2 3 DEVIATIONS FROM IDEAL BEHAVIOR OF REAL GASES
Dening the specic gas constantR as the ratioR/M,
P = RspecificT
This form of the ideal gas law is very useful because itlinks
pressure, density, and temperature in a unique for-mula independent
of the quantity of the considered gas.Alternatively, the law may be
written in terms of thespecic volume v, the reciprocal of density,
as
Pv = RspecificT
It is common, especially in engineering applications,
torepresent the specic gas constant by the symbol R. Insuch cases,
the universal gas constant is usually given adierent symbol such as
R to distinguish it. In any case,the context and/or units of the
gas constant should makeit clear as to whether the universal or
specic gas constantis being referred to.[5]
1.3 Statistical mechanicsIn statistical mechanics the following
molecular equationis derived from rst principles:
PV = NkBT
where P is the absolute pressure of the gas measured inpascals;
N is the number of molecules in the given volumeV. The number
density is given by the ratioN/V; kB is theBoltzmann constant
relating temperature and energy; andT is the absolute
temperature.The number density contrasts to the other
formulation,which uses n, the number of moles and V, the vol-ume.
This relation implies that R=NAkB where NA isAvogadros constant,
and the consistency of this resultwith experiment is a good check
on the principles of sta-tistical mechanics.From this we can notice
that for an average particle massof times the atomic mass constant
m (i.e., the mass is u)
Y =m
mu
and since = mn, we nd that the ideal gas law can berewritten
as:
P =1
V
m
mukT =
k
muT:
In SI units, P is measured in pascals; V in cubic metres;Y is a
dimensionless number; and T in Kelvin. k has thevalue 1.381023 JK1
in SI units.
2 Applications to thermodynamicprocesses
The table below essentially simplies the ideal gas equa-tion for
a particular processes, thus making this equationeasier to solve
using numerical methods.A thermodynamic process is dened as a
system thatmoves from state 1 to state 2, where the state numberis
denoted by subscript. As shown in the rst columnof the table, basic
thermodynamic processes are denedsuch that one of the gas
properties (P, V, T, or S) is con-stant throughout the process.For
a given thermodynamics process, in order to spec-ify the extent of
a particular process, one of the prop-erties ratios (which are
listed under the column labeledknown ratio) must be specied (either
directly or indi-rectly). Also, the property for which the ratio is
knownmust be distinct from the property held constant in
theprevious column (otherwise the ratio would be unity, andnot
enough information would be available to simplify thegas law
equation).In the nal three columns, the properties (P, V, or T)
atstate 2 can be calculated from the properties at state 1using the
equations listed.^ a. In an isentropic process, system entropy (S)
is con-stant. Under these conditions, P1 V1 = P2 V2, where is dened
as the heat capacity ratio, which is constant fora calorically
perfect gas. The value used for is typi-cally 1.4 for diatomic
gases like nitrogen (N2) and oxygen(O2), (and air, which is 99%
diatomic). Also is typi-cally 1.6 for monatomic gases like the
noble gases helium(He), and argon (Ar). In internal combustion
engines varies between 1.35 and 1.15, depending on
constitutiongases and temperature.
3 Deviations from ideal behavior ofreal gases
The equation of state given here applies only to an idealgas, or
as an approximation to a real gas that behaves suf-ciently like an
ideal gas. There are in fact many dierentforms of the equation of
state. Since the ideal gas law ne-glects both molecular size and
intermolecular attractions,it is most accurate for monatomic gases
at high temper-atures and low pressures. The neglect of molecular
sizebecomes less important for lower densities, i.e. for
largervolumes at lower pressures, because the average
distancebetween adjacent molecules becomes much larger thanthe
molecular size. The relative importance of inter-molecular
attractions diminishes with increasing thermalkinetic energy, i.e.,
with increasing temperatures. Moredetailed equations of state, such
as the van der Waalsequation, account for deviations from ideality
caused bymolecular size and intermolecular forces.
3A residual property is dened as the dierence between areal gas
property and an ideal gas property, both consid-ered at the same
pressure, temperature, and composition.
4 Derivations
4.1 EmpiricalThe ideal gas law can be derived from combining two
em-pirical gas laws: the combined gas law and Avogadroslaw. The
combined gas law state that
PV
T= C
where C is a constant which is directly proportional to
theamount of gas, n (Avogadros law). The proportionalityfactor is
the universal gas constant, R, i.e. C = nR.Hence the ideal gas
law
PV = nRT
4.2 Theoretical4.2.1 Kinetic theory
Main article: Kinetic theory of gases
The ideal gas law can also be derived from rst principlesusing
the kinetic theory of gases, in which several simpli-fying
assumptions are made, chief among which are thatthe molecules, or
atoms, of the gas are point masses, pos-sessing mass but no
signicant volume, and undergo onlyelastic collisions with each
other and the sides of the con-tainer in which both linear momentum
and kinetic energyare conserved.
4.2.2 Statistical mechanics
Main article: Statistical mechanics
Let q = (q, q, q) and p = (p, p, p) denote the positionvector
and momentum vector of a particle of an ideal gas,respectively. Let
F denote the net force on that particle.Then the time-averaged
potential energy of the particle is:
hq Fi =Dqxdpxdt
E+Dqydpydt
E+Dqzdpzdt
E=
Dqx@H
@qx
EDqy@H
@qy
EDqz@H
@qz
E= 3kBT;
where the rst equality is Newtons second law, andthe second line
uses Hamiltons equations and theequipartition theorem. Summing over
a system of N par-ticles yields
3NkBT = NXk=1
qk Fk:
By Newtons third law and the ideal gas assumption, thenet force
of the system is the force applied by the wallsof the container,
and this force is given by the pressure Pof the gas. Hence
NXk=1
qk Fk= P
Isurface
q dS;
where dS is the innitesimal area element along the wallsof the
container. Since the divergence of the position vec-tor q is
r q = @qx@qx
+@qy@qy
+@qz@qz
= 3;
the divergence theorem implies that
P
Isurface
q dS = PZvolume
(r q) dV = 3PV;
where dV is an innitesimal volume within the containerand V is
the total volume of the container.Putting these equalities together
yields
3NkBT = NXk=1
qk Fk= 3PV;
which immediately implies the ideal gas law for N
parti-cles:
PV = NkBT = nRT;
where n = N/NA is the number of moles of gas and R =NAkB is the
gas constant.
5 See also Van der Waals equation Boltzmann constant Conguration
integral Dynamic pressure Internal energy
59 Text and image sources, contributors, and licenses9.1
Text
Ideal gas law Source:
https://en.wikipedia.org/wiki/Ideal_gas_law?oldid=703373279
Contributors: CYD, Vicki Rosenzweig, Bryan Derk-sen, Tarquin, Andre
Engels, William Avery, SimonP, FlorianMarquardt, Patrick,
JakeVortex, BrianHansen~enwiki, Mark Foskey, Vivin,Tantalate,
Ozuma~enwiki, Robbot, COGDEN, Wereon, Isopropyl, Mattaschen,
Enochlau, Alexwcovington, Giftlite, Bensaccount, Alexf,Karol
Langner, Icairns, ELApro, Mike Rosoft, Venu62, Noisy,
Discospinster, Hydrox, Vsmith, Femto, Grick, Bobo192,
Avathar~enwiki,Larryv, Riana, Lee S. Svoboda, Shoey, Gene Nygaard,
StradivariusTV, Kmg90, Johan Lont, Mandarax, MassGalactusUniversum,
Jan vanMale, Eteq, Rjwilmsi, Sango123, FlaBot, Intersoa,
Jrtayloriv, Fresheneesz, Scroteau96, SteveBaker, Physchim62,
Krishnavedala, ARAJ,YurikBot, Huw Powell, Jimp, Quinlan Vos~enwiki,
Gaius Cornelius, CambridgeBayWeather, LMSchmitt, Bb3cxv, Dhollm,
Ruhrsch,Acit, Zwobot, T, Someones life, User27091, Smaines, WAS
4.250, 2over0, U.S.Vevek, Nlitement, Junglecat, Bo Jacoby, Bwiki,
SmackBot,Mitchan, Incnis Mrsi, Sal.farina, Pennywisdom2099,
Dave19880, Kmarinas86, Bluebot, Kunalmehta, Silly rabbit,
Tianxiaozhang~enwiki,CSWarren, DHN-bot~enwiki, Metal Militia,
Berland, Samir.Mesic, Ollien, PiMaster3, G716, Foxhunt king, Just
plain Bill, SashatoBot,Esrever, Mbeychok, JorisvS, IronGargoyle,
Ranmoth, Carhas0, Peter Horn, Sifaka, Majora4, Mikiemike, MC10,
Astrochemist, Kimtaeil,Christian75, Thijs!bot, Headbomb, Jakirkham,
Electron9, RedWasp, Hmrox, AntiVandalBot, KMossey, Nehahaha,
Seaphoto, Prolog,Coolhandscot, Fern Forest, AdamGomaa,
Magioladitis, VoABot II, Baccyak4H, Kittyemo, ANONYMOUS
COWARD0xC0DE, JaGa,Nirupambits, MartinBot, Rock4p, Mbweissman,
J.delanoy, SimpsonDG, P.wormer, Nwbeeson, Habadasher, Juliancolton,
KudzuVine,Nasanbat, VolkovBot, Drax Conqueror, Barneca, Philip
Trueman, Rbingama, TXiKiBoT, Malinaccier, Comtraya, Rexeken,
LanceBar-ber, Hanjabba, Riick, Brianga, Hoopssheaer, Givegains,
SieBot, Flyer22 Reborn, Baxter9, Oxymoron83, Smaug123,
123ilikecheese,Lightmouse, JerroldPease-Atlanta, Nskillen, COBot,
Adamtester, Thomjakobsen, Pinkadelica, ClueBot, GorillaWarfare,
Kharazia, The888th Avatar, Vql, Jmk, Excirial, Pdch, DumZiBoT,
Hseo, TZGreat, Frood, RP459, QuantumGlow, Dj-dios-del-sol,
SkyLined, Dnvr-fantj, Addbot, Power.corrupts, LaaknorBot, Eelpop,
CarsracBot, LinkFA-Bot, Lightbot, Loupeter, Legobot, Yobot,
Ptbotgourou, DanielePugliesi, JackieBot, Materialscientist,
Nickkid5, Quark1005, Craftyminion, GrouchoBot,
ChristopherKingChemist, RibotBOT, , Dougofborg, Kamran28,
Khakiandmauve, StephenWade, EntropyTrap, Lambda(T), Happydude69 yo,
Mrahner, Michael93555,D'ohBot, RWG00, Zmcdargh, Citation bot 1,
Kishmakov, I dream of horses, RedBot, Pbsouthwood, Cramyourspam,
Orenburg1, Ger-aldo62, Jade Harley, RjwilmsiBot, MagnInd, Steve
Belkins, EmausBot, Tdindorf, Razor2988, RA0808, Jerry858, Dcirovic,
Ssp37097,JSquish, ZroBot, Susfele, MarkclX, Stovl, SporkBot,
YnnusOiramo, Donner60, Odysseus1479, Theislikerice, RockMagnetist,
GeorgeMakepeace, ClueBot NG, BubblyWantedXx, Helloimriley,
Wrecker1431, Rezabot, Widr, Bibcode Bot, Mariansavu, MusikAnimal,
Avoca-toBot, MarkArsten, Trevayne08, F=q(E+v^B), Klilidiplomus,
TechNickL1, Egm4313.s12, NJITHUMrudyh,
NJITHUMNV,Waterproof-breathable, AlanParkerFrance, Dexbot,
Epicgenius, I am One of Many, Blackbombchu, Zenibus, JustBerry,
Ginsuloft, Keojukwu, Dude-WithAFeud, Whizzy1999, Fuguangwei,
Evanrelf, Monkbot, Nojedi, ChaquiraM, Smanojprabhakar, riugena,
Pwags3147, The Master6969, Qzd and Anonymous: 394
9.2 Images File:Folder_Hexagonal_Icon.svg Source:
https://upload.wikimedia.org/wikipedia/en/4/48/Folder_Hexagonal_Icon.svg
License: Cc-by-
sa-3.0 Contributors: ? Original artist: ?
File:Ideal_gas_isotherms.svg Source:
https://upload.wikimedia.org/wikipedia/commons/9/92/Ideal_gas_isotherms.svg
License: CC0
Contributors: Own work Original artist: Krishnavedala
File:Portal-puzzle.svg Source:
https://upload.wikimedia.org/wikipedia/en/f/fd/Portal-puzzle.svg
License: Public domain Contributors: ?
Original artist: ? File:SpongeDiver.jpg Source:
https://upload.wikimedia.org/wikipedia/commons/8/81/SpongeDiver.jpg
License: Public domain Contrib-
utors: Own work Original artist: Bryan Shrode
File:Symbol_list_class.svg Source:
https://upload.wikimedia.org/wikipedia/en/d/db/Symbol_list_class.svg
License: Public domain Con-
tributors: ? Original artist: ?
9.3 Content license Creative Commons Attribution-Share Alike
3.0
EquationCommon formMolar formStatistical mechanics
Applications to thermodynamic processes Deviations from ideal
behavior of real gasesDerivations EmpiricalTheoreticalKinetic
theoryStatistical mechanics
See alsoReferencesFurther readingExternal links Text and image
sources, contributors, and licensesTextImagesContent license
