thecontact surface between the bodies rubbing against eachother. The same force is needed in order to set in motion,or to keep moving with a constant velocity, a wide sheet

6. Molecular Mechanics

Figure 6.t

11-0389

1.61.

it*

it costs too much. This circumstance greatly impeded thedevelopment of technology at the end of the past centu~y.It was then that the remarkable idea arose of replacmgthe sliding friction in bearings by rolling friction. Ballbearings have accomplished this replacement. Smallballs were fitted in between the axle and the bush. Asthe wheel turns, the balls roll along the b~sh,.nd ~heaxle rolls on the balls. The construction of thIS mechamsmis shown in Figure 6.2. Sliding friction wa~ repl~ce~ byrolling friction in this manner, thus reducmg frIctIOnalforces tens of times over.

The role played by ball and roller bearings in moderntechnology can scarcely be exaggerate.d. The~ are madewith balls, with cylindrical rollers, WIt~ comcal. rollers.All machines, large and small, are eq"?-I~ped w~th ~uchbearings. There exist ball bearing~ a mIllI~etre m dIam­eter; some bearings for large machmes weIgh over a to~.Balls for bearings (you have seen them, ~f course, mcertain store windows) are manufactured WIth the mostvarying diameters-from fractions of a millimetre toseveral centimetres.

163

Figure 6.2

6. Molecular MechanicsMolecules 162

of steel weighing a kilogram and a kilogram weight whichis supported by the surface of a small area.

And one more remark about forces of sliding friction.It is somewhat more difficult to set a body in motion thanto keep it moving: the force of friction overcome at thefirst instant of motioQ (static friction) exceeds the sub­sequent values by 20-30%.

What can be said about the force of rolling friction,say, for a wheel? Just as for sliding friction, the greaterthe force that passes a wheel to a surface, the greater therolling friction. Furthermore, the force of rolling frictionis inversely proportional to the radius of the wheel. Thisis un~erstandable: the larger the wheel, the less percepti­ble wIll be the roughness of the surface along which it isrolling.

If we compare the forces which must be overcome inmaking a body slide and roll, the difference we obtainis very impressive. For example, in order to pull a steelbar weighing 1 tonf along an asphalt pavement, a forceof 200 kgf must be applied-only an athlete is capable ofdoing this. But even a child can roll this very same baron a cart-a force not greater than 10 kgf is needed forthis.

It's no wonder that rolling friction has "defeated"sli?ing friction. It was not without reason that humanitysWItched to wheel transport a very long time ago.

The replacement of runners by wheels is not yet a com­plete victory over sliding friction, for a wheel must beattached to an axle. At first glance it seems impossibleto avoid the friction of the axle on the bearings. So peo­ple thought during the course of centuries and tried todecrease the sliding friction in bearings only by meansof various lubricants. The services rendered by lubricantshave not been small-sliding friction has been reducedby a factor of 8-10. But in a great many cases, the slid­ing friction, even with lubrication, is so considerable that

Viscous Friction in Liquids and Gases

Until now we have been speaking of "dry" friction,Le. of the friction arising from the contact of two solidobjects. But floating and flying bodies are also subject tothe action of frictional 'forces. The source of friction haschanged-dry friction is replaced by "wet" friction.

The resistance experienced by a body moving in wateror air is subject to other laws, essentially different fromthe laws of dry friction, which we have spoken of above.

The rules for the behaviour of a liquid and a gas withrespect to friction cannot be distinguished. Thereforeeverything said below pertains to liquids and gases t~the same degree. When we speak of a fluid, we will meana liquid or a gas.

One of the distinctions between "wet" and dry frictionconsists in the absence of "wet static friction": an objectsuspended in water or air can, generally speaking, bestarted by an arbitrarily small force. But as for the fric­tional force experienced by a moving body, it will dependon the speed of the motion, on the form and dimensionsof the body and on the properties of the fluid. The studyof the motion of bodies in flUids has shown that there isno single law for "wet" friction, but there are two differ­ent laws: one is valid for motions with low speeds, andthe other for motions with high speeds. The existenceof two laws implies that the flow of a medium around abody moving in it takes place differently for motions ofsolids with high and low speeds.

For motions with low speeds, the force of resistance ordrag, is directly proportional to the speed and the lin'eardimensions of a body:

F ex vL

How should we understand the proportionality todimension if we aren't told what shape the body has?

Molecules 164 6. Molecular MechanIcs 165

What is meant is that for two bodies completely similarin shape (Le. all of whose corresponding dimensions arein the same ratio), the ratio of the drag is the same asthat of the linear dimensions of the bodies.

Drag depends to an enormous degree on the propertiesof a fluid. Comparing-the frictional forces experienced bythe same objects moving with identical speeds in variousmedia we see that the thicker or, as we say, the moreviscou~ the medium, the greater the drag experienced bya body. It is therefore appropriate to call the frictionunder discussion viscous friction. It is quite understand­able that air creates a negligible viscous friction, approx­imately 60 times less than water. A liquid can be "thin",like water or very viscous, like sour cream and honey.

We can 'judge the degree of viscosity of a liquid eitherby the speed with which solid bodies fall in.it or by thespeed with which it pours through an openmg.

It will take water several seconds to pour out of a half­litre funnel. A very viscous liquid will trickl~out of itin hours, or even days. It is possible to give examples ofeven more viscous liquids. Geologists have called atten­tion to the fact that in the craters of certain volcanoesspherical pieces are found in accumulations of lava on theinner sides. At first sight, it was completely impossibleto understand how such a sphere might be formed out oflava inside a crater. This would be incomprehensible iflava were regarded as a solid. But if lava behaves likea liquid, it will drop down the crater just li~e any ot~erliquid. But only a drop will !>e formed not m a fractIOnof a second but in the course of decades. When the dropbecomes too heavy, it will break away and fall at thebottom of the crater.

It is clear from this example that real solid bodies andamorphous bodies, which, as we know, resemble liquidsmuch more than crystals, should not be put on the samelevel. Lava is precisely such an amorphOUS body. It seems

Forces of Resistance at High Speeds

But let us return to the laws of "wet" friction. As weexplained, at low speeds the drag depends on the viscosityof a liquid, the speed of the motion and the linear dimen­sions of a body. Let us now consider the laws of friction forhigh speeds. But first of all, we must say what speeds areto be regarded as low, and what speeds as high. We areinterested not in the numerical value of the speed butin whether the speed is sufficiently low for the law ofviscous friction considered above to hold.

It turns out that it is impossible to name a number ofmetres per second such that the laws of viscous friction beapplicable in all cases with lower speeds. The bounds ofapplicability of the law we have studied depends on thedimensions of a body and on the degree of viscosity anddensity of a liquid.

to be solid, but is actually a very viscous liquid.What do you think, is sealing wax a solid? Take two

corks and place them at the bottom of two cups. Pourany melted salt (for example, saltpeter-it is easilyobtained) into one, and pour sealing wax into the othercup with a cork. Both liquids will harden and bury thecorks. Put these cups away in the cupboard and forgetabout them for a long time. After several months, youwill see the difference between sealing wax and salt. Thecork drowned in salt will be at rest, as before, at the bot­tom of its cup. But the cork drowned in sealing wax willturn out to be on the top. But how did this occur? Verysimply: the cork came to the surface in quite the sameway as it would do in water. The only difference being isthat of time: when the force of viscous friction is small,a cork comes to the surface instantly, but in very viscousliquids floating up takes months.

167

°i75cmls

for water, less than

°i05cmls

and for viscous liquids like thick honey, less than

100L cmls

Consequently, the laws of viscous friction are scarcelyapplicable to air and especially to water: even for lowspeeds, of the order of 1 cm/s,. they.will apply o~ly fortiny bodies with millimetre dImensIOns. 'Fhe reSIstanceexperienced by a person diving into water IS to no extentsubject to the law of viscous friction.

But how can we explain the fact that t~e law gover~­ing the resistance of a medium changes WIth a change Inspeed? The causes must be sought in the change in !hecharacter of the flow of a liquid around a body mOVIngin it. Two circular cylinders moving in a li~uid are de­picted in Figure 6.3 (their axes are perpendIcular to thedrawing). For a slow motion, the liqu~d fl.ows smoothlyaround the moving object-the drag WhICh It has to over­come is the force of viscous friction (Figure 6.3a). Fora high speed, behind the moving ~od! th~re arises a com­plicated irregular motion of the lIqUId (FIgu~e 6.3b). Va;­ious streams appear and disappear, formIng fantastIcfigures, rings and eddies. The picture made by t?e stre~msis changing all the time. The appearance of thIS ~otIOn,called turbulent, radically changes the law of resls~ance.

Turbulent resistance depends on the speed and dIme~­sions of an object in an entirely different way than VIS­cous resistance: it is proportional to the square of the

6. Molecular MechanIcs

For air, "low" is a speed less than

166Molecules

Streamline Shape

Motion in the air, as has been said above, is almostalways "speedy", Le. the basic role is played by turbulentand not viscous resistance. Airplanes, birds and parachut­ists experience turbulent resistance. If a person fal.lsthrough the air without a parachute, then after a ~ertamtime he begins falling uniformly (the force of resIstancebalances the weight), but with quite a considerable speed,of the order of 50 m/s. The opening of a parachute leadsto a sharp deceleration of the fall-the same weight isnow balanced by the resistance of the canopy. Since theforce of resistance is proportional to the speed of the mo­tion as well as to the linear dimensions of the falling o~ject, the speed will decrease as many times as. the lineardimensions of the falling body increase. The dIameter ofa parachute is about 7 m, whereas the "diameter" of. aman's body is about one metre. The speed of the fall wIllbe decreased to 7 m/s. One can land safely with such aspeed.

We must say that the problem of increasing the re-sistance is far more easily solved than the converse prob­lem. Reducing the air resistance on a car or an airplane,or the water resistance on a submarine, is one of the mostimportant and difficult technological problem~.

I t proves possible to decrease turbulent resIstance b.ya large factor by changing the shape of a body. For thISone must reduce to a minimum the turbulent motion,which is the source of the resistance. This is achieved by,as they say, streamlining the object.

But what shape is best in this sense? It would appearat first sight that the body should be shaped so that theforward side comes to a point. Such a point, it seems,should "cleave" the air most successfully. But it provesimportant not to cleave the air but to disturb it as l~ttleas possible so that it flows smoothly around the obJect.

Molecules

Figure 6.3

speed and the square of the linear dimensions. The vis­cosity. of a liquid during t~is motion ceases to play anesse~tlal ~ole; the determming property becomes itsdensIty, .wIth the dra.g proportional to the first power ofthe densIty of the flUId. Therefore, the following formulaholds for the force F of turbulent resistance:

F 0:: pv2L2

where v is the speed of the motion, L are linear dimensionsof the object, and p is the density of the medium. Thenumerical proportionality .factor, which we haven'twritten down, has various values depending on the shapeof the body.

6. Molecular Mechanics 169

The best profile for a body moving in a fluid'wh~se s~ape is obtuse in front and pointed behi:d;h~~~:su~ a s ape, the fluid would flow smoothly off th~ ~ tan turbulence would be reduced to am' . SPh°

In

edges should not i . Immum. arpthey would create~:~:uf::~e~e put In front, since there

The streamline shape of . I .only.the least resistance to t~~ ~~fi~~eh:;l~f crt~ates not

~~~e~~~~n,:~e~:hr:o~~~~n~ls~nclined upwar~~ fr~;ret~~pushes against it maini i OWIng a:oun.d a WIng, the airto its plane (Figure 6 4{ I~ .thel dIre~lon perpendicularwing this force is dire~ted u IS cdear t at for an inclined

Th l'f pwar s.But r~a~o~f:~~sa:;~t~o~~increase in t~e angle of attack.would lead us to the flY on Igeo.metncal considerations

a se conc USlon that the greater the*The pointed prows of b t d .for "breaking" waves i e °oanl a\ sehgomg vessels are neededalong the surface. ' .. y w en t e motion is taking place

angle of attack, the better. But as a matter of fact, asthe angle of attack is increased, the smooth flow of theplane becomes more and more difficult, and at a certainvalue for this angle, as illustrated in Figure 6.5, violentturbulence arises; the drag sharply increases and the liftdecreases.

171

Disappearance of Viscosity

Very often, in explaining some phenomenon or describ­ing the behaviour of one or another group of bodies, werefer to well-known examples. It is quite obvious thatthis object should be moving in such a manner, we say,for other bodies also move according to the very samerules. The explanation reducing the new to that which wehave already come across in the course of our lives alwayssatisfies the majority. We therefore did not have any par­ticular difficulty in explaining to the reader the laws inaccordance with which liquids move-for everyone hasseen how water flows, and the laws governing this motionseem quite natural.

Figure 6.5

6. Molecular MechanicsliO

PRESSURE

Molecules

Figure 6.4

Molecules 172

However, there is one perfectly amazing liquid whichdoes not resemble any other liquid and moves in accor­dance with special laws characteristic of it alone. It isliquid helium.

We have already said that liquid helium remains in theliquid phase right down to the temperature of absolutezero. However, helium above 2 K (more precisely, 2.19 K)a~d heliu~ b.elow this temperature are two completelydIfferent lIqUIds. Above two degrees, the properties ofhelium in no way distinguish it fro)m other liquids. Belowthi~ te~perature, h~lium. becomes a miraculous liquid.ThIs mIraculous hellUm IS called helium II.~~e m?st striking property of helium II is its super­

flmd~ty, I.e. complete lack of viscosity, discovered byP. L. Kapitza in 1938.

In order to observe superfluidity, a vessel is made witha very fine slit at the bottom-of only half a micrometrein. widt~. An ordinary liqui? hardly seeps through such aslIt; hellUm also behaves thIS way at temperatures above2.19 K. But as soon as the temperature becomes barelyless than 2.19 K, the speed with which helium flows outof such a vessel grows by leaps and bounds-by a factorof at least several thousand. Helium II almost momen­tarily flows through the narrowest clearance, Le. it com­pletely loses its viscosity. The superfluidity of heliumleads to an even stranger phenomenon. Helium II is ca­pable of "climbing out" of the glass or test tube into whichit was poured. A test tube with helium II is placed ina Dewar vessel over a helium bath. "Without rhyme orreason", helium rises along the wall of the test tube inthe form of a very fine, completely unnoticeable film andoverflows; drops fall from the bottom of the test tube.

I t should be recalled that thanks to capillary forces~hi~h were. discussed on p. 45, the molecules of ever;lIqUId wettIng the wall of a vessel climb up this wall andform the finest film on it whose width is of the order of

6. Molecular MechanIcs 173

10-8 em. This film cannot be seen by an eye, and in gener­al does not manifest itself in any way for an ordinaryviscous liquid.

The picture changes completely if we are de~ling withhelium devoid of viscosity. In fact, a narrow slIt does nothinder the movement of superfluid helium, and a thinfilm on a surface is just the same as a narrow slit. A liquidwithout viscosity flows in a very fine layer. The film cov­ering the surface forms a siphon over the edge of theglass or test tube through which the helium overflowsthe vessel.

It is obvious that we do not observe anything of thekind in the behaviour of an ordinary liquid. A liquid ofstandard viscosity is practically unable to "make its way"through a siphon of negligible thickness. Such a motionis so slow that the outflow would last millions of years.

Thus helium II is devoid of any viscosity. The conclu­sion th~t a solid should move without friction in such aliquid would appear to follow from this with iron logic.Let us take a disc on a string, place it in liquid helium IIand twist the string. Leaving this uncomplicated devicealone we create something like a pendulum-the stringwith 'the disc will oscillate and periodically twist firstin one and then in another direction. If there were nofriction we should expect the disc to oscillate perpet­ually. But nothing of the kind! After a c~mparati,:"elyshort time, approximately the same as for ordInary heln~mI (Le. helium at a temperature above 2.19 K), the dISCcomes to a halt. But how will this happen? When flowingthrough a slit, helium behaves li~e a liq~id wit~ou~ vi.s­cosity, but behaves like an ordInary VISCOUS lIqUId Inrelation to bodies moving in it. It is this, to be sure, thatis in fact completely extraordinary and incomprehensible.

It now remains for us to recall that helium does notsolidify right up to absolute zero. The crux of the problemis that the ideas about motion which we are accustomed

to are not suitable any more. If helium has remainedliquid "illegally", should we be surprised by the lawlessbehaviour of this liquid?

I t is only possible to understand the behaviour ofliquid helium from the point of view of the new concep­tions of motion, quantum mechanics. Let us try to givea general idea of how quantum mechanics explains thebehaviour of liquid helium.

Quantum mechanics is an extremely intricate theory,which is very hard to understand,! and so the reader shouldnot be surprised that the explanation looks even strangerthan the phenomena themselves. It turns out that everyparticle of liquid helium participates simultaneously intwo motions: one motion is superfluid, unrelated to vis­cosity, and the other is ordinary.

Helium II behaves as if it consisted of a mixture oftwo liquids moving completely independently, "onethrough the other". One liquid behaves normally, Le.possesses an ordinary viscosity, while the other componentpart is superfluid.

When helium flows through a slit or overflows a glass,we observe the effect of superfluidity. But during theoscillation of a disc submerged in helium, friction iscreated because it is inevitable in the normal part ofhelium.

The ability to participate in two distinct motions alsogives rise to completely unusual heat-conducting proper­ties of helium. As has been already said, liquids in generalconduct heat rather poorly. Helium I also behaves like anordinary liquid. But when the transformation into he­lium II takes place, its thermal conductivity grows abouta thousand million-fold. Therefore, helium II conductsheat better than the best ordinary heat conductors, suchas copper and silver.

The fact is that the superfluid motion of helium doesnot participate in the heat transfer. Consequently, when

Molecules 174 6. Molecular Mechanics 175

there is a difference in temperature within helium II,there arise two currents in opposite directions, and oneof them-the normal one-carries heat with it. This doesnot at all resemble ordinary thermal conductivity. Inan ordinary liquid, heat is transferred by means of mo­lecular collisions. In helium II, heat flows together withthe ordinary part of helium-flows like a liquid. Hereat last the term "heat flow" is fully justified. It is pre­cisely such a method of transferring heat that leads to animmense thermal conductivity.

This explanation of the thermal conductivity of he­lium may seem so strange that you will refuse to believeit. But it is possible to convince oneself first-hand of thevalidity of what has been said by means of the followingconceptually simple experiment.

In a tub with liquid helium there is a Dewar vessel filledto the brim with helium. The vessel is connected to thetub by means of a capillary branch piece. The helium in­side the vessel is heated by an electric spiral, but heatdoes not pass to the surrounding helium through the wallsof the vessel, since they do not transmit heat. Near theend of the capillary tube there is a vane suspended on afine thread. If the heat is flowing like a liquid, it shouldturn the vane. This is precisely what happens. Moreover,the amount of helium in the vessel does not change. Howcan this miraculous phenomenon be explained? In onlyone way: during heating there arises a current of the nor­mal part of the liquid from the heated place to the cold,and a current of the superfluid part in the opposite di­rection. The amount of helium at each point does notchange, but since the normal part of the liquid movestogether with the heat transfer, the vane turns as a re­sult of the viscous friction of this part and remains de­flected as long as heating continues.

Another conclusion also follows from the fact that su­perfluid motion does not transfer heat. We have spoken

177Molecules 176

above about the "creeping" of helium over the brim of aglass. But the superfluid part "climbs out" of the glass,while the normal part remains there. Heat is connectedto only the normal part of helium and does not accompanythe superfluid part which is climbing out of the glass.Hence, as helium climbs out of the vessel, one and thesame amount of heat will be shared by a smaller andsmaller amount of helium-the helium remaining in thevessel should warm up. This is actually observed duringexperiments.

The masses of the helium partaking in superfluid andnormal motion are not identicah Their ratio depends onthe temperature. The lower the temperature, the greaterthe superfluid part. All of the helium becomes superfluidat absolute zero. As the temperature rises, a larger andlarger part of the helium begins to behave normally, andat the temperature of 2.19 K, all of the helium becomesnormal, acquiring the properties of an ordinary liquid.

But the reader already has some questions in his mind:What precisely is called superfluid helium, how can aparticle of liquid participate simultaneously in two mo­tions and how can the very fact of two motions of a singleparticle be explained?... Unfortunately, we are obligedto leave all these questions unanswered. The theory ofhelium II is too complicated, and it is necessary to knowa great deal in order to understand it.

Plasticity

Elasticity is the ability of a body to recover its formafter a force ceased acting on it. If a kilogram weight ishung on a 1-m steel wire with a cross-sectional area of1 mm2

, the wire will be stretched. The stretching is ne­gligible, only 0.5 mm in all, but it is not difficult to ob­serve. If the load is removed, the wire will contract by the

6. Molecular Mechanics

same 0.5 mm, and so its length will return to its formervalue. Such a deformation is called elastic..

Let us note that a wire of 1-mm2 cross-sec~lOnunder ~heaction of a force of 1 kgf and a wire of 1-cm cross-sectIOnunder the action of a force of 100 kgf are, as one says, underthe same conditions of mechanical stress. Theref?re, thebehaviour of a material must always be ~escrlb?d byindicating not the force (which would be pomtless If ~hecross-section of a body is unknown), but the stress, I.e.the force per unit of area. Ordinary bodies-metals,glass, stones-can be stretched elastically by o?ly sever~lper cent at best. Rubber possesses outstandmg elastIcproperties. Rubber can be stretched elast~cally by severalhundred per cent (Le. made two or three tImes longer thanit was originally), and when we let such a rubber band go,we see that it returns to its initial state.

All bodies without exception behave elastically underthe action of small forces. However, a limit of elasticityappears earlier for some bodies and considerably laterfor others. For example, the elastic limit for such softmetals as lead has aheady been reached when ~ load ?f0.2-0.3 kgf is hung on a wire of 1-mm2 cross-se~tlOn. ThISlimit is approximately 100 times as great, l.e. about25 kgf, for such hard materials as steel. ., .

With respect to large forces exce~d~ng t.he elastIc IImI~the various bodies can be roughly dIVIded m~o two clas~es.those like glass, Le. fragile, and those lIke clay, l.e.plastic. . '11

If you press a piece of clay with you: flI~ger, It ~Ileave its imprint, containing even the mtrI?ate WhIrlSdrawn on its skin. If you strike a piece of soft Iron or. leadwith a hammer, a clear trace will be left. T~ere I.S ~olonger any force, but the defor~ation remams-:-It IScalled plastic or permanent. You wIll no~ succeed I~ o~­taining such residual effects with glass: If you perSIst III

this endeavour, the glass will break. Certain metals and

12-0389

179MOlecules 178

alloys, such as_cast iron, are just as fragile. An iron pailwill flatten, but a cast iron pot will crack, under the blowsof a hammer. One can judge the strength of fragile bodies

. by the following figures. In order to convert a piece ofcast iron into powder, one must act with a force of about50-80 kgf per square millimetre of surface area. This figurefalls to 1.5-3 kgf for a brick.. . As for every classifica,tion, the division of bodies intofragile and plastic is, to a fair degree, relative. First ofall, a body which is fragile at a low temperature can be­come plastic at higher temperatures. Glass can be superb­ly processed as a plastic material if it is heated to atemperature of several hundred degrees. Soft metals.. ,such as lead, can be forged cold, but hard metals yieldto forging only when burning hot. A rise .in temper­ature sharply increases the plasticity of a material.

One of the essential features of metals, which has madethem irreplaceable building materials, is their hardnessat room temperatures and plasticity at high temperatures:an incandescent metal can be easily given the requiredform, but it is only possible to change this form at roomtemperature by means of very substantial forces.

The internal structure of a material influences its me­chanical properties in an essential way. It is obvious thatcracks and holes weaken the apparent strength of a bodyand make it more fragile.

The ability of plastically deformable bodies to strength­en is remarkable. A single crystal of a metal is very softwhen it has just grown up out of a melt. Crystals of manymetals are so soft that they are easily bent with one'sfingers; however, one will not succeed in straighteningthem out. Strengthening has taken place. This same spec­i!l!en can now be plastically deformed only by means ofa considerably greater force. It turns out that plasticityis not only a property of a material, but also a prop­erty of its treatment.

6. Molecular Mechanics

Why are instruments made not by casting metals, butby forging them? The reason is obvious-a metal subjectedto forging (or rolling, or drawing) is much stronger thancast metal. No matter how much we forge a metal, we'cannot increase its strength beyond a certain limit, whichis called the yield stress. For steel it is between 30 and50 kgflmm2 • This figure has th~ following. meaning.. ~fa one-pood weight (below the YI~ld st,ress) I~ hung o~ a,wire of 1-mm2 cross-section, the WIre WIll begm stretchmgand strengthening simultaneously. The stretching willtherefore quickly cease-the weight will hang calmly onthe wire. But if a two- or three-pood weight (above theyield stress) is hung on such a wire, .the pic~u~e will bedifferent. The wire will keep stretchmg untIl It breaks.Let us emphasize once again that the mechanical behav­iour of a body is determined not by the force, but by thestress. A wire of a cross-section of 100 f1m2 will yield underthe action of a load of (30-50) X 10-4 kgf, Le. 3-5 gf.

Dislocations

Proving that plastic deformation is a phenomenon ofimmense significance in engineering practice would beforcing an open door. Smith and die forging, rollin~ sheets,drawing wire are all techniques based on the plastic work­ing of metals leading to their deformation.

We could understand nothing about plastic deforma­tion if we considered the crystallites of which a metal con­sists to be ideal fragments of space lattices.

The theory of the mechanical properties of an idealcrystal was worked out long ago, at the beginning of thecentury. This theory disagrees with experimental dataabout a thousand-fold. If all crystals were ideal, theirtensile strength would be many orders of magnitude high­er than that actually observed. Then plastic deformationwould require huge forces.12*

upper part of the crystal is in compression, and the lowerpart in tension. The effect of the dislocation on the crystalis rapidly dissipated as seen in Figure 6.6b which is asort of "top" view of the left-hand drawing

Another type of dislocation, frequently found in crys­tals, is the screw dislocation. It is illustrated schemati­cally in Figure 6.7. Here the lattice is divided into twoblocks, one of which has slipped, as it seems, by one latticeconstant with respect to the other. Maximum distortionis concentrated at the vertical axis shown in the diagrams.The region adjoining this axis is what is called a screwdislocation.

We shall understand the nature of this distortion betterif we look at the other diagram (Figure6.7b), which illus­trates two adjacent atomic planes on the two sides ofthe vertical plane passing through the axis (the one alongwhich slip has occurred). With respect to the three-di­mensional view, this is a projection showing the planefrom the right. Here we see the axis of the screw disloca­tion, the same as in the other drawing. The atomic planebelonging to the right-hand block is shown by full lines;that belonging to the left-hand block by dash lines. The

Molecules

- - ,"71//

~, I

/

'" , IliP ~ ~ , tt:. fit ~p p~ : I: I 4. . . If-/, I I I I I J, / , I I I I I I

- :- ~_.. ' ./ I I I I I I Jf.- I I I I I I I

~ : ,-- ) I I I J I I

· :,

I I I I I ' I, I I I I I I II I I I I I I· (6) 4 HH~ 4~j ~ ~. JJ d I(a · :

Figure 6.6

. Hypotheses had been suggested much earlier than suffi­Cle~t fact~ accumulated to support them. It was obviousto. mvestI~ators that the only way to reconcile theorywI.th practICe was to assume that crystallites have cer­tam defects. Various conjectures could be made, of course,on the nature of these defects. But the picture began to?lear ~p o?-ly after the most sophisticated equipment formv~stlgatmg the structure of matter had been madea.vallable to .physicists. It was found that an ideal por­tion of a lattI~e.(called a block) has dimensions of the ordero~ se~eral mIllIonths of a centimetre. These blocks aredlsonented with ~espect to one another by angles withina second or a mInute of arc.

Many. facts accumulated by the end of the twenties ledto the Important contentio~ that the chief (though notonly) defect of a real crystal IS a regular displacement that~a~ named a dislocation. A simple, or linear, dislocationIS Illustrated by the model in Figure 6.6. The essence ofthe defect, as is evident, is that there are places in thecrystal th~t se~m to contain an "extra" atomic plane(fro;ID whICh ItS name "extraplane" is derived). ThehorIzontal dash line in Figure 6.6a divides two blocks. The

6. Molecular Mechanics

Figure 6.7

181

is between rows 2 and 3; the compressed rows are 2' and3'. As soon as the shifting force is exerted, row 2 movesinto the "fissure"; now row 3' can "breathe freely", butrow I' will have to squeeze into less space. What hap­pened? The whole dislocation moved to the left, and its mo­tion continues in the saine way until it (the dislocation)"emerges" from the crystal. The result is a shift over onerow of atoms, i.e. the same as in the shear of a perfectcrystal. .

It is evidently unnecessary to prove that shear by dIS"location motion requires much less force. In the shear ofa perfect crystal we had to overcome the interaction be­tween atoms in rolling over all the rows simultaneously.In dislocation shear, only a single row of atoms is in­volved at each instant.

Calculations indicate that the strength of crystals inshear, assuming there are no dislocations, should. be ahundred times more than that actually observed m ex­periments. The presence of even a slight number of dis­locations can reduce the strength of the material by asubstantial factor.

The preceding discussion poses a question that must becleared up. It is evident from the drawing (see Figure 6.8)

Molecules 182

bla.ck c~rcles are closer to t~e reader than the light ones.It IS eVIdent from the drawmg that a screw dislocation isanother type of distortion differing from that of a lineardistortion. There is no extra row of atoms. The essenceof the distortion is that the rows of atoms change theirnearest neighbours near the dislocation axis. They arebent downwards to align themselves with the neighbourslocated one storey below.

Why is this called a screw dislocation? Imagine thaty.ou are strollin~ a~ong the atoms (after reducing yourSIze to subatomIc dImensions) and have decided to passaround th~ dislocation axis. It is obvious that if youstart your Journe~ on the lowest plane you will find your­self one storey hIgher after each turn. Finally, you willemerge on the top surface of the crystal, as if you hadbeen climbing winding stairs in a helix like the threadof a screw. In our drawing, the climb is in the counter­clockwise direction. If the slip of the blocks had been inthe opposite direction, your journey would have beenclockwise.

We are now ready to answer the question of how plasticdeformation takes place.

.Assume that we want to shift the upper half of a crystalwIt.h respect to the lower one by one interatomic distance.EVIdently, we must roll all the atoms in the shear planeover one another. It is an entirely different matter whena s~ear force acts on a crystal containing a dislocation.

FIgure 6.8 illustrates a close packing arrangement ofspheres (only the end spheres of the rows of atoms ares~own), c?ntaining a linear dislocation, consisting of almear vOId (something like a fissure) between two ofthe upper rows. We begin to shift the upper block to theright with r~spect to .the lower one. To help you under­stand what IS happemng, we have given numbers to therows, with primed numbers for the spheres of the com­pressed (lower) layer. At some initial moment, the "fissure"

6. Molecular Mechanics

Figure 6.8

183

by mineralogists. Ten definite minerals are arranged in aseries. The first is diamond, followed by corundum, thentopaz, quartz, feldspar, apatite, fluor-spar, calcite, gyp­sum and talc. The series is sorted out in the followingmanner: diamond leaves its mark on all minerals, butnone of these minerals can scratch diamond. This meansthat diamond is the hardest mineral. The hardness ofdiamond is evaluated by the number 10. Corundum, whichfollows diamond in the series, is harder than all the min­erals below it-corundum can scratch them. Corundum isassigned the hardness number 9. The numbers 8,7 and 6are assigned to topaz, quartz and feldspar, respectively,on the same grounds. Each of them is harder than (Le.can scratch) all the minerals below it, and softer than(can itself be scratched by) the minerals having greaterhardness numbers. The softest mineral, talc, has hard­ness number 1.

A "measurement" (this word must be taken in quotationmarks) of hardness with the aid of this scale consists infinding the place for the mineral of interest to us in aseries containing the ten chosen standards. If the unknownmineral can be scratched by quartz, but leaves its mark onfeldspar, its hardness is .equal to 6.5. . .

Metallurgists use a different method for determllllllghardness. A dent is made on the material being tested bypressing a steel ball 1 cm in diameter against. it with astandard force (ordinarily 3000 kgf). The radIUs of thesmall pit so formed is taken as th~ hardness number..

Hardness with respect to scratchlllg and hardness withrespect to pressing do not necessarily correspond, and onematerial may prove to be harder than another when testedby scratching but softer when tested by pressing.

Consequently, there is no universal concept of hardnessindependent of the method of measurement. The conceptof hardness is a technological and not a physical con­cept.

Molecules 184

that the applied force "drives" the dislocation out of the~rystal. This implies that the crystal becomes strongerand stronger as we increase the degree of deformation.Finally, when the last dislocation is eliminated, the crys­tal should, according to the theory, become about ahundred times stronger than a perfect regular crystal.The crystal actually is strengthened in the course ofdeformation, but far from a hundred-fold. The screw dis­locations save the day. It was established (and here thereader will just have to take our word for it because itis 'so difficult to illustrate with a drawing) that screw dis­locations cannot be easily "driyen" out of a crystal. More­over, shear of the crystal [may take place with the aidof dislocations of both types. Dislocation theory satis­factorily explains the features of shear phenomena alongcrystal planes. According to up-to-date views, plasticdeformation of crystals consists in the motion of disorderalong the crystal.

Hardness

Strength and hardness do not go hand in hand witheach other. A rope, a scrap of cloth and a silk thread canpossess a great deal of strength-a considerable stress isneeded to tear them. Of course, nobody will say that ropeand cloth are hard materials. And conversely, the strengthof glass is not great, but glass is a hard material.

The concept of hardness used in technology is takenfrom everyday practice. Hardness is the resistance to pen­etration. 'A ;body is hard if it is difficult to scratch it,difficult to leave an imprint on it. These definitions mayseem somewhat vague to the reader. We are accustomed tophysical concepts being expressed in terms of numbers.But how can this be done in relation to hardness?

One rather primitive method, which is nevertheless use­ful in practice, has already been employed for a long time

6. Molecular Mechanics 185

Sound Vibrations and Waves

We have already given the reader a lot of informationabout oscillations. How a pendulum and a ball on a springoscillate and what regularities there are in the oscillationof a string, all of this was studied in one of the chaptersof the first book. We haven't spoken about what takesplace within the air or some other medium when a bodylocated in it performs oscillations. There is no doubt thatthe medium cannot remain indifferent to vibrations. Anoscillating object pushes the air, displacing the air par­ticles from the positions in which they were previouslylocated. It is also obvious that this cannot be limited tothe influence on only the adjacent layer of air. The bodywill push the nearest layer, this layer presses againstthe next one, and so layer after layer, particle after par­ticle, all of the surrounding air is brought into motion.We say that the air has come to a vibrating state, or thatsound vibrations are taking place in the air.

We call the vibrations of the medium sound vibrations,but this does not mean that we hear all of them. Physicsuses the concept of sound vibrations in a broader sense.The question as to which sound vibrations are heard willbe discussed below.

We are dealing with air only because sound is most of­ten transmitted through air. But there are, of course, nospecial properties of air which would give it a monopolyto perform sound vibrations. Sound vibrations arise inany medium capable of being compressed, but since thereare no incompressible bodies in nature, the particles ofany material can therefore find themselves in this state.The study of such vibrations is usually called acoustics.

Each particle of air remains at one place, on Tthe av­erage, during sound vibrations-it only performs oscilla­tions about its equilibrium position. In the simplest case,a particle of air can perform a harmonic oscillation, which,

Molecules 186 6. Molecular Mechanics 187

as we recall, takes place in accordance with the sinu~oidallaw. Such an oscillation is characterized by the maXImumdisplacement of a particl~ from its equ~libr~um J?osition(amplitude) and the perIod of the oscI!latI?n, I.e. thetime required to perform a complete OSCIllatIOn.

The concept of the frequency of vibration is used moreoften than that of the period for describing the propertiesof sound vibrations. The frequency v = 1fT is the recip­rocal 'of the period. The unit of frequency is the inversesecond (S-1). If the frequency of vibration is eq~al to100 S-1 this means that during one second a particle ofair performs 100 complete vibrations. Since in physics wemust deal rather often with frequencies which are manytimes greater than a hertz, the units kilohertz and megahertzare widely applied; 1 kHz = 103 Hz, 1 MHz = 106

H~.The speed of a vibrating particle is maximum when It

is passing through its equilibrium position. On the con­trary, in positions of maximum displacement, the speedof a particle is, naturally, equal to ze~o. ~e ha~e alreadysaid that if the displacement of a particle IS subject to thelaw of harmonic oscillation, the change in its speed ofvibration obeys the same law. If we denote the maximumvalue (amplitude) of the displacement by so, and of thespeed by vo, then Vo = 2,nso/T.. or v~ = ~nvso' .Loud con­versation brings air partIcles mto VIbratIOn WIth an am­plitude of only several millionths of a centimetre. Themaximum value of the speed will be of the order of0.02 cm/s.

Another important physical quantity v~ryi~g togetherwith the displacement and speed of a particle IS the excesspressure, also called the sound press~re. A sound vi?rationof air consists in a periodic alternatIOn of compreSSIOn andrarefaction at each point in the medium. The air pressureat any place is now higher, now lower than the pressurewhich would be there in the absence of sound. ThIS excess(or insufficiency) of pressure is just what is called the

sound pressure. Sound pressure is a small fraction ofstandard atmospheric pressure. For a loud conversationthe sound pressure amplitude will be equal to approxi­mately one-millionth of the atmospheric pressure. Soundpressure is directly proportional to the speed of the vibra­tion of a particle, where the ratio of these physical quanti­ties depends only on the properties of the medium. Forexample, a speed of vibration of 0.025 cm/s correspondsto a sound pressure in air of 1 dyn/cm2•

A string vibrating in accordance with a sinusoidal lawbrings air particles into harmonic oscillation. Noises andmusical sounds lead to a considerably more complicatedpicture. A graph of sound vibrations, namely of the soundpressure as a function of time, is shown in Figure 6.9.This curve bears little resemblance to a sine wave. Itturns out, however, that any arbitrarily complicatedvibration can be represented as the result of superposinga large number of sine waves with different amplitudesand frequencies. These simple vibrations make up, as issaid, the spectrum of the complex vibration. Such a su­perposition of vibrations is shown for a simple examplein Figure 6.10. .

189

If sound were propagated instantaneously, .all the airparticles would vibrate in unison. But sound ~s no~ prop­agated instantaneously, and the m~sses of ~lr lpng onthe lines of propagation are brought mto motIo~ m turn,as if caught up by a wave coming from some source. I~

exactly the same way, a chip lies calmly on the water untilthe circular waves from a pebble thrown into the watercatch it up and make it vibrate. . . .

Let us confine our attention to a smgle vibratmg par­ticle and compare its behaviour with the motion ~f otherparticles lying on t~e same li~e of. sound propagatIOn. Anadjacent particle wIll start vibratmg s?mewhat .later, ~he

next particle still later. The del~y WIll. keep I~creasmg

until we meet a particle which IS ~agg~ng .behm~ by awhole period and is therefore vibratmg m tIme WIth theinitial particle. So an unsuccessful runner, who has fal~en

behind the leader by an entire lap, can cross the fimsh

Figure 6.10

6. Molecular Mechanics188Molecules

Figure 6.9

line simultaneously with the leader. But at what distancewill we meet the point which is vibrating in time withthe initial particle? I t is not hard to see that this distanceIv is equal to the product of the speed of propagation ofsound c by the period of vibration T. The distance Ivis called the wavelength:

Iv = cT

We will find points vibrating in time after intervals oflength Iv. Points separated by a distance of 1./2 will moverelative to each other like an object vibrating perpendic­ularly to a mirror moves with respect to its image.

If we depict the displacement (or speed, or sound pres­sure) of all the points lying on a:line of propagation of aharmonic sound, then a sine wave is again obtained.

The graphs of the wave motion and the vibration shouldnot be confused. There is a strong resemblance betweenFigures 6.11 and 6.12, but the distance is plotted alongthe horizontal axis in the former, while the time in thelatter. One figure represents the development of the vibra­tion in time, and the other an instantaneous "photograph"of the wave. It is evident from the comparison of these

1916. Molecular Mechanics

two graphs that the wavelength may also be c~lled itsspatial period: the quantity Iv plays the same role III spaceas T plays in time.

In a graph of a sound wave, the displacements of aparticle are plotted along the vertical axis, an.d the hor­izontal axis is the direction of the propagatlOn of thewave along which the distance is marked off. This mightsuggest the false idea that the particles are disJ?laced per­pendicularly to the direction of the propagatlOn of thewave. In reality, air particles always vibrate along thedirection of the propagation of the sound. Such a wave iscalled longitudinal.

The speed of light is incomparably greater than thatof sound; light is propagated practically instantaneously.Thunder and lightning occur at the same instant; we seea flash of lightning practically at the instant of discharge,but the sound of thunder reaches us at a speed of approx­imately one kilometre in three seconds (the speed ofsound in air is 330 m/s). Hence, when we hear the thunder,there is no longer any danger in being struck by that par­ticular bolt of lightning.

Knowing the speed of the propagation of sound, we canusually determine how far away a thunderstorm is raging.If 12 seconds have passed from the moment of the flashof lightning to that of the peal of thunder, the storm istherefore 4 kilometres away.

The speed of sound in gases is approximately equal tothe average speed of the motion of their molecules. Itis also independent of the density of a gas and prop~rti~n­

al to the square root of its absolute temperature. LIqUIdspropagate sound faster than gase~. Sound. is propagated ~n

water with a speed of 1450 mis, I.e. 4.5 times as fast as III

air. The speed of sound in solids is even greater; for exam­ple, it is about 6000 mls in iron.

When sound passes from one medium to another, thespeed of its propagation changes. But another interesting

190

a

Figure 6.12

Molecules

a

Figure 6.t1

phenomenon also occurs simultaneously-the partialreflection of sound from the boundary between the twomedia. The fraction of sound reflected depends mainly onthe ratio of the densities. In the case when sound passingthrough air is incident upon a solid or liquid surface, orvice versa, from dense media into air, the sound is almostcompletely reflected. When sound passes from air to wa­ter or, conversely, from water to air, only one-thousandthof the sound passes into the latter medium. If both mediaare dense, the ratio between the transmitted and the reflect­ed sound can be small. For example, 13%.of the sound willpass from water into steel or from steel into water, and87 % will be reflected.

The reflection of sound is widely applied in navigation.fhe construction of an instrument for measuring depths,the sonic depth finder, is based on it. A source of sound isplaced under water on one side of a ship (Figure 6.13).A discontinuous sound creates sound ray's, which willmake their way through the watery thickness to the bottomof the sea or river, be reflected from the bottom and re­turn in part to the ship, where sensitive instruments willcatch them. Accurate clocks will show how much timethe sound needs for this journey. The speed of sound inwater is known, and it is possible to obtain precise in­formation about the depth by means of a simple calcula­tion.

By aiming the sound forward or sideways instead ofdownwards, it is possible to determine with its aid whetheror not there are dangerous reefs or deeply submerged ice­bergs near the ship.

All the particles of air surrounding a vibrating bodyare in a state of vibration. As we explained in the firstbook, a mass point vibrating in accordanceLwith the si­nusoidallaw possesses a definite and constant total energy.

When a vibrating point passes through its equilibriumposition, its speed is waximum. Since the displacement

of the point is equal to zero at this instant, the entireenergy is kinetic:

193

mV~ax

2E=

Consequently, the total energy is proportional to thesquare of the amplitude of the speed of vibration.

This is also true for particles of air vibrating in a soundwave. However, a particle of air is something indefinite.Therefore, sound energy is given per unit of volume. Thismagnitude can be called the density of sound energy.

Since the mass of a unit of volume is the density p,

Figure 6.13

6. Molecular Mechanics192Molecules

13-0389

Above we have spoken about another important phys­ical quantity which vibrates according to the sinusoidallaw with the same frequency as the speed. This is thesound, or excess, pressure. Since these quantities areproportional, we can say that the energy density is pro­portional to the square of the amplitude of sound pressure.

The maximum value of the speed of sound vibrationfor loud conversation is equal to 0.02 cm/s. One cubic cen­timetre of air weighs about 0.001 g. Therefore, the energydensity is equal to

~ X 10-3 X 0.022 erg/cm3 = 2 X 10-7 erg/cm3

Suppose that a source is emitting sound. It is radiatingsound energy in t~e surrounding air. It is as though theenergy were "flowing" from the vibrating body. Duringa second, a definite amount of energy flows through eacharea element, which is perpendicular to the direction ofthe propagation of the sound. This quantity is called theenergy flux passing through the area element. Furthermore,if we take an area element of 1 cm2, the amount of energywhich has passed through it is called the intensity of thesound wave.

It is not difficult to see that the intensity I of sound isequal to the product of the energy density w by the speedof sound e. Imagine a small cylinder of l-cm height andl-cm2 base area whose generatrices are parallel to thedirection of propagation of a sound. The energy w con­tained in such a cylinder will completely leave it duringa time of lie. Therefore, an energy of w/(1Ie), Le. we, willpass through a unit of area during a unit of time. It is

as though the energy- were moving with the speed ofsound.

During a loud conversation, the intensity of the soundnear the talkers will be approximately equal to (we aremaking use of the number obtained above)

2 X 10-7 X 3 X 104 = 0.006 erg/cm2 .s

195

Audible and Inaudible Pitches

But what sounds can be perceived by the human ear?It turns out that the ear is capable of perceiving only thevibrations lying within the interval from approximately20 to 20 000 Hz. Sounds with a arge frequency are calledhigh-pitched, and those with a small frequency low­pitched.

But what wavelengths correspond to the limiting au­dible frequencies? Since the speed of sound is approximate­ly equal to 300 mis, using the formula ').. = eT = el'v,we find that the lengths of audible pitches correspondingto wavelengths that lie within the limits of 15 m for thelowest tones to 3 cm for the highest ones.

And how do we "hear" these vibrations?The way the organ of hearing functions has not as yet

been fully clarified. The crux of the matter is that theinternal ear (the cochlea, a canal of few centimetres longand filled with fluid) contains several thousand sensorynerves capable of perceiving sound vibrations transmittedto the cochlea from the air through the tympanic mem­brane. This or that part of the cochlea vibrates more stronglydepending on the frequency of sound. Although the sen­sory nerves are situated along the cochlea so closely thata large number of them are stimulated simultaneously,man (and animals) is capable, particularly in childhood,of distinguishing minute changes in frequency (thousandsof a fraction). It is still not known precisely just how this

6. Molecular MechanIcsHl4Molecules

the density of sound energy

PlJ~axw=-2-

13*

occurs. The only obvious fact is that the analysis by thebrain of the stimuli arriving from many different nervesis of utmost importance here. A mechanical model of thesame design as the human ear that could be capable ofdiscerning sound frequencies just as well has yet to beinvented.

A sound frequency of 20 000 Hz is the value beyondwhich the human ear does ;not perceive the mechanicalvibrations of a medium. It is possible to create vibra­tions of a higher frequency in various ways; a person willnot hear them, but instruments can record them. Inciden­tally, not only instruments record such vibrations. Manyanimals-bats, bees, whales and dolphins (as can be seen,it isn't a question of the dimensions of a living being)"­are capable of perceiving mechanical vibrations withfrequencies up to 100 000 Hz.

We are now able to obtain vibrations with frequenciesup to a thousand million hertz. Such vibrations, althoughof inaudible pitch, are called ultrasonic or supersonic inorder to affirm their affinity to sound. Supersounds of thehighest frequencies are obtained with the aid of quartzplates. Such plates are cut out of monocrystals of quartz.

Molecules 19& 7. Transformations

of Molecules

Chemical Readions

Physics is the foundation of all natural scie~ces. Itis therefore absolutely impossible to separate phYSICS fromchemistry, geology, meteorology, biology and all the otherbranches of science. The basic laws of nature come underthe topic of physics. It is no .mere cha,nce t~en t~at bookshave been written on geologIcal phySICS, bIOlogIcal phys­ics, chemical physics, construction physics, etc. He~ce,we find it appropriate to devote a section to chemICalreactions in this book which deals with the basic laws ofnature.

Strictly speaking, chemistry begins where a mol~culeis broken down into its component parts, or when a smglemolecule is formed of two others, or when two collidingmolecules are converted into two other molecules. Ifwe find that from the beginning to the end of a phenome­non a change has occurred in the chemical composition ofthe bodies involved, then a reaction has taken place.

Chemical reactions can oecur "on their own", Le. dueto the motions of the molecules at the given temperature.Hence, we often say that a substance decomposes: Thismeans that the internal vibrations of the atoms m themolecules rupture the bonds between the atoms and themolecule "falls apart".

A chemical reaction is most often the result of encoun-ters between molecules. A metal rusts. This is a chemicalreaction: an atom of the metal encounters a molecule of

Molecules

combustion and explosion, which we are to discuss inthe next section.

Next we shall consider the reaction rate from the molec-ular point of view, Le. at the molecular level. Just abouteveryone knows that some reactions are completed, fasterthan you can blink an eye (say, explosions), while otherstake years. Assume that we are again concerned withreactions in which two molecules collide and form twoother molecules. The following assumption is evidentlytrue. Of significance, in the first place, is collision energyof an amount sufficient to break up the molecules and torearrange them. Also important, in the second place, iswhether the reaction proceeds regardless of the angle atwhich the molecules meet in collision or whether theymust meet at certain angles.

The minimum energy required to implement a rear-tion is called the activation energy. I t plays the chief rolein the course of the reaction, but we must give its due toa second factor as well-the percentage of "lucky" colli­sions of particles having a given energy.

A heat-producing chemical reaction can be modelledas shown in Figure 7.1. The ball rolls uphill, over thebarrier and drops down. Since the initial energy levelis higher than the final one, more energy is produced thanhas been expended.

This mechanical model visually illustrates the criticaldependence of the reaction rate on the temperature. Ifthe temperature is low, the "speed" of the ball is insuffi­cient to surmount the barrier. As the temperature increas­es, the number of balls leaping over the barrier increasesfaster and faster. The rate of chemical reactions dependsto an exceptionally great extent on the temperature. Asa rule, a temperature increase of 10 degrees speeds upa reaction from two- to four-fold. If the reaction rate is,say, tripled when we raise the temperature 10 degrees,then a temperature increase of 100 degrees leads to a rate

198

water, and an oxide is form d A . h .. .a spoon of soda are m' d.e . pmc of CItrIC aCId andof gas begin to evo] IX~ mto a glass of water. Bubblestwo kinds of molec~;e;~~~ousl)'t ~h~ encounter of thesenew substances . I' resu e m the formation ofrise to the suria~~ca~~mbg catrbon dioxide whose bubbles

Hurs.

ence, spontaneous d' . .collision of molecules are ;smtegratIOn of ~olecules and

But chemical reactions : ca~se~of chemIcal reactions.well. You are vexed h ay e ~e to other causes asthat you took al w en you examme the suit of clothesThe cloth has fa~~~ ~~l~ur v?cation trip to the south.intense action of the S ,ost Its true colour. Owing totion has occurred in t~n: rays, a chemical transforma-

R.. eye.

eactIOns mduced by li ht .Investigations in this fiefd are saId to be ~hotochemical.conducted to distinguish he~~.stdbe ~specIa~IY carefullyheat generated by light (wh' h ~n uce reactIOns, due togy of motion of I I IC mcreases the kinetic ener-

mo ecu es so that th' II' .more frequent and mo' ell' co ISIOns areinduced by light I ;~ v~olent), froI? reactions directlytons-"rupture" thench e . atltebr, particles of light-pho-Th' emICa onds

ical :e:~::~::So:h~rht is respon~ible for the chain of chem­is called photosynth~~fse t~ce 1: green p~ants, and whichtion occurring in I' .' e p otochemical transforma­cycle without whic~v~~g pla~~ brea.lizes t~e great carbon

Chemical bonds ma1 ebwou e I~possI~le.reactions by other er:er \:upture.d In varIOus chemicalelectrons, protons etc ge IC partIcles as well, such as

Chemical react' ' .tion of heat or w~~~si~ay )ro~eed either with the absorp-at the molecular level? e~otutlO~ What does this signifyform two fast ones th~n h wo s ow molecules meet andwe know that an i~creas .eat has been generated becausean acceleration of the ;o~~ telmpesratuhre is e~uiv~lent to

eu es, uc reactlOns lllclude

7. Transformations of Molecules199

ignition point. For example, theo ignition ,Point of woodis 610°C, of benzine about 200 C, of whIte phosphorus50°C.

The combustion of wood, coal or oil is a chemical reac­tion uniting the substance with the oxygen of the ai:.-Therefore, such a reaction proceeds at the surface: un~Ilthe outside layer burns up, the next layer cannot partIc­ipate in the combustion. This is. what e.xplains. the rela­tive slowness of such combustIOn. It IS not dIfficult toconvince oneself in practice of the validity of what wehave said. If fuel is broken up into small pieces, the rateof combustion may be considerably increased. This iswhy the crushing of coal is carried out in furnaces.

Fuel fed to the cylinder of an internal-combustion en­gine is broken down and mixed with air in exactly thesame way. More complex substances than coal are usedfor engine fuel, for instance, gasoline. A molecule of s'!-cha substance is shown in Figure 7.2' at the left. It conSIsts

201

Figure 7.2

7. Transformations of Molecules

--tACTIVATIONi ENER<TY

increase of 310 ~ 60000 times, 200 degrees to 320 ~

~ 4 X 109, and 500 degrees to 350, i.e. approximately

1024 times. No wonder then that a reaction proceedingat a normal rate at 500°C does not take place at allat room temperature.

Molecules

Figure 7.f

Combustion and Explosion

For combustion to begin, it is necessary, as is wellknown, to bring a burning match to an inflammable object.But neither does a match light up by itself; it must bestruck against a match-box. Therefore, in order that achemical reaction start, a preliminary heating is necessary.The ignition creates the required temperature for thereaction at the initial moment. A high temperature isfurther maintained by the heat which is liberated in thecourse of the reaction.

The initial local heating should be sufficient for theheat liberated by the reaction to exceed that transferredto the cold surrounding medium. Therefore, each reac­tion has its own, as one says, ignition point. Combustiononly begins if the initial temperature is higher than the

of 8 carbon atoms and 18 hydrogen atoms bonded togetheras shown. As it burns, this molecule is subject to colli­sions from oxygen molecules. These collisions break upthe gasoline molecules. The forces that join one or twocarbon atoms to a hydrogen atom in a molecule, and thosethat join two oxygen atoms to make up an oxygen mole­cule, cannot withstand the stronger "affinity", as chemistssay, between oxygen atoms, on the one hand, and carbonand hydrogen atoms, on the other. Consequently, the oldbonds between the atoms in a molecule are ruptured andthe atoms are rearranged into new molecules. As is indi­cated in Figure 7.2 at the right, the new molecules-com­bustion products-consist of carbon dioxide and water.Water is produced in the form of steam.

The situation is entirely different in the case wherethe atmosphere is not needed, where everything that isnecessary for the reaction is contained within the sub­stance. An example of such a substance is a mixture ofhydrogen and oxygen (it is called a detonating gas). Thereaction occurs not at the surface but within the substance.Unlike the case of combustion, all the energy formingin the course of the reaction is liberated almost instanta­neously, as a result of which the pressure rises sharply andan explosion takes place. A detonating gas doesn't burn­it explodes.

Thus, an explosive should contain within it the atomsor molecules needed for a reaction. It is evident that wecan prepare explosive gas mixtures. There also existsolid explosives. They are explosive precisely because intheir compositions all atoms occur which are necessarvfor chemical reactions giving off heat and light. .

The chemical reaction taking place during an explosionis a decomposition reaction of the splitting up of mole­cules into parts. An example of an explosive reaction isshown in Figure 7.3-the splitting up of a nitroglycerinemolecule into its parts. As is evident from the right-hand

Figure 7.3

part of the diagram, molecules of ca~b~m dioxide, waterand nitrogen are formed out of the oflglllal molecule. Wefind the ordinary combustion products among the endproducts, but combustion has occurred wit~out the par­ticipation of oxygen molecules from the au; all atomsnecessary for the combustion are contained within thenitroglycerine molecule. .

How is an explosion propagated through an explosIve,for example, a detonating gas? When we set fire to ~n

explosive, a local heating arises. A reaction occurs IIIthe heated region. But heat is liberated in the course ofthe reaction, and passes into the adjacent layers of themixture by means of heat transfer. This heat is sufficientfor the reaction to take place in the neighbouring layer,too. New amounts of the heat being liberated enter thenext layers of the detonating gas, and so with the rate. ofheat transfer the reaction spreads throughout the entuesubstance. The rate of such a transfer is of the order of

2037. Transformations of Molecules202Molecules

20-30 m/s. This is, of course, very rapid. A metre-longtube of gas explodes within one-twentieth of a second, Le.almost instantaneously, while the rate of the combus­tion of wood or pieces of coal that takes place at the sur­face, and not within the volume, is measured in centime­tres per minute, Le. is several thousand times less.

Nevertheless, such an explosion may also be called slow,since a different kind of explosion, hundreds of timesfaster than the one described, is possible.

A shock wave gives rise to a fast explosion. If the pres­sure rises sharply in some layer of a substance, a shockwave will begin propagating from this place. A shockwave leads to a considerable jump in temperature whichis transmitted from layer to layer. The rise in tempera­ture gives an impetus to an explosive reaction, and theexplosion leads to a rise in pressure and maintains theshock wave, whose intensity would otherwise fall offrapidly with its propagation. Therefore, a shock wavegives rise to an explosion, while the explosion in turnmaintains the shock wave. The type of explosion de­scribed by us is called a detonation. Since a detonation ispropagated through a substance with the speed of a shockwave (of the order of 1 km/s), it is actually hundreds oftimes faster than a "slow" explosion.

But which substances explode "slowly" and whichexplode "rapidly"? The question should not be posedin this fashion: one and the same substance can explode"slowly" and also detonate under different conditions,while a "slow" explosion turns into a detonation in cer­tain cases.

Some substances, for example, nitrogen iodide, explodeon contact with a straw, on being slightly heated or asa result of a flash of light. An explosive substance liketrotyl does not explode when it is dropped or even shotout of a rifle. A powerful shock wave is required in orderto· explode it.

There exist substances even less sensitive to externalinfluences. The fertilizing mixture of ammonium nitrateand ammonium sulfate had not been considered to beexplosive until a tragic accident occurred in ~921 at achemical factory in Oppau, Germany. An explosIve meth­od was used there for pulverizing mixtures which had'become caked. As a result, a warehouse and the entirefactory were blown up. The engineers at the factory couldnot be blamed for the catastrophe: approximately twen­ty thousand blasts had proceeded normally, and only oncehad the conditions favourable for detonation been creat­ed.

Substances which explode only when subjected to ashock wave but exist in a stable state and are not evenafraid of fire under ordinary conditions are quite conve­nient for explosion technology. Such substances can bemanufactured and stored in large amounts. However, inorder to bring these inert explosives into action, initia­tors of the explosion are needed. Such initiating explo­sives are absolutely necessary as sources of shock waves.

Mercury fulminate can serve as an example of initia­tors. If a grain of it is placed on a sheet of tin and lighted,an explosion takes place, making a hole in the tin. Anexplosion of such a substance under any conditions is adetonation.

If a bit of mercury fulminate is placed on the charge ofa secondary explosive and lighted, the explosion of theinitiator creates a shock wave which is enough to deto­nate the secondary explosive. Explosions are produced inpractice with the aid of a detonating capsule (1-2 gof an initiator). The capsule can be ignited at a distance,for example, with the aid of a long cord (Bickford fuse);the shock wave coming from the capsule will blow upthe secondary explosive.

In a number of cases arising in technology, we mustcontend with detonations. "Slow explosions" of a mixture

Molecules 204 7. TransforMations of Molecules 205

Steam engines have become things of the past. A steamlocomotive is fast becoming a rarity to be exhibited inmuseums. This has occurred because the efficiency of aheat engine is much too low.

This does not imply that steam turbines have becomeobsolete. But there as well the cOIl-version of the energyof expanding steam into mechanical motion of the rotoris only an intermediate stage. The final stage is the pro­duction of electrical power.

As to aircraft and automobiles, there is obviously nopoint in powering them with a steam boiler or steam tur­bine. The total weight of the engine and heating deviceproves too great when calculated on a kilograms per horsepower basis.

We can, however, eliminate the separate heating plant.In a gas turbine, the immediate working fluid consistsof the incandescent combustion products of a high-ener­gy fuel. In such engines, we use chemical reactions, i.e.transformations of molecules, to produce mechanicalenergy. This is what determines both the significant ad­vantages of gas turbines over the steam-powered type,and the great design and manufacturing difficulties thathave to be overcome to ensure their reliable perfor­mance.

The advantages are obvious: the combustion chamber,where the fuel is burned, has a small size and can beplaced under the casing of the turbine. The components ofcombustion consisting, for example, of atomized kero­sene and oxygen have a temperature unattainable forsteam. The heat flow formed in the combustion chamberof a gas turbine is very intense, which makes it possibleto obtain a high efficiency.

But these advantages turn into shortcomings. The steelblades of the turbine function in streams of gas havinga temperature up to 1200 °C and inevitably saturatedwith microscopic ash particles. It is easy to imagine

Molecules 208

of gasoline and air occur in the motor of a car under ordi­nary conditions. However, sometimes a detonation alsoarises. Shock waves in a motor are absolutely unaccept­able as a systematic phenomenon, since th~will soon putthe walls of the cylinders of the motor out of commis­sion.

In order t? cope with detonations in engines, it is nec­essary to either use a special gasoline with a high oc­tane number or else mix the gasoline with special sub­stances, antiknocks, which do not let a shock wave de­velop. One of the most widespread antiknocks is tetra­et~yllead (TEL). This substance is highly toxic, and sodrIvers are warned of the need to be careful when usingsuch gasoline.

Cannons must be constructed so as to avoid detona­tions. Shock waves should not be formed inside the barrelwhen the gun is being fired, otherwise, the gun will beput out of commission.

Engines Operated by Transformations of Molecules

Peo~le living in the 20th century use a great varietyof engmes and motors to help them do their work andto increase their power about ten-fold.

In the simplest case, it may prove expedient to convertmechanical energy into another kind of mechanical ener­gy. For instance, we may have the wind or a stream ofwater rotate the vanes of a windmill or the water wheelof a mill.. In a hydroelectric power station, the process of convert­mg the energy of a stream of water into rotary motionof the turbine runner is an intermediate one. The tur­bine drives an electrical machine, called a generator whichproduces electricity. Such conversion of energy ~ill bediscussed further on.

7. Transformations of Molecules 207

what great demands have to be made of the materialsout of which gas turbines are manufactured.

During an attempt to construct a gas turbine with apower of about 200 hp for an automobile, it became nec~

essary to deal with quite a peculiar difficulty: the tur­bine was of such small dimensions that the usual engi­neering solutions and the customary materials simplycould not be used. However, the technological difficul­ties are already being overcome. The first automobilespowered by gas turbines have been designed and built,but it is difficult to foresee their future.

It turned out easier to utilize the gas turbine in rail­way transport. Locomotives with gas turbines, gas-tur­bine locomotives, have already received universal rec­ognition.

But entirely different engines in which the gas turbineis a subordinate although necessary component part pavedthe way for the widespread use of the gas turbine.We are speaking of the turbojet engine, the basic type ofengine at the present time in jet aviation.

The principle on which the jet engine is based is simple.A gas mixture is burned in a durable combustion cham­ber; the combustion products, having an extraordinarilygreat speed (3000 mis, when hydrogen is burned in oxy­gen, somewhat less for other types of fuel), are thrustthrough a smoothly expanding nozzle in the direction op­posite to that of motion. Even relatively small amountsof combustion products will carry a large momentumaway from the engine.

With the creation of jet engines, people received therealistic opportunity of carrying out interplanetaryflights.

Liquid propellant rocket engines have become verywidespread. Definite portions of a fuel (for example,ethyl alcohol) and an oxidizer (usually liquid oxygen)are injected into the combustion chamber of such an

Molecules 208 7. Transformations of Molecules 209

engine. The mixture burns, creating traction. In high­altitude rockets, such as the V-2, the traction is of theorder of 15 ton£. The figures are sufficiently eloquent:8.5 tons of fuel and oxidizer which take 1.5 minutes toburn up are poured into the rocket. The use of liquid pro­pellant rocket engines is only expedient for flights atgreat heights or beyond the Earth's atmosphere. It makesno sense to pour large amounts of a special oxidizer intoan aircraft destined for flights in the lower layers of theatmosphere (up to 20 km), where there is enough oxygen.But then the problem arises of forcing huge amounts ofair necessary for intensive burning into the combustionchamber. This problem is solved in a natural fashion:part of the energy of the gas stream created in the com­bustion chamber is taken away to rotate the powerfulcompressor forcing air into the chamber.

We have already said what engine can do work at theexpense of the energy of a stream of incandescent gases;of course, this is the gas turbine. The whole system iscalled a turbojet engine (Figure 7.4). A turbojet engine hasno competitodor flights with speeds from 800 to 1200 km/h.

For long-distance flights with speeds of 600-800 km/han ordinary aircraft propeller is added to the shaft of aturbojet engine. This is a turboprop engine.

At flying speeds of about 2000 km/h or more, the airpressure developed by the aircraft is so great that theneed for a compressor no longer arises. I t is then only nat­ural that neither is a gas turbine needed. The engineis converted into a pipe of variable cross-section, inwhich fuel is burned at a very definite place. This is aramjet engine. It is clear that a ramjet engine cannot liftan aircraft off the ground: it becomes capable of function­ing only at very high flying speeds.

Jet engines are completely inexpedient for flights atsmall speeds owing to the large expenditures of fuel.

For motion on land, on water or in air with speeds from

14-0389

Molecules 210 7. Transformations of Molecules

Figure 7.5

211

Figure 7.4

o to 500 kill/h, people are reliably served by gasoline ordiesel intelnal-combustion piston engines. As indicatedby the name, the main part of such an engine is the cyl­inder inside which the piston can move. The back-and­forth motion of the piston is transformed into a rotarymotion of the shaft with the aid of the connecting rod andcrank system (Figure 7.5).

The motion of the piston is transmitted through the con­necting rod to the crank, which is part of the crankshaft.The motion of the crank sets the shaft into rotation. Con­versely, if the crankshaft is turned, this will move theconnecting rods and a displacement of the pistons insidethe cylinders will set in.

The cylinder of a gasoline engine is equipped with twovalves, one of which is meant for the inlet of the fuel mix­ture, and the other for the exhaust of waste gases. Inorder to start the engine, we must turn it over, using

the energy of some outside source. Assume that at a cer­tain moment the piston is moving down and the inletvalve is open. A mixture of atomized gasoline and air issucked into the cylinder. The inlet valve is connected tothe shaft of the engine in such a way that it closes at themoment when the piston reaches its lowest position. Asthe shaft continues to turn, the piston starts coming up.The automatic drive of the valves keeps then closed dur­ing this stroke, and the fuel mixture is therefore compres­sed. When the piston reaches its highest position, thecompressed mixture is ignited by an electric spark, whichjumps between the electrodes of the spark plug. Themixture ignites and the expanding combustion productsact, pushing the piston down. The engine shaft receives apowerful thrust, and the flywheel on the shaft storesc~)ll~iderable kinetic energy. All of the next three pre­lImmary strokes proceed at the expense of this energy:first the exhaust, when the exhaust valve is open and thepiston is moving up, driving the exhaust out of the cyl-14*

inder, next the intake and compression that we alreadyknow about, then a new ignition. The engine is started.

Gasoline engines have powers from fractions of a horse­power to 4000 hp, an efficiency of up to 40% and weighup to 300 gf per horsepower. Their widespread utiliza­tion in automobiles and aircrafts is explained by such agood showing.

How might we increase the efficiency of a gasolineengine·? The major means is by raising the degree of com­pression.

If the fuel mixture is more highly compressed beforeignition, it will have a higher temperature. And why isit of advantage to raise its temperature? The point is thatwe can rigorously prove that the maximum attainable ef­ficiency equals i-TITo, where T is the temperature ofthe working body, and To is the ambient temperature.This proof is tedious and uninteresting, and we omit it.In numerous instances we have asked our readers to justtake our word for various statements. Since the ambienttemperature, Le. the temperature of the surrounding me­dium, is something we cannot do much about, we makeevery effort to raise the temperature of the working bodyas high as is feasible. But-there, unfortunately, is a but­a highly compressed fuel mixture detonates (see p.204). Then the working stroke acquires the features ofa heavy explosion which may damage the engine.

One has to take special measures to decrease the deto­nation properties of the gasoline, and this would greatlyraise the cost of a fuel that is already quite expen­sive.

The problems of raising the temperature during theworking stroke, of eliminating detonation and reducingthe cost of the fuel have been successfully solved in thediesel engine.

A diesel engine resembles a gasoline engine to a greatextent in its construction, but is designed for products

of oil distillation which are cheaper than and inferiorin quality to gasoline. . . . .

The cycle begins with the admIsSIOn of pure ~Ir mtothe cylinder. The air is then compressed .by the PISto~ toapproximately 20 atm. It would be .very dIfficult to a?hIeVesuch' a high compression by turnmg over t~e engme ?yhand. Therefore, diesel engines are started wIth a slJeClalstarting·motor, usually a gasoline one, or by eompressed

air. f h ..When highly compressed, the temperature 0 t e aIr m

a cylinder rises so much that it beco~es suffieie~t for. thefuel mixture to ignite. But how can It be admItted mtothe cylinder, where a high pressure has been. attained?An inlet valve would not be suitable here. It IS re-placedby a sprayer which forces the fuel into the cyl.in~er ~hrougha tiny opening. It ignites as it enters, ehmI~atmg .thedanger of a detonation considerable for a gasolme engme.

Eliminating the danger, of a detonation permits us toconstruct slow diesels with many thousands of horsepow­ers. Naturally, they acquire quite considerable dim.en­sions but remain more compact than an aggregate conSIst­ing ~f a steam boiler and a turbine.

A ship in which there are .direct-current ge~erator andmotor between a diesel engme and a blade IS called adiesel-electric motor 'ship.

Diesel locomotives, now being widely introdueed ~nrailroads are built along the same lines; they may ma sense 'be called diesel-electric locomotives.

Internal-combustion piston engines, considered lastby us, have borrowed their basic const.ructive elements­the cylinder, the piston, the connectmg r?d and crankmechanism which helps obtain rotary motIon-from thesteam engine now gradually leaving the scene. The steamengine could' have been called a~ "exter~al-c.ombustionpiston engine". It is precisely thIS combm~tlOn of theunwieldy steam boiler with the no less unWIeldy system

Molecules 212 7. Transformations of Molecules 213

of transforming translatory motion into rotary one thatdeprives the steam engine of the possibility of success­fully competing with more modern engines.

Modern steam engines have an efficiency of about 10%.The locomotives which have been taken out of productionreleased up to 95 % of the fuel they burned through thesmoke-stack to no !advantage.

This "record-breaking" low efficiency is explainedby the inevitable deterioration in the properties of a steamboiler designed for installation on locomotive as com­pared to a stationary steam boiler.

But why were steam engines employed so widely intransportation for such a long time? Besides adherence tocustomary solutions, the circumstance that a steam en­gine has a very good tractive characteristic also playeda role: the greater the force with which the loadresists a displacement of the piston, the greater theforce exerted on it by the steam, Le. the torque developedby a steam engine grows under difficult conditions, whichis important in transportation. But, of course, the factthat the steam engine does not need a complicaterlsystemof variable transmission to the driving axles cannot inthe least compensate for its basic defect, a low efficiency.This is what explains the supplanting of the steam engineby other engines.

Molecules 214 8. Laws of Thermodynamics

Conservation of Energyat the Molecular Level

The laws of thermodynamics belong to the basic lawsof nature. There are not many such laws. You can countthem on the fingers of one hand.

The chief aim of science, including physics, of courseis the search for the rules, regularities, general laws andthe basic laws that govern nature. This search beginswith observations or experiments. That is why we saythat all of our knowledge is of an empirical or experimen­tal nature. Investigations and observations are followedby a search for generalization. Persistent brain work,meditation, calculations and inspiration enable us tofind laws of nature. Next comes the third stage: from thegeneral laws, we derive, in a strictly logical manner, theensuing effects and special laws that can be checked .byexperiments. This, by. the way, is wh~t the expl anatlOnof a phenomenon conSIsts of. To explam means to corre­late the particular with the general law.

A cherished goal of science has been, of course, suc­cessful attempts to reduce laws to the minimum numberof postulates. Physicists tirelessly search for such oppor­tunities; they seek to express the whole sum of knowledgeabout nature in several concise and elegant formulas.Albert Einstein tried for about thirty years to unify thelaws of gravitational and electroma;gnetic fields..Thefuture will show whether or not thIS proves pOSSIble.

What, then} are these laws of thermodynamics? Too

brief a difinition must necessarily be somewhat inac­curate. Maybe we approach the essence of the matter clos­est of all if we contend that thermodynamics is a study ofthe rules according to which bodies flxchmge energy.The laws of thermodynamics enable us to find strictlylogical relations by mathematical means between thethermal and mechanical properties of bodies, and to es­tahlish a number of vital principles concerned with thechange of state of bodies. Perhaps, the most precise def­inition of the branch of physics we are about to give ourattention to is the trivial statement: thermodynamics isthe totality of knowledge that follows from the first andsecond laws of thermodynamics.

The first law of thermodynamics was formulated in aconcise and expressive form back in the time when phys­icists preferred not to mention molecules. Such a formu­lation (which does not require us to "get inside" a body)is said to be phenomenological, Le. one simply referringto the phenomena. The first law of thermodynamics con­stitutes a certain refinement and expansion of the energyconservation law.

We have established that a body has kinetic and po­tential energies, and that in a closed system, the sum ofthese energies-the total energy-can neither be de­stroyed nor created. Energy is conserved.

Except with respect to the motion of celestial bodies,we can contend without exaggeration that there are nophenomena in which mechanical motion is not accompa­nied with heating or cooling surrounding bodies. When abody is stopped by friction, its kinetic energy would seem,on the face of it, to be lost. This first impression is mis­leading, however. Actually, it can be proved that con­servation is maintained with absolute precision: the me­chanical energy of the body has been utilized to heat thesurrounding medium. What does this mean at the molec­ular level? Simply that the kinetic energy of the body has

been transformed into kinetic energy of molecules.This seems clear enough, but what happens when we

crush ice in a mortar with a pestle? We keep pounding andthe thermometer reading remains at zero all the time.It would seem that the mechanical energy we exert hasvanished. If not, what has happened to it? Again, wefind a clear answer: the ice has been converted into water.This means that the mechanical energy was expended inrupturing the bonds between the molec~les, and theirinternal energy has been changed. Each time we observethat the mechanical energy of a body has disappeared, wereadily discover that this only seems to be so, and thatactually mechanical energy has been converted into theinternal energy of the bodies being observed.

In a closed system, certain bodies can lose and otherscan gain internal energy. But the sum of the internal ener­gies of all the bodies added to their mechanical energiesremains constant for the given system.

Now let us turn our attention away from mechanicalenergy. We consider two instants in time. At the firstinstant, the bodies were at rest, then certain events oc­curred, and at the second instant the bodies are againat rest. We are certain that the internal energy of all thebodies in the system remained constant. But some bodieshave lost and others have acquired energy. This could hap­pen in two ways. Either one body performed mechanicalwork on another (for example, by compressing orstretching it), or one body transferred heat to anotherbody.

The first law of thermodynamics states that the changein the internal energy of a body is equal to the sum ofthe work done on the body plus the heat transferred toit.

Heat and work are two different forms in which energycan be transferred from one body to another. Heat ex­change is accomplished by disordered collisions of mole-

Molecules 216 B. Laws of Thermodynamics 217

cules. Mechanical energy is transmitted with the mole­cules of one body aligned in orderly "rows and ranks"transferring their energy to another body.

How Heat Is Converted into Work

The word "heat" in this heading has been employedsomewhat carelessly. As mentioned immediately aboveheat is a form of energy transfer. Consequently, ~more proper formulation of the question would be:How to convert thermal energy, Le. kinetic energy ofmolecular motion, into work. But the word "heat" iscustomary, concise and expressive. We hope the reader willnot be confused if we continue to apply the word in thesense we have just accurately defined.

There is certainly plenty of heat around us. But, un­fortunately, all of this energy of molecular motion is ab­solutely useless insofar as it cannot be converted intowork. Such energy cannot be ranked in any way among ourenergy resources. Let us look into this matter.

A pendulum which has been deflected from its equilib­rium position will sooner or later come to rest a wheel ofa~ upside-down bicycle, which has been spu'n by hand,wIll.make a lot.of turns, b~t it, too, will eventually stopmovmg. There IS no exceptIOn to the following importantlaw: all the spontaneously moving bodies surroundingus will eventually come to rest. *

If there are two bodies, one heated and the other cOldheat will be transferred from the former to the latter untdtheir temperatures are equalized. Then the heat trans­fer will cease and the states of the bodies will stopchanging. Thermal equilibrium will set in.

There is no phenomenon whereby a body spontaneous­ly leaves a state of equilibrium. A wheel on an axle at

*He!e, of cours~, we do not have in mind a uniform translatorymotion or a umform rotation of a system of bodies as a whole.

rest never starts turning by itself. Neither does it happenthat an ink-well on a table warms up by itself.

The tendency towards equilibrium implies that eventstake a natural course: heat passes from a hot body to acold one, but cannot spontaneously pass from a cold bo­dy to a hot one.

As a result of air resistance and friction at the suspen­sion, the mechanical energy of an oscillating pendulumwill be converted into heat. However, not under any con­ditions will a pendulum begin swinging at the expense ofthe heat of the surrounding medium. Bodies come to astate of equilibrium, but cannot leave it spontaneously.

This law of nature shows at once what part of the energysurrounding us is absolutely useless. This is the energy ofthe thermal motion of the molecules of those bodies whichare in a state of equilibrium. Such bodies are incapableof converting their energy into mechanical motion.

This part of the energy is immense. Let us calculatethe value of this "inaccessible" energy. If the tempera­ture is lowered by one degree, then a kilogram of earth,having a heat capacity of 0.2 kcal/kg, loses 0.2 kcal.A relatively small figure. However, let us estimate howmuch energy we would obtain if we were able to coolby only one degree a substance with the mass of theEarth, Le. 6 X 1024 kg. Multiplying we obtain an im­mense figure: 1.2 X 1024 kcal. In order that you mightpicture this value, let us state at once that at the presenttime, the yearly energy output of all the electric powerstations in the world is equal to 1015_1016 kcal, Le. abouta thousand million times less.

We needn't be astonished by the fact that such calcu­lations act hypnotically on poorly informed inventors.We have spoken above of attempts to construct a perpet­ual motion machine ("perpetuum mobile") creating workout of nothing. Operating with the principal proposi­tions of physics following from the law of conservation

Molecules 218 8. laws of Thermodynamics 219

of energy, one cannot possibly refute this law with thecreat~on of a perpetual motion machine (we shall nowcall It a perpetual motion machine of the first kind).

The same sort of error is also committed by certaincleve.rer invent~rs, wh~ create models of machines per­for~mgmecham~almotion at the expense of nothing but bycoolmg the medmm. Such, alas, unrealizable machinesare called perpetual motion machines of the second kind.Here, t?O, a logical error is committed, since the inventorbases hImself on laws of physics, which are consequencesof the law by which all bodies tend towards a state ofequilibrium, and with the aid of these> laws, tries to refutethe foundations on which they are based:l

Thus, it is impossible to produce work by merelytaking heat from a medium. In other words, a system ofbodies in equilibrium with each other is energeticallvbarren. .

Hence, in order to obtain work, it is first of all neces­sary to find bodies which are not in equilibrium with theirneighbours. Only then will one succeed in realizing aprocess .of transfe.rring heat from one body to another orconvertmg heat mto mechanical energy.

The c~e~tion of an energy flux is a necessary conditionfor obtammg work. In the "path" of such a flux a conver­sion of some of the energy of bodies into work is possible.Therefore, the energy of only those bodies which are notin equilibrium with the surrounding medium is rankedamong the en~rgy reserves .which are of use to 'people.

.~he law ~hl.ch we have Just expounded, the impossi­blhty of bmldmg a perpetual motion machine of the sec­ond kind, is called the second law of thermodynamics.?o far we have ~xpressed it in the form of a phenomenolog­Ical rule. But smce we know that bodies consist of mole­~ules and that .the internal energy is the sum of the kinet­IC and potential energies of the molecules, the suddenappearance of some kind of "additional" law may seem

Molecules 220 8. Laws of Thermodynamics 221

strange. Why is the law of conservation of energy formu­lated for molecules insufficient to explain all phenomenain nature?

In short, this poses the following question: Why domolecules behave so that when left to themselves theytend towards a state of equilibrium?

Entropy

This question is very important and interesting. Inorder to answer it, we will have to begin from afar.

Frequently encountered events that occur at every turnare said to be highly probable. On the contrary, eventsthat occur thanks to a rare coincidence are regarded asimprobable.

An improbable event does not require a display of anysupernatural forces whatsoever. There is nothing impos­sible about it, nothing contradicting the laws of nature.And nevertheless, in many cases we are perfectly surethat the improbable is identical with the impossible.

Consider a lottery prize-list. Count the number of win­ning tickets whose numbers end with a 4, 5 or 6. You willnot be the least bit surprised when you find that approxi­mately one-tenth of the winning tickets correspond to eachdigit. Well, but perhaps tickets with numbers ending with a5 were to make up one-fifth of the winners, instead ofone-tenth? Unlikely, you say. Well, and if half of thewinning tickets were to have such numbers? No, thatwould be absolutely improbable ... , and therefore, alsoimpossible.

H.eflecting on what conditions are necessary for an eventto be probable, we arrive at the following conclusion:the probability of an event depends on the number ofways in which it can be realized. The greater this num­ber, the more frequently will such an event occur.

More precisely, the probability is the ratio of the num­ber of ways of realizing a given event to the number ofways of realizing all possible events.

Write down the numbers from 0 to 9 on ten cardboarddiscs and place them in a sack. Now pull out a disc notei~s number and pu~ it back in the sack. This is very 'muchhke a lottery drawmg. It can be confidently said that you~ill not draw one and the same number, say, seven timesm. a ro,w, even i~ you devote an entire evening tot~IS bormg occ,upatIOn, Why? The drawing of seven iden­tIcal numbers IS an event which is realizable in only tenways (7 zeros, 7 ones, 7 twos, etc.). But there are a totalof 107 possible ways of drawing seven discs. Thereforethe probability of drawing seven discs in a row withidentical numbers is equal to 10/107 = 10-6, Le. onlyone-millionth.

If black and white grains are poured into a box andmixed with a shovel, the grains will very soon be distri­buted uniformly throughout the entire box. Scooping upa handful of grains at random, we shall find approxi­mately the same number of white and black grains in it.No matter how much we mix them, the result will alwaysbe the same-uniformity is preserved. But why doesn'tthe separation of the grains take place? Why won't wesucceed in driving the black grains to the top and the~hite grains to the bottom by means of a prolonged mix­mg? Here; too,. it is enti~ely a matter of probability.:rhe state m whIC~ the ~rams are distributed disorderly,l.e. black and whIte grams are uniformly intermingledcan be realized in an enormous number of ways and, con~sequently, possesses the greatest probability. On thecontrary, the state in which all the white grains are on thetop and all the black grains at the bottom is unique.Therefore, the probability of its realization is negligiblysmall.

Molecules 222 8. Laws of ThermodynamicsWe can easily pass from grains in a sac~ to the mole­

cules that bodies are made of. The, behaVIOur o~ mole­cules is subject to chance. Thisean be seen partIcularlyclearly in the case of gases. As we know; gas ~ole~ulescollide randomly and move in all possIble d~rectIOns,first with one speed and then with another. ThIs eternalthermal motion continually reshuffles. th~ molecules,mixes them like a shovel mixes the grams I.n a ~ox.

The room in which you are now is filled WIth aIr .. Whycan't it happen at some moment that the molecules m thelower half of the room pass into the upper h~lf-:-under~heceiling? Such a process is not impossible-It IS very Im­probable But what does very improbable mean? Ifsuch a phenomenon were even a t~ou.san~ million times lessprobable than a disordered dlstnbutIOn of molecules,someone might nevertheless observe it. Will we reallyobserve such a phenomenon? 3

A computation shows that such an event for a 1-cmvessel in volume takes place once in 103x to" cases. Ithardly pays to make a dist,~~ction. be~~een the words"extremely improbable" and ImpossIble.,For th~ .nu~­ber written above is unimaginably great; If we dIvIde .Itby the number of atoms not ~nly ~n the,Earth b~t also mthe entire solar system, It WIll still remam enor-

mous.But what will the state of the gas molecules be~ The

most probable one. And the most probable state WIll bethat which is realizable in the greatest number of ways(the molecules will all be distributed at random) forwhich there are approximately the same numbers of mol­ecules moving to the right as to ,the ~eft, upwards asdownwards, for which one finds IdentICal n~mbers ofmolecules in equal volumes, the same proportIOn of fastand slow molecules in the upper an~ lower ~alves of thevessel. Any deviation from such a dls?rder, l.e. from t~erandom intermingling of molecules WIth respect to pOSI-

Molecules 224

tion and velocity, is linked with a decrease in probabilityor, more concisely, is an improbable event. '

On the contrary, phenomena linked with an intermin­gling, with the creation of disorder out of order increase theprobability of state. Hence, it is these phenomena thatwill determine the natural course of events. The law ofthe impossibility of a perpetual motion machine of thesecond kind and the law by which all bodies tend towardsan equi~ibrium ~tate receive their explanation. Why ismechamcal motion transformed into thermal? Simplyb?cause mechanical motion is ordered and thermal isdIsordered. The transition from order to disorder in­creases the probability of state.

Physicists called the quantity characterizing the degreeof order and related by a simple formula to the number ofways of creating states entropy. We shall not give the for­mula, but shall only say that the greater the probability,the greater the entropy.

The law of nature which we are now discussing asserts:all. ~atural proc~sses proceed in such a way that the prob­abIlity of state mcreases. In other words, this same lawof nature can be formulated as the law of increasing en­tropy.

The law of increasing entropy is one of the most im­port~nt la,:"s. ?f nature. From it follows, in particular,th? ImpossIbIlity of constructing a perpetual motion ma­chul;e of the second kind, or, which is the same, the as­sert~o.n ~hat bodies left to themselves tend to a state ofeqUllibrlUm. The law of increasing entropy is the sames~cond law of thermodynamics. Only its formulation isdIfferent, but its content is the same. What is more im­portant is that we have given an interpretation of thesecond law of thermodynamics at the molecular level

In a certain sense, the union of these two laws "unde;a sing~e banner" was quite fortunate. For the law of con­servatIOn of energy is an absolute law. But as for the law

8. Laws of Thermodynamics 225

of increasing ent~opy, as follows from what has been saidabove, it is applicable only to sufficiently large collectionsof particles and is simply impossible to formulate forindividual molecules.

The statistical (this means pertaining to a large col-lection of particles) character of the second law of ther­modynamics does not in the least diminish its signifi­cance. The law of increasing entropy predetermines the di­rection of processes. In this sense, entropy may be calledthe managing director of natural resources, while energyserves as its bookkeeper.

FluctuationsWe have seen that spontaneous processes bring a sys­

tem to its most probable state, i.e. to the growth in en­tropy. After the entropy of a system has become maximum,equilibrium is reached.

But a state of equilibrium does not by any means im-ply internal rest. An intensive thermal motion takesplace within the system. Therefore, strictly speaking,any physical body "stops being itself" at each instant:the mutual distribution of its molecules at every succes­sive moment is not the same as it was at the preceding one.Consequently, the values of all physical quantities areconserved only "on the average"; they are not exactlyequal to their most probable values, but vary aroundthem. Deviations from the most probable values at equi­librium are called fluctuations. The values of the variousfluctuations are extremely negligible. The greater thevalue of a fluctuation, the less probable it is.

The average value of a relative fluctuation, i.e. thefraction of the magnitude of the physical quantity ofinterest by which it can change as a result of the chaoticthermal motion of molecules, can be approximatelyrepresented by the expression 1IV N, where N is the num-

1/ 2 t5-0389

227Molecules

ber of molecules in the body, or in that part of it, whichwe are investigating. Therefore, fluctuations are appre­ciable for systems consisting of a small number of mole­cules, and completely insignificant for large bodies con­taining millions of millions of millions of molecules.

The formula 1/V N shows that in one cubic centimetreof a gas, the density, pressure, temperature and also anyother properties can change by 11V 3 X 1019 , i.e. by notmore than 10-8 %. Such fluctuations are too small to bedetected experimentally. However, things are entirelydifferent for a volume of a cubic micrometre. Here N == 3 X 107 and fluctuations will attain measurable val­ues of the order of hundredths of a per cent.

A fluctuation is an "abnormal" phenomenon in the sensethat it implies a transition from a more probable state toa less probable one. In the course of a fluctuation, heattransfers from a cold body to a hot one, the uniform dis­tribution of molecules is violated and an ordered motionsets in.

Will someone perhaps succeed in constructing a perpet­ual motion machine of the second kind on the basis ofthese violations?

Let us imagine, for example, a tiny turbine placed ina rarefied gas. Can't we arrange things in such a waythat this small machine would react to all fluctuations inan arbitrary but fixed direction? For example, so thatit would rotate if the number of molecules flying to theright became greater than the number of molecules mov­ing to the left? Such small tremors might be accumulated,eventually giving rise to work. The law asserting the im­possibility of a perpetual motion machine of the secondkind would be refuted.

But, alas, such a mechanism is impossible in princi­ple. A detailed examination, taking into account the factthat the turbine would have its own fluctuations (the

8. Laws 01 Thermodynamics

greater, the smaller its dimensions), shows that fluctua­tions can never perform any work wh~t.so~ver. Alt~oughviolations of the tendency towards eqUIlIbrIUm ~ontmual­ly arise around us, they cannot change t~e me~orablecourse of physical processes in the direction mcreasmg theprobability of state, i.e. entropy.

d . IWho Discovered the Laws of Thermo ynamlcs.

Here it is impossible to limit oursel~es to ~ singlename. The second law of thermodynamICs has ItS ownhistory. Here, too, just as in the history of the first la,:of thermodynamics, the name of the Frenchman SadlCarnot should be mentioned first of all. In 1~24, he pub­lished a work entitled Reflections on the Motwe Power ?fFire at his own expense. It was first demonstrated mthis work that heat cannot pass from a cold body to awarm one without the consumption of work. Carno~ als.oshowed that the maximum efficiency of a heat engme ISdetermined only by the difference in the temperatures ofthe heater and the cooling medium: ..

Only after Carnot's death in 1832 d~d other. phys~Clstspay attention to this work. However, It had lIttle mflu­ence on the further development of scien~~ because ~ll ofCarnot's writings were based on the recogmtIOn o~ an mde­structible and uncreatable "substance" -calOrIC.

Only after the work of Mayer, Joule and Helmholtz,who established the law of equivalence of heat ~nd work,did [the great German physicist Rudolf ClausIUS. (1822­1888) arrive at the second law of t~lerm.odynamics andformulate it mathematically. ClausIUS mtroduced theconcept of entropy and showed that the essence of thesecond law of thermodynamiCS can be reduced to theinevitable growth of entropy in al~ real pr~cesses.

The second law of thermodynamICs permIts one to for­mulate a number of general laws which all bodies should

15*

Rudolf Clausius 11822·18881 .. . . -an outstandmg German theoretical

IPhYSlcflst

h. ClausIUs was the first to accurately formulate the second

aw 0 t ermodynam' .' 1850 .. . . . ICS. In , m the form of the thesis of theIml~ossl~Il~ty of heat being spontaneously transmitted from aco er 0 y to a hotter one, and in 1865, with the aid of the

concept of entropy which he introduced. Clausius was one of thefirst to consider the questions of the heat capacity of polyatomicgases and the thermal conductivity of gases. Clausius' work in thekinetic theory of gases contributed to the development of statis­tical concepts of physical processes. A series of interesting in­vestigations into electrical and magnetic phenomena are due to

Clausius.

2298. Laws of Thermodynamics

obey, regardless of their structures. However, there stillremains the problem of finding the relationship betweenthe structure of a body and its properties. The branch ofphysics, which is called statistical physics, gives an answerto this question.

It is clear that in calculating physical quantities de-scribing a system consisting of millions of millions ofmillions of particles, a new approach is absolutely nec­essary. In fact, it would be pointless, not to say complete­ly impossible, to follow the motions of all the particlesand describe this motion with the aid of formulas of me­chanics. However, it is precisely this enormous number ofparticles which enables us to apply new "statistical"methods to the study of bodies. These methods widelyuse the concept of the probability of events. The founda­tions of statistical physics were laid down by the outstand­ing Austrian physicist Ludwig Boltzmann (1844-1906).In a series of papers, Boltzmann showed how the indicatedprogram can be carried out for gases.

The statistical interpretation of the second law of ther­modynamics given by Boltzmann in 1877 was the logi­cal culmination of these investigations. The formularelating the entropy and probability of state of a systemwas carved out on Boltzmann's tombstone.

It would be difficult to overestimate the scientificachievement of Boltzmann, who discovered completelynew paths in theoretical physics. Boltzmann's inves-

228Molecules

Molecules 230

tigations we~e subjected to ridicule during his lifetimeby conservatIve German professors: at that time at .a d 1 l' ' omlCn. mo ecu ar conceptlOns were regarded by many asnaIve. B~ltzm~nn committed suicide, and the role playedby the sltuatlOn just described was undoubtedly ffrom the least important. arT~e edifice of statistical physics was perfected to a

~onslderab.le. degree. by th~ work of the outstanding Amer­lCan P~YSICISt JosIah WIllard Gibbs (1839-1903). Gibbsgenera~lzed Boltzmann's methods and showed howone .mlg?t extend the statistical approach to all bodies.

GIbbs last paper appeared at the beginning of the2?th centur.y. A very modest researcher, Gibbs publishedhIS ~apers III the proceedings of a small provincial uni­V?rslty. A consi~erabl~ number of years had passed untilhIS r.e~arkable lllvestIgations were made known to allphYSICIStS.

Statistical physics has shown the way in which one cancalculate p~operties of bodies consisting of a given num­ber of particles. Of course, it should not be thought thatthese computatio~al methods are all-powerful. If thena.ture of the. motlOn of the atoms in a body is very com­p.hcated, as IS the ca~e for. liquids, the actual computa­tlOn becomes unfeasIble III practice.

9. Giant Molecules

Chains of Atoms

Chemists and technologists have been dealing for along time with natural substances consisting of longmolecules in which the atoms are bound like the links ofa chain. We find examples on every hand: abundantsubstances as rubber, cellulose and protein constitutechain-type molecules made up of many thousands ofatoms. The structural conceptions of such molecules wereformed and developed in the twenties, when chemistslearned to produce such substances in their laborato-

ries.One of the first steps in producing substances built of

long-chain molecules was the creation of synthetic caout­chouc. This brilliant research was completed in 1926by the Soviet chemist Sergei Vasilyevich Lebedev (1874­1934). The industrial production of synthetic caoutchouc(as crude rubber is called) was a vital problem in theUSSR as it was in great demand for the manufactureof automobile tires (rubber is made of caoutchouc, orcrude rubber) and because the rubber tree requires atropical climate for its cultivation.

The hevea, or simply rubber, tree flourishes in theBrazilian jungle and from it oozes latex, a milky liquidthat is a suspension of crude rubber. The BrazilianIndians made balls of crude rubber and also used it formaking footwear. But in 1839, Charles Goodyear (1800­1860) originated the process for vulcanizing crude rub-

,...

Molecules 2329. Giant Molecules 233

?er. Treating crude rubb~r with sulphur and subjectingI t to heat convert the sticky plastic crude rubber intoelastic rubber.

At first the demand for rubber was low, but today man­kind requires millions of tons annually. As we have al­l eady mentioned, rubber trees grow only in tropicalforests so that countries in cold or temperate zones mustproduce synthetic rubber in plants on an industrial scaleto avoid dependence on crude rubber imports.

To produce synthetic crude rubber, it is necessary, ofcourse, to know what crude rubber (caoutchouc) is.When Lebedev began his investigations, the chemicalformula of crude rubber was already known. It has thefollowing form:

CHz-CH2 / CH2-CH2 /CH2-CH2 // "'/C=CH '> C=CH "',C=CH

/ /CH3 CH3 CH3

The chain shown here has neither beginning nor end. Wesee that the molecules are built up of identical links.We can therefore write the formula for crude rubber ina more concise form:

[

- CH2-,=CH- CH2- ]'

CH3 n

The quantity n may reach many thousands. Long-chainmolecules built of repeating groups are called polymers.

An enormous number of synthetic polymers have foundmost extensive application in engineering and in thetextile industries. They include nylon, polyethylene,capron, polypropylene, polyvinyI chloride and many, man yothers.

H

Figure 9.1

Of simplest structure are the molecules of polyethylene.Bags of this material are found in the kitchen tables ofhouses and apartments almost all over the world. Ifa molecule of polyethylene is stretched as far as it willgo, it resembles the illustration in Figure 9.1. As youcan see, physicists were able to determine the distancesbetween the atoms and the angles between the valencebonds.

Long-chain molecules are not necessarily polymers,Le. they do not necessarily consist of repeating groups.Chemists have learned to "design" molecules built upof two or more different groups. If the groups alternatein a definite order, for example, according to the ar­rangement ABABABABAB, it is still appropriate to callsuch a molecule a polymer. We often deal, however, withmolecules not having such an ordered sequence. Can wecan a molecule with the arrangement ABBBABABAAA­BBBBABABAAB BA a polymer? This, however, is a matterof taste and tastes in names may differ.

A natural protein molecule is rarely called a polymer.Proteins are built up of some twenty pieces of differentkinds. These pieces, or building blocks, are called aminoacid radicals.i6-038~

J

234Molecules

There is one essential difference between protein mole­cules and synthetic molecules built up of several piec~sarranged in disorder. No two molecules .are the same ma piece of synthetic polymer. Such a cham molecule. mayconsist of one random sequence of component pIeces,with another, different random sequence in another mol­ecule. This usually has a detrimental effect on the prop­erties of the polymer. If the molecules do not resemble oneanother, they cannot be efficiently p~cked together. Inprinciple, such molecules cannot bUIld up a crystal.Substances of this type produce amorphous glasses.

In the last decades, chemists have learned to buildregular polymers, and have thus made many valuablematerials available to industry.

As to natural proteins of a definite type (say the hem­oglobin of a bull), the molecules are all the same eventhough they have a disordered structure. A moleculeof protein of a given sort can be compared to a page ofa book: the letters follow one another in a random, butquite definite order. All the molecules of a protein arecopies of the same page of the book.

9. Giant Molecules

Figure 9.2

235

Flexibility of Molecules

A long-chain molecule can be likened to a rail. A line0.1 mm long can accommodate a million atoms. T~ecrosswise dimension of a molecule of polyethylene ISabout 3 or 4 A. Thus the length of the molecule is great­er than its width by several hundred thousand times.Since a rail of the kind used in railways has a thickness ofabout 10 em, with the same length-to-thickness ratio asthe molecule, the rail would have to be 10 km long.

This does not imply, of course, that there are no short­er polymer molecules. Unless special measures are. taken,we find in a polymer substance molecules of dIfferent

lengths, from one consisting of several groups to thosebuilt up of thousands of groups.

We likened a long-chain molecule to a rail. In anothersense, this is not a very apt comparison. A rail is diffi­cult to bend, while a molecule bends easily. The flexibil­ity of a macromolecule is not similar to that of a willowswitch. The molecule owes its flexibility to a capacitypossessed by all molecules: one part of the molecule canswivel about another part if they are joined by bondsthat. chemists call single (monovalent) bonds. We canreadIly understand that this property of polymer mol­ecules enables them to assume the queerest shapes.Illustrated in Figure 9.2 is a model of a flexible molecule16*

Globular CrystalsMany molecules are capable of winding up into tight

coils or, as they are sometimes called, globules. Very neatand quite identical globules make up a protein molecule.There is one subtle reason for this. As a matter of fact,a protein molecule contains parts that "like" water, andother parts that have an aversion to water. The partsthat do not "like" water are said to be hydrophobic. Thecoili,ng of the protein molecules is governed by a singletendency: all the hydrophobic parts are to be hidden in­side the globule. This is why the globules in a solutionof protein resemble one another like identical twins.

Protein globules are more or less spherical. The sizeof a globule ranges from 100 to 300 A, making it read-

ip three positions. 1£ a molecule is suspended in a solu­tion, it is usually coiled into a ball.

The stretching of a rubber band consists in the uncoil­ing of its molecules. Thus, the flexibility of polymers is ofan entirely different nature than that of metals. If thestretched band is released, it contracts to its former length.Hence, the molecule tends to go over from its linear(stretched) shape to a coiled one. Why does this occur?There are two possible reasons. In the first place, we canassume that the coiled shape is more advantageous fromenergy considerations. In the second place, we can sup­pose that coiling contributes to an increase in entropy.Hence, which law of thermodynamics governs this be­haviour: the first or the second? Evidently, both. But thecoiled state is doubtlessly advantageous from the entropyviewpoint as well. Obviously, the sequence of the atomsof molecules coiled into a ball is in more disorder thanwhen the molecules are stretched out in a straight line.We also know that disorder and entropy are closely re-lated.

2379. Giant Molecules

figure 9.3

ily visible under an electron microscope. The first elec­tron micrographs of globular crystals were obtainedseveral decades ago, when electron microscopy tech­niques were considerably less advanced than they are to·day. Such a photograph of the tobacco mosaic virus isshown in Figure 9.3. A virus is more complex than aprotein, but this example is quite suitable to illustrateour point, the tendency of biological globules to an ex­ceptionally high degree of order.

But why haven't the authors provided a micrographof a protein crystal? The answer is simple. Protein crys­tals are absolutely extraordinary. They contain an im­mense amount of water (sometimes up to 90%). Thismakes it impossible to photograph them in an electronmicroscope. Protein crystals can be investigated only bymanipulating them in a solution. A tiny flask contains thesolution and a monocrystal of the protein. This objectcan then be studied by all available physical methods,including X-ray structure analysis, which we have re­peatedly mentioned.

Notwithstanding the huge amount of water-ordinarywater, no different from tap water-the globular proteinmolecules are arranged in absolutely strict order. Their

236Molecules

Bundles of Molecules

If molecules can be closely packed when they arestretched to the limit, the solid polymer material they com­prise can form various quite complex structures that pos­sess one common property. The solid body contains moreor fewer regions in which the molecules adjoin one ano­ther like a bundle of pencils held in the fist.

Depending upon the percentage of such bundle-typeregions in the body, and on how neatly the molecules com­prising each such region are packed, the polymer pos­sesses a certain "degree of crystallinity". Most polymersobject to being simply classified as either amorphous orcrystalline solids. There is nothing strange in this becausewe are dealing with giant molecules and, moreover, most-

orientation to the axis of the crystal is the same for allthe molecules. We have already mentioned that the mol­ecules are identical. This superior order enables thestructure 9f the protein molecule to be determined. Thisis no simple task and the investigators Max FerdinandPerutz (b. 1914) and John Cowdery Kendrew (b. 1917),the first to determine the detailed structure of hemoglo­bin and myoglobin, were awarded a Nobel prize for theirwork.

The structure of about a hundred protein moleculesis known today. Research continues in this field. Alto­gether, there are about ten thousand different proteins ina living organism. The activity of the living organismdepends upon the way in which the proteins are coiled andin which order the different amino acid radicals followone another in the molecule. No doubt that research con­cerning the structure of protein molecules will continueuntil all the details of the ten thousand kinds of moleculesthat govern life processes have been completely clearedup.

239

Figure 9.4

ly of different kinds. Ordered ("crystalline") regions inpolymers can be roughly divided into three classes:bundles, spherulites and crystals of folding molecules.

A typical microstructure of a polymer is. shown in .Fig­ure 9.4. This photograph has been magmfied 400 timesand was made from a film of polypropylene. The star-shapedfigures are kinds of crystallites. The growth of the spher­ulite began from the centre of the star as the polymer wascooled. Then the spherulites met and interfered with oneanother's growth. Consequently, they did not acquire aperfect spherical shape (if you succeed in observing thegrowth of a separate spherulite, you will actually see asphere, so that the name is justified). Inside the spheru­lite, the long molecules are arranged with sufficient neat­ness. Most likely, a spherulite can be envisioned as aneatly coiled rope. The rope represents a bundle of mol­ecules. Thus the long axis of the molecules is positionedperpendicular to a radius of the spherulite. On the same

9. Giant Molecules238Molecules

241

than that actually observed. This contradiction betweentheory and practice was finally solved: such rapid ~rystalgrowth was found possible when ther~ are screw d~sloca­tions in the growing crystal. If there IS a screw dIsloca­tion, the faces grow so that the steps at which new atomscan be readily attached never grow out to the edge of. thecrystal where they disappear. Physicists were certaml.yrelieved when screw dislocations extricated them from thendilemma. The growth rates became clear as did the essenceof the winding staircase feature indicated above for pa­raffin. Such spiral pyramids are frequently observed, ~ndthere is nothing strange about the fact that they eXIst.

Nothing strange while we are referring to crystals

Figure 9.6

9. Giant Molecules240Molecules

Figure 9.5

photograph we see lamellar regions. These may consist?f bundles of molecules, or they may be crystals of fold­mg molecules. The existence of such crystals is, per­haps, one of the most interesting and absolutely authen­tic facts concerning the structure of polymers.

The following outstanding discovery was made twentyyears ago. Small crystals of various polymer substanceswere .separated out of solutions. The investigators wereastolllshed to find that the same kind of crystals, with as~uface resembling a winding staircase, grew from solu­tIons of various paraffins. What is the cause of spiralgrowth of c~ystals which look like they had been prepar­ed by a skIlled pastry-cook (Figure 9.5)?

When we discussed crystal growth on p. 119, we omit­ted one important circumstance. Imagine that a growino­plane (face) of a crystal is completely filled with atoms~No positio~s re~ain that attract new atoms stronglyenough. ThIS bemg the case, it can be calculated thatgrowth should proceed at an inconceivably slower rate

-7; ~~Z7rr,---;'-··

built up of small molecules. The explanation is appropri­ate for such crystals: the size of the molecules, height ofthe steps and the thickness of the crystal are all data thatdo not contradict one another.

But, when such a picture of crystal growth is observedin a polymer, it is, at first, puzzling. The point is thatthe thic}mess of the layers of polyester ranges from 190to 120 A, while the length of the molecule is 6000 A.What conclusion can we reach on the basis of these fig­ures? There is only one feasible explanation: the moleculesare folded in these crystals. The flexibility of the mole­cules enables them to fold without difficulty. All that re­mains is to ponder (and such pondering continues to thepresent day) over the choice of the most suitable of thethree models illustrated in Figure 9.6. The differencesbetween the models are only minor ones. However, a spe­cialist in this field will resent such a statement. "How canyou ~all them minor?" he will object. "In the upperdrawmg, the molecules are folded over haphazardly, skip­ping their nearest neighbours; in the second model themolecules become their own neighbours. The diffe;encebetween the second and third models is that the surfaceof the crystal is smoother in the middle drawing than inthe lower one."

The specialist is right: the packing or stacking arrange­ment of polymer molecules is of exceptional significanceand has a fundamental effect on the properties of the sub­stance. Although polyethylene, nylon and like materialswere first synthesized several decades ago, the study oftneir supramolecular structure and investigation in thetechniques required to pack the molecules in different waysare being continued to this day by many scientists in thisfield.

Molecules 242 9. Giant Molecules

Muscular Contraction

We shall complete our discussion on giant moleculesby considering an example that demonstrates how macro­molecules behave in a living organism.

Biologists considered it to be their task to explain theconformity of the shapes of living organs-for instance,the shape of the hand or a leaf of a tree-to the functionsof the organs.

Physicists, having decided to apply methods for in­vestigating the structure of matter and the laws of naturein studying processes that occur in living organisms, aremaking efforts to understand life at the molecular level.The structure of tissues can be investigated today ingreat detail. After establishing the structure, it becomesfeasible to develop a model of biological events.

A quite significant achievement in this line is the advance­ment of the theory of muscular contraction. The mus­cle fibre consists of two types of threads: thin ones andthick ones (Figure 9.7a). The thick threads consist of pro­tein molecules called myosin. Physicists have estab­lished that a myosin molecule has the shape of a rod witha thickening at the end. In the thick thread, the mole­cules join with their thickened tails in the middle (Figure9.7c). The thin threads consist of actin whose structureresembles two strings of beads forming a double helix.Muscle contraction consists in the thick threads slidiBginto the helixes of the thin threads.

The details of this mechanism are known, but we cannotdiscuss them here. The signal for a muscle contraction isgiven by a nerve pulse. Its arrival releases atoms of cal­cium which transfer from one part of the thread to another.As a result, the molecules turn towards one another sothat it becomes advantageous from an energy viewpointfor one set of molecules to slide into the other set. The

Molecules 244

(a) MUSCLEFIBRES

.' ,MYOFIBRIL

(b). I>,~ I 4 ',' ....

.~\_ ~ _~~( ~l~ •

,,~, '\"'-' ';<;'i.ii.~~:~ ;I i:ir~ - ": • " -:. ~ ;':1 'R,!.

TIl .~ ~:~ "'\ ~ ::"ilJ.·~i- ... ,::: \ i"rf"'~.~ .. _•• ,. "1';

;<~:~"'-~ 1t~~;~.."o .' J~,~~ Ii p:..&':.... i,1U::-n::,~:~,~:' -. '. __."c· '..'

: ~'\\'~J~ ~ ~: ":~~.6-~~l"- • "'; -,}. II' '~1:!-~~~;..&

THIN THREADS

(C) ,...- ~ - ':':'. -- :-- .'

:cL - ' ."-.... ./v-

THICK THREADS

Figure 9.7

illustration gives two schematic diagrams with an elec­tron photomicrograph (Figure 9.7b) between them.

I fear that these pages provide only a faint idea of thedegree of detail to which the mechanism of muscle con­traction has been studied. But our aim was only to in­terest the reader, Consider the last page of this book de­voted to molecules as only a preview of a detailed discus­sion on biological physics which we hope to include inone of the subsequent books of Physics tor Everyone,

To the lleader

Mir Publishers would be grateful fo~ your co.mmentson the content, translation and desIg~ of thIS book.We would also be pleased to reCelve any othersuggestions you may wish to make.

Our address is:USSR, 129820, Moscow I-110, GSPPervy Rizhsky Pereulok, 2Mir Publishers

Printed in the Union of SOViet Socialist Republics

OTHER BOOKS IN THE SERIESTO BE PUBLISHED

ElectronsA. KITAIGORODSKY, D.Sc.

The book continues the fam .Everyone by the late renowneod

us ~erl~s Phlfsics forthe Nobel nd L" sCientist, willner of

r.~~df~~. ~~:~~*ri~~;~~~~Iia~~:L!\~~the b::i~an rea er III an easy-to-understand mannerfield of ph;~i~~~ts and latest developments in the

The book provides information thof electric currents and el t on e general theorywell as on current flow in ll·eqc ~domagnetilc fields, asd t . Ul s, meta 9' and sem'

t~bfnd~le~~~ic: a~dc~:~:etf:s )~es~::i~d~d ktnoOWl~dd;;e scope of the book. . WI en

Photons and NucleiA. KITAIGORODSKY t D.Sc.

The book concludes the famous series Physics forEveryone by the late renowned scientist, winnerof the Nobel and Lenin prizes, Academician LevLandau, and the distinguished physicist AlexanderKitaigorodsky. This book discusses in a simple andeasy-to-understand manner the phenomena of electro­magnetic waves, thermal radiation, and the mostrecent treatment of spectroscopic analysis. It providesan introdllction to nuclear physics and explainsthe most common types of lasers. It outlines theprincipal. aspects of the special theory of relativityand quantum mechanics.

•

Physical BodiesL. LANDAU, A. KITAIGORODSKY

The aim of the authors, the late renowned scientist,winner of the Nobel and Lenin prizes, Academician

,Lev Landau, and the distinguished physicist Alexan­der Kitaigorodsky, was to provide clear conceptionsof the fundamental ideas and latest discoveries ofmodern physics, set forth in readily understandablelanguage for the nonexpert. As a whole, the fourbooks of the new edition of Physics for Everyone(Physical Bodies, Molecules, Electrons, and Photonsand Nuclei) cover the fundamentals of physics.This is a new edition of the first half of Physics forEveryone: Motion and Heat. In Book I, the motionof bodies is dealt with from the points of view oftwo observers: one in an inertial and the other ina non-inertial reference frame. The law of universalgravitation is taken up in detail, including its ap­plication to calculating space velocities and tointerpreting lunar tides and various geophysicalphenomena.

\

{I

. ,

,~6·~tNJ.- , . . /.

•