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ArithmeticWhenmultiplyingordividingpositiveandnegativenumbersthesignoftheresultisgivenby:

positivepositive=positivepositivenegative=negativenegativepositive=negativenegativenegative=positive

positive

positive=positive

positive

negative=negative

negative

positive=negative

negative

negative=positive

TheBODMASruleremindsusoftheorderinwhichoperations

arecarriedout.BODMASstandsfor:

Brackets()Firstpriority

OfSecondpriority Division

Secondpriority

MultiplicationSecondpriority

Addition+Thirdpriority

SubtractionThirdpriority

Fractions:fraction=

numerator

denominator

Addingandsubtractingfractions:toaddorsubtracttwofrac-

tionsfirstrewriteeachfractionsothattheyhavethesamede-

nominator.Then,thenumeratorsareaddedorsubtractedasappropriateandtheresultisdividedbythecommondenomina-

tor:e.g.4

5+

3

4=

16

20+

15

20=

31

20.

Multiplyingfractions:tomultiplytwofractions,multiplytheirnumeratorsandthenmultiplytheirdenominators:e.g.

3

7

5

11=

35

711=

15

77

Dividingfractions:todividetwofractions,invertthesecond

andthenmultiply:e.g.3

5

2

3=

3

5

3

2=

9

10.

ProportionandPercentage:

Toconvertafractiontoapercentagemultiplyby100andlabeltheresultasapercentage.

5

8asapercentageis5

8100%=62.5%

1

3asapercentageis1

3100%=33

1

3%

Somecommonconversionsare:1

10=10%,

1

4=25%,

1

2=50%,

3

4=75%

Ratiosareanalternativewayofexpressingfractions.Consider

dividing200betweentwopeopleintheratioof3:2.Forevery

3thefirstpersongets,thesecondpersongets2.Sothefirstgets

3

5ofthetotal,andthesecondgets2

5ofthetotal;thatis120and80.

Generally,tosplitaquantityintheratiom:n,thequantityis

splitintom

m+nofthetotalandn

m+nofthetotal.

TrigonometryDegreesandradians:

360

=2radians,1

=2

360=

180radians

1radian=180

degrees57.3

Trigratiosforanacuteangle:

sin=sideoppositeto

hypotenuse=

b

c

cos=sideadjacentto

hypotenuse=

a

c

tan=sideoppositeto

sideadjacentto=

b

a

hypotenuse

opposite

adjacent

c

a

b

Pythagorastheorem:

a2

+b2

=c2

Standardtriangles:2

1

1

1

45

60 2

3

30

sin45

=12,cos45

=12,tan45

=1

sin30

=1

2,cos30

=

32

,tan30

=13

sin60

=

32

,cos60

=1

2,tan60

=3

BraggsLaw:afundamentallawofx-raycrystallography.

n=2dsin

=wavelengthofthex-raysincidentonacrystallatticewithplanesadistancedapart,nisaninteger(wholenumber),and,theBraggangle,istheanglebetweentheincidentbeamand

thereflectingplanes.

MatricesandDeterminants

The22matrixA=ab

cd

hasdeterminant

|A|=

ab

cd

=adbc

The33matrixA=

a11a12a13a21a22a23a31a32a33

hasdeterminant

|A|=a11

a22a23a32a33

a12

a21a23a31a33

+a13

a21a22a31a32

Theinverseofa22matrix:IfA=

abcd

thenA

1=

1

adbcdb

ca

providedthatadbc=0.Matrixmultiplication:for22matrices

ab

cd

=

a+ba+bc+dc+d

RememberthatAB=BAexceptinspecialcases.

AlgebraRemovingbrackets:

a(b+c)=ab+ac,a(bc)=abac(a+b)(c+d)=ac+ad+bc+bd

ab

c=

ac

bFormulaforsolvingaquadraticequation:

ifax2

+bx+c=0thenx=bb24ac2a

LawsofIndices:

am

an

=am+na

m

an=amn

(am)

n=a

mn(ab)

n=a

nbn

a0

=1am

=1

ama1/n

=naamn=(na)m

Logarithms:foranypositivebaseb(withb=1)logbA=cmeansA=b

c.

Logarithmstobasee,denotedlogeoralternativelylnarecallednaturallogarithms.Theletterestandsfortheexponentialcon- stantwhichisapproximately2.718.

logeAorlnA=cmeansA=ec.

c,thenaturallogarithmofanumberA,isthepowertowhichewouldhavetoberaisedtoequalA.Note:

elnA

=AifA>0;ln(eA)=A

Logarithmstobase10:log10A=cmeansA=10c.

pH:ofasolutionmeasuresitsacidityorbasicity.

pH=log10([H+]/c

)so[H+]=10

pHc

where[H+]=hydrogenionconcentrationinmoldm

3and

c

=1moldm3

.Equivalently,pH=log10aH3O+whereaH3O+=hydroniumionactivity.LawsofLogarithms:foranypositivebaseb,withb=1,

logbA+logbB=logbAB,logbAlogbB=logbA

B,

nlogbA=logbAn,logb1=0,logbb=1.

Formulaforchangeofbase:logax=logbxlogba.Specifically,

log10x=lnx

ln10.

Inequalities:a>bmeansaisgreaterthanba 0(2) b2 4ac= 0 (2) b2 4ac= 0(3) b2 4ac > 0 (3) b2 4ac < 0

StatisticsPopulation values, or parameters, are denoted by Greek letters.Population mean = . Population variance = 2. Population

standard deviation =. Sample values, orestimates, are denotedby roman letters.

Themean of a sample ofn observations x1, x2, . . . xn is

x= n

i=1xi

n

=x1+ x2+ +xn

nThe sample mean x is an unbiased estimate of the populationmean. The unbiased estimate of the varianceof thesen sample

observations is s2 =

n

i=1(xi x)

2

n 1 which can be written as

s2

= 1

n 1

ni=1

x2

i

nx2

n 1

The sample unbiased estimate of standard deviation, s, is the

square root of the variance:s =

n

i=1(xi x)2

n 1 . The standard

deviation of the sample mean is called the standard error of the

mean and is equal to n

, and is often estimated by sn

.

Differentiation

Differentiating a function,y = f(x), we obtain its derivative

dy

dx .This new function tells us the gradient (slope) of the original

function at any point. When dydx

= 0 the gradient is zero.

y= f(x) dydx

= f(x)

k, constant 0x 1

x2 2xxn, constant n nxn1

sin(kx) kcos(kx)cos(kx) ksin(kx)ex ex

ekx kekx

ln kx= loge

kx 1x

minimum point

dy

dx0

y

x

The linearity rules:d

dx(u(x)v(x)) =

d u

dx

dv

dxd

dx(k f(x)) = k

df

dxfork constant.

The product and quotient rules:

d

dx (uv) = u

dv

dx+ v

du

dx

d

dxu

v

=

v dudx u dv

dx

v2 .

The chain rule:

Ify = y(u) where u = u(x) then dydx

= dydu

dudx

.

Higher derivatives: f(x), or

d2f

dx2, means differentiate

df

dx

with respect to x. That is, d2f

dx2 =

d

dx

df

dx

.

Partial derivatives: I ff= f(x, y)is a function of two (or more)independent variables, f

xmeans differentiatefwith respect to

x treating y as if it were a constant. fy

means differentiate f

with respect to y treating x as if it were a constant.

Integration

f(x)f(x) dx

k, constant kx+c

x x2

2 +c

x2 x3

3 +c

xn, (n=1) xn+1

n+1

+c

x1 = 1x

lnx+c or ln cxex ex +c

ekx ekx

k +c

sin kx 1kcos kx+c

cos kx 1ksin kx+c

The linearity rule: (af(x) +bg(x)) dx= a

f(x) dx + b

g(x) dx, (a, b constant)

Integration by parts:

ba

udv

dxdx= [ uv]b

a

ba

du

dxvdx.

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Mixtures

Raoults law: states that the partial vapour pressure, pA, in aliquid mixture, A , is proportional to its mole fraction, xA, andits vapour pressure when pure,pA: pA= xAp

A.

Henrys law: states that the vapour pressure,pB, of a volatilesolute, B, is proportional to its mole fraction, xB, in a solution:pB= xBKB. HereKB is Henrys law constant.Chemical potential of a solvent:

A=

A+ RT lnxA

where A = chemical potential of pure A and xA is the molefraction.

Properties of mixtures: suppose an amount nA of substanceA is mixed with nB of substance B . The total volume of themixture is

V =nA Vm,A+ nBVm,B

where Vm,A= partial molar volume ofA and Vm,B = partial

molar volume ofB . More generally, V =i

niVm,i, where

Vm,i is the partial molar volume of the ith substance.Total Gibbs energy for the mixture is G = nAGA+ nBGBwhere GA and GB are the partial molar Gibbs energies of sub-

stancesA and B respectively. The partial molar Gibbs energiesare also denotedA and B so thatG = nA A+nBB. More

generally, with mixtures of several substancesG =i

nii.

Reaction ThermodynamicsStandard state: The standard state of a substance is the pure

substance at a pressure of 1 bar. The standard state value is

denoted by the superscript symbol , as in G .

Reaction Gibbs energy: rG = G

is the slope of the

graph of Gibbs energy against the progress of the reaction.

Here, = nJ/J for all species J in the reaction. ReactionGibbs energy at any composition of the reaction mixture can

be written

rG= rG + RTln Q whereQ =

J

avJJ

where aJ is the activity of species J and J is its stoichiometricnumber.

At equilibrium Q = K, rG = 0 and rG = RTln K

where

K=J

(avJJ )equilibrium

vant Hoff equation: d

dT lnK=

rH

RT2 .

ln

K2K1

=

rH

R

1

T2

1

T1

Vectors

Ifr = xi+yj +zk then|r|=x2 +y2 +z2.

Scalar product:

ab= |a| |b|cos b

a

Ifa = a1i+a2j+a3k and b = b1i+b2j+b3k then

ab= a1b1+ a2b2+a3b3

Vector product:

ab= |a| |b| sin eb

a

e

e is a unit vector perpendicular to the plane containing a and

b in a sense defined by the right hand screw rule.

Ifa = a1i+a2j+a3k and b = b1i+b2j+b3k then

ab = (a2b3 a3b2)i+ (a3b1a1b3)j+ (a1b2 a2b1)k

=

i j k

a1 a2 a3b1 b2 b3

KineticsArrhenius equation: The rate at which most chemical reac-

tions proceed depends upon the temperature. The amount of

energy necessary for the reaction to take place at all is called

the activation energy. These quantities are related by the

Arrhenius equation: k= AeEa/(RT)

where k = rate constant, Ea = the activation energy for thereaction, R = ideal gas constant, T = absolute temperature,

and A is a constant.

By taking logarithms this can be expressed as

ln k

k= ln

A

k Ea

RT

wherek is a chosen standard rate constant. Together,A and

Ea are called the Arrhenius parameters.

Rate Laws

In the table,[A] = molar concentration of reactant A at timet. [A]0 = concentration of reactant A at timet = 0 .

Order Rate Law Rate Law Half-life Common

Differential form Integrated form unit of k

0 d[A]

dt =k [A]0[A] = kt [A]0

2k moldm3 s1

1 d[A]

dt =k[A] [A] = [A]0ekt ln 2

k s1

2 d[A]

dt =k[A]2 1

[A] 1[A]0

=kt 1k[A]0 mol1 dm3 s1

2 d[A]dt =k[A][B] 1[B]0[A]0

ln [B][A]0[A][B]0 - mol1 dm3 s1

=kt

(* A + B Preaction.)

The Greek alphabet

A alpha I iota P rhoB beta K kappa sigma

gamma lambda T tau

delta M mu upsilonE epsilon N nu phiZ zeta xi X chiH eta O o omicron psi

theta pi omega

Physical constantsAvogadro constant NA = 6.022 10

23mol

1

Boltzmann constant kB = 1.381 1023

JK1

Planck constant h = 6.626 1034 J s

Elementary charge e = 1.602 1019 C

Ideal gas constant R = 8.314 JK1mol1

Vacuum permittivity 0 = 8.854 1012

J1

C2m1

Speed of light (vacuum) c= 2.998 108ms1

Faraday constant F =e NA = 96.485 kC mol1

General ThermodynamicsFirst Law: For a closed system, U= q + w. Here U is the

change in internal energy of a system, w is the work done on

the system, and q is the heat energy transferred to the system.Enthalpy: H= U+pVwhere U= internal energy,p=pressureand V = volume.Heat capacity at constant volume: CV =

UT

V

Heat capacity at constant pressure: Cp =HT

p

In general Cp depends upon T. Values ofCp at temperatures

not much different from room temperature can be estimatedfrom Cp = a + bT+

c

T2

wherea, b and c are experimentally determined constants.Second Law of thermodynamics:

During a spontaneous change, the total entropy of an isolated

system and its surroundings increases: S > 0. For a reversibleprocess, at constant temperature, T, change in entropy

S= qrev

T

whereqrev= energy reversibly transferred as heat.Boltzmann formula: S = kB ln W where W = weight ofthe most probable configuration of the system and kB is the

Boltzmann constant.Helmholtz energy: A = U T S.Gibbs energy: G= H T S.Change in Gibbs energy: G = H TS (at constanttemperature).Entropy change for isothermal expansion of an ideal gas:

S= nR ln

Vfinal

Vinitial

whereVfinal and Vinitial are the final and initial volumes.

Gibbs-Helmholtz equation:

T

G

T

p

= H

T2 .

Units&ConversionsScientificnotation:isusedtoexpresslargeorsmallnumbers

concisely.Eachnumberiswrittenintheforma10n

wherea

isusuallyanumberbetween1and10.Wemakeuseof

...0.01=102

,0.1=101

,...,100=102

,1000=103

,...

Then,forexample,6859=6.8591000=6.859103

and

0.0932=9.320.01=9.32102

.SIbaseunits:formostquantitiesitisnecessarytospecifytheunitsinwhichtheyaremeasured.

QuantitySIunitSymbol

lengthmetremmasskilogramkg

timesecondstemperaturekelvinK

amountofsubstancemolemolcurrentampereA

luminousintensitycandelacd

Derivedunitsareformedfromthebaseunits.Forexample,the

unitofforceisfoundbycombiningunitsofmass,lengthandtimeinthecombinationkgms

2.Thiscombinationismore

usuallyknownasthenewton,N.

Propertyunitnameunitsymbols

frequencyhertzHz=s1

forcenewtonN=kgms2

pressurepascalPa=Nm2

energyjouleJ=Nm=kgm2

s2

chargecoulombAs

potentialdifferencevoltV=JC1

powerwattW=Js1

CelsiustemperaturedegreeCelsius

CCapacitancefaradF=CV

1

Resistanceohm

Commonprefixes:aprefixisamethodofmultiplyingtheSI

unitbyanappropriatepowerof10tomakeitlargerorsmaller.

MultiplePrefixSymbolMultiplePrefixSymbol1012teraT101decid

109

gigaG102

centic

106

megaM103

millim

103

kilok106

micro

102

hectoh109

nanon

101

decada1012

picopInterconversionofunits:

Sometimesalternativesetsofunitsareusedandconversionbetweentheseisneeded.PressureisoftenquotedinunitscorrespondingtotheStandardatmosphere(atm).

Then1atm=1.01325105

Pa=1.01325bar=760mmHg=

760Torr.(1bar=1105

Pa).Length(ofbonds)issometimesquotedinAngstroms,Awhere

1A=1010m.1nm=10

9m.0.1nm=100pm.

Energyisoftenmeasuredincalories(cal):1cal=4.184J.

Amountofsubstanceetc.Themoleistheamountofsubstancethatcontains6.022141510

23(Avogadroconstant/mol

1)atomsormoleculesofthe

puresubstancebeingmeasured.Forexample1mole(mol)of

potassiumwillcontainNAatoms.1moleofwatercontainsNAwatermolecules.Amoleofanysubstancecontainsasmany

atomsormolecules(asspecified)asthereareatomsin12gofthecarbonisotope

126C.(Toavoidconfusion,alwaysspecifythe

typeofparticle.)

Themolarmass,M,ofasubstanceisthemassdividedbychemicalamount.Forpurematerials,molarmass,M=molec-

ularweight(orrelativeatomicormolecularmass)gmol1

.Theconcentrationormolarityofasoluteisc=

n

Vwhere

n=amountofsolute,V=volumeofsolution.Molarityis

usuallyquotedinmoldm3

.Theunit1moldm3

iscommonlydenoted1Mandreadasmolar.

Themolality,b,ofasoluteistheamountofsolutedividedbythemassofsolvent:b=

n

m.So,molalityrepresents1moleof

asolutein1kgofsolvent.

Themolefraction,x,ofasoluteistheamountofsolutedividedbythetotalamountinthesolution:

x=n

ntotal.

Themassdensity(orjustdensity),,ofasubstanceisits

massdividedbyitsvolume,thatis=m

V.Themasspercentageisthemassofasubstanceinamixture

asapercentageofthetotalmassofthemixture.Themass-volumepercentageisthenumberofgramsofsolute

in100millilitresofsolution.

Thevolume-volumepercentageisthenumberofmillilitresofliquidsolutein100millilitresofsolution.

Partspermillion(ppm):mg/kg.Partsperbillion(ppb):g/kg.

Yield:%yield=Massobtained

Theoreticalyield100

Gases Thegeneralformofanequationofstateisp=f(T,V,n)

wherep=pressure,Tistemperature,V=volume,n=amount

ofsubstance.BoylesLawrelatesthepressureandvolumeofagaswhenthe

temperatureisfixed:p1V1=p2V2.

Anequivalentformis:pV=1

3nMc

2,wheren=numberof

molecules,m=massofamolecule,M(=mNA)isthemolarmassofthemolecules,c

2=meansquarespeed.

CharlessLawrelatesthevolumeandtemperatureofagasat

constantpressure:V1

V2=

T1

T2.

DaltonsLawofpartialpressures:statesthatthepressure

exertedbyamixofidealgasesisthesumofthepartialpressureseachwouldexertiftheywerealoneinthesamevolume:

ptotal=p1+p2+...+pnorptotal=ni=1

pi

Thepartialpressure,pi,ofoneofthegasescanbecalculatedbymultiplyingthegasmolefraction,xi,bythetotalpressure

ofallthegases,ptotal.

PerfectGasLaw:pV=nRTwhereR=idealgasconstant.

CombinedGasLaws:p1V1

T1=

p2V2

T2.

Virialequationofstate:Thisimprovestheperfectgaslawbecauseittakesintoaccountintermolecularforces.

pVm=RT(1+B

Vm+

C

V2m

+...)

Vm=Vn=molarvolume,B,Cetcarethevirialcoefficients.

Observethatwhenthemolarvolumeisverylarge,thetermsBVmand

CV2m

becomeincreasinglylessimportant,andinthelimit

weobtaintheidealgaslaw.VanderWaalsequationtakesintoaccountthefinitedistance

betweenmoleculesandinterparticleattractions:

p=RT

Vmb

a

V2m

or

p+

an2

V2

(Vnb)=nRT

aisameasureofattractionbetweenparticles,bisthevolume

excludedbyamoleofparticles.

PhasesGibbsphaserule:F=CP+2,whereFisthenum-berofdegreesoffreedom,C=numberofindependentcom-

ponents,P=thenumberofphasesinequilibriumwitheachother.Thisisarelationshipusedtodeterminethenumberof

statevariables,F,chosenfromamongsttemperature,pressureandspeciescompositionsineachphase,whichmustbespecified

tofixthethermodynamicstateofasysteminequilibrium.

Clapeyronequationrelateschangeinpressuretochangeintemperatureataphaseboundary.Theslopeofthephase boundaryis

dp

dT

.dp

dT=

H

TV

HereH=molarenthalpyoftransition,V=changeinmolarvolumeduringtransition.

Clausius-ClapeyronequationisanapproximationoftheClapeyronequationforaliquid-vapourphaseboundary.Plotting

vapourpressureforvarioustemperaturesproducesacurve.Forpureliquids,plottingln

p

pagainst

1

Tproducesastraightline

withgradientHR.(Here,p

=anystandardpressure).

lnp1

p2=

H

R

1T1

1

T2

HereH=molarenthalpyofvaporisation.Thisequationre-

latesthenaturallogarithmofthevapourpressuretothetem-

peratureataphaseboundary.