Properties of Gas Manik

Properties of gases: gas laws, ideal gas equation, dalton’s law of partial pressure, diffusion of gases, kinetic theory of gases, mean free path, deviation from ideal gas behavior, vander wails equation, critical constants, liquefaction of gases, determination of molecular weights, law of corresponding states and heat capacity

1. Properties of Gases
Md. Imran Nur Manik
Lecturer
Department of Pharmacy
Northern University Bangladesh

2. Allmattersexistinthreestates:gas,liquidandsolid.Ofthethreestatesofmatter,
thegaseousstateistheonemoststudiedandbestunderstood.
Inagaseousstate,moleculesremainseparatedwideapartinemptyspace.
Themoleculesarefreetomoveaboutthroughoutthecontainerinwhichtheyareplaced.
General Characteristics of Gases
1.Expansibility
Gaseshavelimitlessexpansibilitybecauseoflittleintermolecularattractionamongthegas
molecules.Theyexpandtofilltheentirevesseltheyareplacedin.
2.Compressibility
Duetolargeintermolecularspace,gasescanbeeasilycompressedbytheapplicationof
pressuretoamovablepistonfittedinthecontainer.
3.Diffusibility
Gasescandiffuserapidlythrougheachothertoformahomogeneousmixture.
4.Pressure
Gasesexertpressureonthewallsofthecontainerinalldirections.
5.EffectofHeat
Whenagas,confinedinavesselisheated,itspressureincreases.Uponheatingina
vesselfittedwithapiston,volumeofthegasincreases.

3. Thevolumeofagivensampleofgasdependsonthetemperatureandpressure
appliedtoit.Anychangeintemperatureorpressurewillaffectthevolumeofthegas.
Asresultsofexperimentalstudiesfrom17thto19thcentury,scientistsderivedthe
relationshipsamongthepressure,temperatureandvolumeofagivenmassofgas.
Theserelationships,whichdescribethegeneralbehaviourofgases,arecalledthe
gaslaws.

4. Boyle’sLaw
In1660RobertBoylefoundoutexperimentallythechangeinvolumeofagiven
sampleofgaswithpressureatroomtemperature.Fromhisobservationshe
formulatedageneralizationknownasBoyle’sLaw.Itstatesthat:Atconstanttemp
erature,thevolumeofafixedmassofgasisinverselyproportionaltoitspres
sure.Ifthepressureisdoubled,thevolumeishalved.
The Boyle’s Law may be expressed mathematically as-(Writing on white board)
V ∝1/P (T, n are constant)
Or, V = k ×1/P
Where, k is proportionality constant.
So, PV =k
If P
1, V
1are the initial pressure and volume of a given sample of gas and P
2, V
2the changed pressure and volume, we can
write;
P
1V
1= k = P
2V
2
Or, P
1V
1= P
2V
2
This relationship is useful for the determination of the volume of a gas at any pressure, if its volume at any other pressureisk
nown.

5. Charles’s Law
In1787JacquesCharlesinvestigatedtheeffectofchangeoftemperatureonthe
volumeofafixedamountofgasatconstantpressure.Heestablishedageneralization
whichiscalledtheCharles’Law.
Itstatesthat:Atconstantpressure,the
volumeofafixedmassofgasisdirectly
proportionaltotheKelvintemperatureor
absolutetemperature.Iftheabsolute
temperatureisdoubled,thevolumeis
doubled.
If V
1
, T
1
are the initial volume and temperature of
a given mass of gas at constant pressure and
V
2
, T
2
be the new values, we can write-
Using this expression, the new volume V
2
can be found
from the experimental values of V
1
, T
1
and T
2
.2
2
1
1
2
2
1
1
T
V
T
V
,
T
V
T
V


Or
k
Charles’ Law may be expressed mathematically as-
V∝T (P, n are constant)
Or, V = kT; Where, k is a constant.K
T
V
,Or

6. Boyle’sLawandCharles’LawcanbecombinedintoasinglerelationshipcalledtheCombinedGas
Law.
Boyle’sLaw,V∝1/P(T,nconstant)
Charles’Law,V∝T(P,nconstant)
Therefore,V∝T/P(nconstant)
Thecombinedlawcanbestatedas:Forafixedmassofgas,thevolumeisdirectlyproportionalto
kelvintemperatureandinverselyproportionaltothepressure.
Ifkbetheproportionalityconstant,
V=kT/P (nconstant)
Or,PV/T=k (nconstant)
Ifthepressure,volumeandtemperatureofagasbechangedfromP
1,V
1andT
1toP
2,T
2andV
2,then-2
22
1
11
2
22
1
11
T
VP
T
VP
,
T
VP
T
VP


Or
k
This is the form of combined law for two sets of conditions. It can be used to solve problems involving
a change in the three variables P, V and T for a fixed mass of gas.

7. Gay Lussac’sLaw
In1802JosephGayLussacasaresultofhisexperimentsestablishedageneral
relationbetweenthepressureandtemperatureofagas.ThisisknownasGayLuss
ac’sLaworPressure-TemperatureLaw.2
2
1
1
2
2
1
1
T
P
T
P
,
T
P
T
P


Or
k
The law may be expressed mathematically as-
P ∝T (Volume, n are constant)
Or, P = kT
Or, P/T = k
For different conditions of pressure and temperature
Knowing P
1
, T
1
, and T
2
, P
2
can be calculated.
Itstatesthat:atconstantvolume,thepressureofa
fixedmassofgasisdirectlyproportionaltothe
Kelvintemperatureorabsolutetemperature.

8. Letustakeaballooncontainingacertainmassofgas.Ifweadd
toitmoremassofgas,holdingthetemperature(T)andpressure
(P)constant,thevolumeofgas(V)willincrease.Itwasfound
experimentallythattheamountofgasinmolesisproportionalto
thevolume.Thatis,
V ∝n (T and P constant)
Or, V = A n
Where, A is constant of proportionality.
Or, V/A = n
For any two gases with volumes V
1, V
2and moles n
1, n
2at
constant T and P,
If V
1= V
2then, n
1= n
2
Thus, for equal volumes of the two gases at fixed T and P, numb
er of moles is also equal.
ThisisthebasisofAvogadro’sLawwhichmaybestatedas:
Equalvolumesofgasesatthesametemperatureand
pressurecontainequalnumberofmolesormolecules.Ifthe
molaramountisdoubled,thevolumeisdoubled.2
2
1
1
n
V
A
n
V


9. The Molar Gas Volume
ItfollowsasacorollaryofAvogadro’sLawthatonemoleofanygasatagiven
temperature(T)andpressure(P)hasthesamefixedvolume.Itiscalledthe
molargasvolumeormolarvolume.Inordertocomparethemolarvolumesof
gases,chemistsuseafixedreferencetemperatureandpressure.
Thisiscalledstandardtemperatureandpressure(abbreviated,STP).Thestandard
temperatureusedis273K(0°C)andthestandardpressureis1atm(760mmHg).
AtSTPwefindexperimentallythatonemoleofanygasoccupiesavolumeof22.4
litres.
Toputitintheformofanequation,wehave
1 mole of a gas at STP = 22.4 litres

10. We have studied three simple gas laws:
Boyle’s Law, V ∝1/P (T, n constant)
Charles’ Law, V ∝T (n, P constant)
Avogadro’s Law, V ∝n (P, T constant)
These three laws can be combined into a single more general gas law:
V ∝
????????????
??????
…………………………………………………………………... (1)
ThisiscalledtheUniversalGasLaw.ItisalsocalledIdealGasLawasitapplies
toallgaseswhichexhibitidealbehaviori.e.,obeysthegaslawsperfectly.
Theidealgaslawmaybestatedas:Thevolumeofagivenamountofgasis
directlyproportionaltothenumberofmolesofgas,directlyproportionalto
thetemperature,andinverselyproportionaltothepressure.

11. IntroducingtheproportionalityconstantRintheexpression(1)wecanwrite-
V=R
????????????
??????
Or, PV=nRT...................................................................................................(2)
Theequation(2)iscalledtheIdealGasEquationorsimplythegeneralGas
Equation.TheconstantRiscalledtheGasconstant.Theidealgasequationholds
fairlyaccuratelyforallgasesatlowpressures.Foronemole(n=1)ofagas,the
ideal-gasequationisreducedto
PV=RT...................................................................................................................(3)
Theideal-gasequationiscalledanEquationofStateforagasbecauseitcontains
allthevariables(T,P,Vandn)whichdescribecompletelytheconditionorstateof
anygassample.Ifweknowthethreeofthesevariables,itisenoughtospecifythe
systemcompletelybecausethefourthvariablecanbecalculatedfromtheideal-gas
equation.

12. The Numerical Value of R
From the ideal-gas equation, we can write
R = PV/nT
We know that one mole of any gas at STP occupies a volume of 22.4 litres. Substituting
the values in the above expression, we have
R = (1 atm×22.4 litres)/(1 mole ×273 K)
= 0.0821 atm. litremol
–1
K
–1
It may be noted that the unit for R is complex; it is a composite of all the units used in c
alculating the constant.

13. JohnDaltonvisualizedthatinamixture
ofgases,eachcomponentgasexerted
apressureasifitwerealoneinthe
container.Theindividualpressureof
eachgasinthemixtureisdefinedasits
PartialPressure. Based on
experimentalevidence,in1807,Dalton
enunciatedwhatiscommonlyknownas
theDalton’sLawofPartialPressures.
Itstatesthat:Thetotalpressureofa
mixtureofgasesisequaltothesum
ofthepartialpressuresofallthegas
espresent.

14. MathematicallythelawcanbeexpressedasP
total=P
1+P
2+P
3...(VandTare
constant)
where,P
1,P
2andP
3arepartialpressuresofthethreegases1,2and3;
andsoon.
Dalton’sLawofPartialPressuresfollowsbyapplicationoftheideal-gasequation
PV=nRTseparatelytoeachgasofthemixture.
Thus,wecanwritethepartialpressuresP
1,P
2andP
3ofthethreegases-
P
1=n
1(RT/V) P
2=n
2(RT/V) P
3=n
3(RT/V)
Where,n
1,n
2andn
3aremolesofgases1,2and3.Thetotalpressure,P
t,ofthe
mixtureis
P
t=(n
1+n
2+n
3)RT
V
Or, P
t=n
t
RT
V
Inthewords,thetotalpressureofthemixtureisdeterminedbythetotalnumberof
molespresentwhetherofjustonegasoramixtureofgases.

15. Whentwogasesareplacedincontact,they
mixspontaneously.Thisisduetothe
movementofmoleculesofonegasintothe
othergas.Thisprocessofmixingofgasesby
randommotionofthemoleculesiscalled
Diffusion.ThomasGrahamobservedthat
moleculeswithsmallermassesdiffused
fasterthanheavymolecules.
In1829,GrahamformulatedwhatisnowknownasGraham’sLawofDiffusion.
Itstatesthat:Underthesameconditionsoftemperatureandpressure,therates
ofdiffusionofdifferentgasesareinverselyproportionaltothesquarerootsof
theirmolecularmasses.

16. Mathematicallythelawcanbeexpressedas-
??????1
??????2
=√
??????2
??????1
Where,r
1andr
2aretheratesofdiffusionof
gases1and2,whileM
1andM
2aretheir
molecularmasses.
Whenagasescapesthroughapin-holeintoaregionoflowpressureorvacuum,the
processiscalledEffusion.Therateofeffusionofagasalsodependsonthe
molecularmassofthegas.
Dalton’slawwhenappliedtoeffusionofagasiscalledtheDalton’sLawof
Effusion.Itmaybeexpressedmathematicallyas-1
2
2
1
M
M
gasofrateEffusion
gasofrateEffusion

Thedeterminationofrateofeffusionismucheasiercomparedtotherateofdiffusion.
Therefore,Dalton’slawofeffusionisoftenusedtofindthemolecularmassofagivengas.
(P,Tconstant)

17. Kinetic Molecular Theory of Gases
MaxwellandBoltzmann(1859)developedamathematicaltheorytoexplainthe
behaviourofgasesandthegaslaws.Itisbasedonthefundamentalconceptthat
agasismadeofalargenumberofmoleculesinperpetualmotion.Hence,the
theoryiscalledthekineticmoleculartheoryorsimplythekinetictheoryofgases
(Thewordkineticimpliesmotion).Thekinetictheorymakesthefollowingassumptions.
Assumptions of the Kinetic Molecular Theory
(1)Agasconsistsofextremelysmalldiscreteparticlescalledmolecules
dispersedthroughoutthecontainer.
Theactualvolumeofthemoleculesisnegligiblecomparedtothetotalvolumeofthe
gas.Themoleculesofagivengasareidenticalandhavethesamemass(m).
(2)Gasmoleculesareinconstantrandommotionwithhighvelocities.
Theymoveinstraightlineswithuniformvelocityandchangedirectiononcollisionwith
othermoleculesorthewallsofthecontainer.

18. (3)Thedistancesbetweenthemoleculesareverylargeanditisassumed
thatVanDerWaalsattractiveforcesbetweenthemdonotexist.
Thus,thegasmoleculescanmovefreely,independentofeachother.
(4)Allcollisionsareperfectlyelastic.Hence,thereisnolossofthekinetic
energyofamoleculeduringacollision.
(5)Thepressureofagasiscausedbythehitsrecordedbymolecules
onthewallsofthecontainer.
(6)Theaveragekineticenergy(1/2mv
2
)ofthegasmoleculesisdirectly
proportionaltoabsolutetemperature(Kelvintemperature).Thisimpliesthat
theaveragekineticenergyofmoleculesisthesameatagiven
temperature.

19. How Does an Ideal Gas Differ from Real Gases?
Agasthatconfirmstotheassumptionsofthekinetictheoryofgasesiscalledan
idealgas.Itobeysthebasiclawsstrictlyunderallconditionsoftemperatureandpressure.
Therealgasesashydrogen,oxygen,nitrogenetc.,areopposedtotheassumptions(1),
(3)and(4)Statedabove.Thus:
(a)Theactualvolumeofmoleculesinanidealgasisnegligible,whileinarealgasitis
appreciable.
(b)Therearenoattractiveforcesbetweenmoleculesinanidealgaswhiletheseexistina
realgas.
(c)Molecularcollisionsinanidealgasareperfectlyelasticwhileitisnotsoinarealgas.
Forthereasonslistedabove,realgasesobeythegaslawsundermoderateconditionsof
temperatureandpressure.Atverylowtemperatureandveryhighpressure,theclauses
(1),(3)and(4)ofkinetictheorydonothold.Therefore,undertheseconditionstherealgas
esshowconsiderabledeviationsfromtheidealgasbehaviour.

20. Different Kinds of Velocities
In our study of kinetic theory we come across two different kinds of molecular velocities:
(1)The Average velocity (V)
(2)The Root Mean Square velocity (µ)
Average Velocity
Let, there be n molecules of a gas having individual velocities v
1, v
2, v
3..... v
n. The ordin
ary average velocity is the arithmetic mean of the various velocities of the molecules,
RootMeanSquareVelocity
Ifv
1,v
2,v
3.....v
narethevelocitiesofnmoleculesinagas,µ
2
,themeanofthesquare
sofallthevelocitiesis-
µisthustheRoot
MeanSquarevelocity
orRMS velocity.
Itisdenotedbyu.

21. ThevalueoftheRMSofvelocityu,atagiventemperaturecanbecalculatedfrom
theKineticGasEquation.
BysubstitutingthevaluesofR,TandM,thevalueofu(RMSvelocity)canbe
determined.RMSvelocityissuperiortotheaveragevelocityconsideredearlier.
Withthehelpofu,thetotalKineticenergyofagassamplecanbecalculated.

22. Derivation of Kinetic Gas Equation
Startingfromthepostulatesofthekineticmoleculartheoryofgaseswecandevelop
animportantequation.ThisequationexpressesPVofagasintermsofthemolecular
mass,numberofmoleculesandmolecularvelocity.Thisequationwhichweshall
nameastheKineticGasEquationmaybederivedbythefollowingclauses.
Letusconsideracertainmassofgas
enclosedinacubicboxatafixedtem
perature.
Supposethat:
Thelengthofeachsideofthebox
=lcm
Thetotalnumberofgasmolecules
=N
Themassofonemolecule=m
Thevelocityofamolecule=v

23. Thekineticgasequationmaybederivedbythefollowingsteps:
(1)ResolutionofVelocity‘v’ofaSingleMoleculeAlongX,YandZAxes
Accordingtothekinetictheory,amoleculeofagascanmovewithvelocity‘v’inany
direction.
Velocityisavectorquantityandcanberesolvedintothecomponentsv
x,v
y,v
zalong
theX,YandZaxes.Thesecomponentsarerelatedtothevelocityvbythefollowinge
xpression.
v
2
= v
x
2
+ v
y
2
+ v
z
2
Nowwecanconsiderthemotionofasinglemoleculemovingwiththecomponent
velocitiesindependentlyineachdirection.

24. (2) The Number of Collisions per Second on Face ‘A’ Due to One Molecule
ConsideramoleculeismovinginOXdirectionbetweenoppositefacesAandB.
ItwillstrikethefaceAwithvelocityv
xandreboundwithvelocity–v
x.Tohitthesame
faceagain,themoleculemusttravellcmtocollidewiththeoppositefaceBandthen
againlcmtoreturntofaceA.
Therefore,thetimebetweentwocollisionsonfaceA=2l/v
xseconds
ThenumberofcollisionspersecondonfaceA=v
x/2l
(3) The Total Change of Momentum on All Faces of the Box Due to One Molecule
Only
EachimpactofthemoleculeonthefaceAcausesachangeofmomentum
(mass×velocity):
Themomentumbeforetheimpact=mv
x
Themomentumaftertheimpact=m(–v
x)
So,Thechangeofmomentum=mv
x–(–mv
x)
=2mv
x

25. But, the number of collisions per second on face A due to one molecule = v
x/2l
Therefore, the total change of momentum per second on face A caused by one
molecule,
ThechangeofmomentumonboththeoppositefacesAandBalongX-axiswouldbed
oublei.e.,2mv
x
2
/l
similarly,thechangeofmomentumalongY-axisandZ-axiswillbe2mv
y
2
/land2mv
z
2
/lr
espectively.
Hence, the overall change of momentum per second on all faces of the box will be

26. (4) Total Change of Momentum Due to Impacts of All the Molecules
on All Faces of the Box
Suppose, there are N molecules in the box each of which is moving with a different
velocity v
1, v
2, v
3, etc. The total change of momentum due to impacts of all the
molecules on all faces of the box
Multiplying and dividing by n, we have

27. (5) Calculation of Pressure from Change of Momentum; Derivation of
Kinetic Gas Equation
Since force may be defined as the change in momentum per second, we can write
Since l
3
is the volume of the
cube, V, we have
Thisisthefundamentalequationofthekineticmoleculartheoryofgases.Itiscalled
thekineticgasequation.Thisequationalthoughderivedforacubicalvessel,is
equallyvalidforavesselofanyshape.Theavailablevolumeinthevesselcouldwell
beconsideredasmadeupofalargenumberofinfinitesimallysmallcubesforeachof
whichtheequationholds.

28. Collision Properties
The Mean Free Path
Inthederivationofkineticgasequationwedidnottakeintoaccountcollisionsbetween
molecules.Themoleculesinagasareconstantlycollidingwithoneanother.Thetrans
portpropertiesofgasessuchasdiffusion,viscosityandmeanfreepathdependon
molecularcollisions.
Atagiventemperature,amoleculetravelsinastraightlinebeforecollisionwith
anothermolecule.Thedistancetravelledbythemoleculebeforecollisionistermed
freepath.Thefreepathforamoleculevariesfromtimetotime.Themeandistance
travelledbyamoleculebetweentwosuccessivecollisionsiscalledtheMean
FreePath.Itisdenotedby??????.
Where,nisthenumberofmoleculeswithwhichthemoleculecollides.Evidently,the
numberofmolecularcollisionswillbelessatalowerpressureorlowerdensityand
longerwillbethemeanfreepath.
Ifl
1,l
2,l
3arethefreepathsforamoleculeofagas,its
meanfreepath-

29. The Collision Diameter
Whentwogasmoleculesapproachoneanother,theycannotcomecloserbeyonda
certaindistance.
Theclosestdistancebetweenthecentresofthetwomoleculestakingpartina
collisioniscalledtheCollisionDiameter.Itisdenotedby??????.
Wheneverthedistancebetweenthecentresoftwomoleculesis??????,acollisionoccurs.
Thecollisiondiameterisobviouslyrelatedtothemeanfreepathofmolecules.
Thesmallerthecollisionormoleculardiameter,thelargeristhemeanfreepath.

30. The Collision Frequency
Thecollisionfrequencyofagasisdefinedas:Thenumberofmolecularcollisions
takingplacepersecondperunitvolume(c.c.)ofthegas.
LetagascontainNmoleculesperc.c.Fromkineticconsiderationithasbeen
establishedthatthenumberofmolecules,n,withwhichasinglemoleculewill
collidepersecond,isgivenbytherelation,
where v = average velocity; σ = collision diameter.
If the total number of collisions taking
place per second is denoted by Z, we have,
Sinceeachcollisioninvolvestwomolecules,thenumberofcollisionpersecondperc.c.
ofthegaswillbeZ/2.
Hencethecollisionfrequency
Evidently, the collision frequency of
a gas increases with increase in
temperature, molecular size and
the number of molecules per c.c.

31. Deviations from Ideal Behaviour
AnidealgasisonewhichobeysthegaslawsorthegasequationPV=RTatallpressuresand
temperatures.However,nogasisideal.Almostallgasesshowsignificantdeviationsfromthe
idealbehaviour.Thus,thegasesH
2,N
2andCO
2whichfailtoobeytheideal-gasequationare
termednonidealorrealgases.
Compressibility Factor
The extent to which a real gas departs from the ideal behaviour may be depicted in terms of a
new function called the compressibility factor, denoted by Z. It is defined as,Z=PV/RT
Foranidealgas,Z=1anditisindependentoftemperatureandpressure.Thedeviations
fromidealbehaviourofarealgaswillbedeterminedbythevalueofZbeinggreaterorless
than1.
Thedifferencebetweenunityandthevalueofthecompressibilityfactorofagasisa
measureofthedegreeofnon-idealityofthegas.
Forarealgas,thedeviationsfromidealbehaviourdependonpressureandtemperature.
Thiswillbeillustratedbyexaminingthecompressibilitycurvesofsomegasesdiscussedbelow
withthevariationofpressureandtemperature.

32. Effect of Pressure Variation on Deviations
Atverylowpressure,forallthese
gasesZisapproximatelyequaltoone.
Thisindicatesthatatlowpressures
(upto10atm),realgasesexhibit
nearlyidealbehaviour.Asthe
pressureisincreased,H
2showsa
continuousincreaseinZ(fromZ=1).
Thus,theH
2curveliesabovetheideal
gascurveatallpressures.
For N
2and CO
2, Z first decreases
(Z < 1).
Itpassesthroughaminimumandthen
increasescontinuouslywithpressure
(Z>1).ForagaslikeCO
2thedipin
thecurveisgreatestasitismost
easilyliquefied.
Thefollowingfigureshowsthecompressibility
factor,Z,plottedagainstpressureforH
2,N
2and
CO
2ataconstanttemperature
Fig. Z versus P plots for H
2, N
2and CO
2at 300 K.

33. Effect of Temperature on Deviations
Itisclearfromtheshapeofthecurvesthat
thedeviationsfromtheidealgas
behaviourbecomelessandlesswith
increaseoftemperature.Atlower
temperature,thedipinthecurveislarge
andtheslopeofthecurveisnegative.That
is,Z<1.Asthetemperatureisraised,
thedipinthecurvedecreases.
Atacertaintemperature,theminimumin
thecurvevanishesandthecurveremains
horizontalforanappreciablerangeof
pressures.
At this temperature, PV/RT is almost unity and the Boyle’s law is obeyed. Hence this
temperature for the gas is called Boyle’s temperature. The Boyle temperature of each gas
is characteristic e.g., for N
2it is 332 K.
Figure shows plots of Z or PV/RT against P
for N
2at different temperatures.
Fig. Z versus P plots for N2 at different temperatures.

34. Pressure Correction
Amoleculeintheinteriorofagasisattractedbyothermoleculesonallsides.Theseattractivefor
cescancelout.Butamoleculeabouttostrikethewallofthevesselisattractedbymoleculeson
onesideonly.Hence,itexperiencesaninwardpull(Figure).Therefore,itstrikesthewallwithred
ucedvelocityandtheactualpressureofthegas,P,willbelessthantheidealpressure.Iftheact
ualpressureP,islessthanP
idealbyaquantityp,wehave,P=P
ideal–p
Or, P
ideal=P+p
pisdeterminedbytheforceofattractionbetweenmolecules(A)strikingthewallofcontainer
andthemolecules(B)pullingtheminward.
Thenetforceofattractionis,therefore,proportionaltotheconcentrationof(A)typemolecules
andalsoof(B)typeofmolecules.Thatis,
Where,nistotalnumberofgasmoleculesinvolumeVand‘a’isproportionalityconstantcharact
eristicofthegas.Thus,thepressurePintheidealgasequationiscorrectedas:
for n moles of gas.

35. Conclusions
Fromtheabovediscussionsweconcludethat:
(1)Atlowpressuresandfairlyhightemperatures,realgasesshownearlyidealbehaviourand
theideal-gasequationisobeyed.
(2)Atlowtemperaturesandsufficientlyhighpressures,arealgasdeviatessignificantlyfrom
idealityandtheideal-gasequationisnolongervalid.
(3)Thecloserthegasistotheliquefactionpointthelargerwillbethedeviationfromtheideal
behaviour.
Explanation of Deviations –Van Der Waals Equation
VanDerWaals(1873)attributedthedeviationsofrealgasesfromidealbehaviourtotwoerroneous
postulatesofthekinetictheory.Theseare:
(1)Themoleculesinagasarepointmassesandpossessnovolume.
(2)Therearenointermolecularattractionsinagas.
Therefore,theidealgasequationPV=nRTderivedfromkinetictheorycouldnotholdforrealgases.
VanDerWaalspointedoutthatboththepressure(P)andvolume(V)factorsintheidealgas
equationneededcorrectioninordertomakeitapplicabletorealgases.

36. Volume Correction
Thevolumeofagasisthefreespaceinthe
containerinwhichmoleculesmoveabout.
Volume(V)ofanidealgasisthesameas
thevolumeofthecontainer.Thedot
moleculesofidealgashavezerovolumeand
theentirespaceinthecontainerisavailable
fortheirmovement.However,VanDerWaals
assumedthatmoleculesofarealgasare
rigidsphericalparticleswhichpossessa
definitevolume.
Thevolumeofarealgasis,therefore,idealvolumeminusthevolumeoccupiedby
gasmolecules.(Fig.)
Ifbistheeffectivevolumeofmoleculespermoleofthegas,thevolumeintheidealgas
equationiscorrectedas:(V–b)
Fornmolesofthegas,thecorrectedvolumeis:(V–nb)
Where,bistermedtheexcludedvolumewhichisconstantandcharacteristicforeachgas.
Fig. Volume of a Real gas.

37.  RTnbV
V
an
p 









2
2 RTbV
V
a
p 









2 Substituting the values of corrected pressure and volume in the ideal gas equation,
PV = nRT, we get-
ThisisknownasVanDerWaalsequationfornmolesofagas.For1moleofagas
(n=1),VanDerWaalsequationbecomes
ConstantsaandbinVanDerWaalsequationarecalledVanDerWaalsconstants.
Theseconstantsarecharacteristicofeachgas.
Van Der Waals Equation

38. Liquefaction of Gases –Critical Phenomenon
Agascanbeliquefiedbyloweringthetemperatureandincreasingthepressure.
Atlowertemperature,thegasmoleculeslosekineticenergy.Theslowmovingmoleculesthenaggregatedueto
attractionsbetweenthemandareconvertedintoliquid.Thesameeffectisproducedbytheincreaseofpressure.
Thegasmoleculescomecloserbycompressionandcoalescetoformtheliquid.
Andrews(1869)studiedtheP–Tconditionsofliquefactionofseveralgases.Heestablishedthatforeverygas
thereisatemperaturebelowwhichthegascanbeliquefiedbutaboveitthegasdefiesliquefaction.This
temperatureiscalledthecriticaltemperatureofthegas.
Thecriticaltemperature,Tc,ofagasmaybedefinedasthattemperatureabovewhichitcannotbeliquefied
nomatterhowgreatthepressureapplied.
Thecriticalpressure,Pc,istheminimumpressurerequiredtoliquefythegasatitscriticaltemperature.
Thecriticalvolume,Vc,isthevolumeoccupiedbyamoleofthegasatthecriticaltemperatureandcritical
pressure.
Tc,PcandVcarecollectivelycalledthecriticalconstantsofthegas.Allrealgaseshavecharacteristiccritical
constants.
Atcriticaltemperatureandcriticalpressure,thegasbecomesidenticalwithitsliquidandissaidtobein
criticalstate.Thesmoothmergingofthegaswithitsliquidisreferredtoasthecriticalphenomenon.

39. Table: The critical constants of some common gases
Gas Critical temperature (K) Critical pressure (atm) Critical volume (ml/mole)
Helium 5.3 2.26 57.8
Hydrogen 33.2 12.8 65.0
Nitrogen 126.0 33.5 90.1
Oxygen 154.3 50.1 74.4
Carbon dioxide 304.0 72.9 94.0
Ammonia 405.5 111.5 72.1
Chlorine 407.1 76.1 123.8
Methods of Liquefaction of Gases
Ifagasiscooledbelowitscriticaltemperatureandthensubjectedtoadequate
pressure,itliquefies.Thevariousmethodsemployedfortheliquefactionofgases
dependonthetechniqueusedtoattainlowtemperature.Thethreeimportantmethods
are:
1.Faraday’smethodinwhichcoolingisdonewithafreezingmixture
2.Linde’smethodinwhichacompressedgasisreleasedatanarrowjet(Joule-Thom
soneffect)
3.Claude’smethodinwhichagasisallowedtodomechanicalwork

40. Faraday’s Method
Faraday(1823)usedfreezing
mixturesoficewithvarioussalts
forexternalcoolingofgases.The
meltingoficeanddissolutionof
saltsbothareendothermic
processes.Thetemperatureofthe
mixtureislowereduptoa
temperaturewhenthesolution
becomessaturated.
FaradaysucceededinliquefyinganumberofgasessuchasSO
2,CO
2,NOandCl
2by
thismethod.HeemployedaV-shapedtubeinonearmofwhichthegaswasprepared.
Intheotherarm,thegaswasliquefiedunderitsownpressure.Thegasesliquefiedby
thismethodhadtheircriticaltemperatureaboveorjustbelowtheordinaryatmospheric
temperature.TheothergasesincludingH
2,N
2andO
2havinglowcriticalpointscould
notbeliquefiedbyFaraday’smethod.

41. Linde’sMethod
Linde(1895)usedJouleThomsoneffectasthe
basisfortheliquefactionofgases.Whena
compressedgasisallowedtoexpandinto
vacuumoraregionoflowpressure,it
producesintensecooling.Inacompressed
gasthemoleculesareverycloseandthe
attractionsbetweenthemareappreciable.Asthe
gasexpands,themoleculesmoveapart.In
doingso,theintermolecularattractionmustbe
overcome.Theenergyforitistakenfromthe
gasitselfwhichistherebycooledandbecome
liquid.
Lindeusedanapparatusworkedontheaboveprinciplefortheliquefactionofair.Puredryairis
compressedtoabout200atmospheres.Itispassedthroughapipecooledbyarefrigeratingliquidsuch
asammonia.Here,theheatofcompressionisremoved.Thecompressedairisthenpassedintoaspiral
pipewithajetatthelowerend.Thefreeexpansionofairatthejetresultsinaconsiderabledropof
temperature.Thecooledairwhichisnowataboutoneatmospherepressurepasseduptheexpansion
chamber.Itfurthercoolstheincomingairofthespiraltubeandreturnstothecompressor.Byrepeating
theprocessofcompressionandexpansion,atemperaturelowenoughtoliquefyairisreached.The
liquefiedairiscollectedatthebottomoftheexpansionchamber.

42. Claude’s Method
Thismethodforliquefactionofgasesis
moreefficientthanthatofLinde.Here,
alsothecoolingisproducedbyfree
expansionofcompressedgas.Butin
addition,thegasismadetodoworkby
drivinganengine.Theenergyforit
comesfromthegasitselfwhichcools.
Thus,inClaude’smethodthegasiscoolednotonlybyovercomingtheintermolecularforces
butalsobyperformanceofwork.
Thatiswhy,thecoolingproducedisgreaterthaninLinde’smethod.
Puredryairiscompressedtoabout200atmospheres.Itisledthroughatubecooledbyrefrigerating
liquidtoremoveanyheatproducedduringthecompression.Thetubecarryingthecompressedairthen
entersthe‘expansionchamber’.Thetubebifurcatesandapartoftheairpassesthroughtheside-tube
intothecylinderofanengine.Hereitexpandsandpushesbackthepiston.Thus,theairdoes
mechanicalworkwherebyitcools.Theairthenenterstheexpansionchamberandcoolstheincoming
compressedairthroughthespiraltube.Theairundergoesfurthercoolingbyexpansionatthejetand
liquefies.Thegasescapingliquefactiongoesbacktothecompressorandthewholeprocessis
repeatedoverandoveragain.

43. Uses of Liquefied Gas
1.To produce low temperature
2.Commercial preparation of Oxygen and Nitrogen
3.Liquid He can be used for intense cold
4.Liquid Cl
2can be used for bleaching purpose
5.Liquid O
2and He can be used for welding purpose
Determination of Molecular Weight
Theidealgasequationis-
PV=nRT
=(w/M)RT
So,P=(w/V).(RT/M)
=d(RT/M)
Where,disthedensityofthegas.
Thus,ifaliquidcanconvenientlybeconvertedintothevapourstatethenthe
measurementofthedensityofvapourwillyieldthemolecularweightofthe
liquidinthevapourstate.
There are some methods to determine the
molecular weight of gas such as-
1.Regnault’smethod
2.Duma’s method
3.Hofmann’s method
4.Victor Meyer’s method
5.The Buoyancy method

44. Victor Meyer’s Method
Thisisperhapsthemostcommonmethodfor
thedeterminationofthevapourdensityof
substanceswhichareliquidorsolidatordinary
temperature.Inthismethodthevolumeofa
knownweightofthesubstanceisdetermined
bymeasuringthevolumeofairdisplacedbyit.
Whentheconditionbecomessaturatedtemperatureisnotedonathermometerinsidethe
vapourjacket.Thevolumeofvapourisnotedinthegraduatedtubeuponthemercury
level.Thedifferenceofmercurylevelbeforeandafterintroductionofvapouristhevolume
ofthevapour.Sincethevolumeofthevapourmaybeeasilyreadfromthegraduatedtube
andtheweightofliquidintroducedisknownaswellasthedensitysothemolecularweight
ofthevapour(thesubstancebeingexamined)maybecalculated.

45. THANK YOU