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Chapter 8
Real Gases
Compression FactorsReal gases do not obey the perfect gas equation exactly. The measure of the deviation from ideality of the behavior of a real gas is expressed as the compression factor Z:
RT
PVTPZ m),( (8.1)
Real Gases
Z
P
200 K500 K
1000 K
200 K
1000 K
0 300 600 900
3
2
1
0
CH4
Z
P
H2
0 100 200 300
1.2
1
0.8
0
0oC
N2
CH4
Physical Chemistry
Real Gas Equations of State
RTbVV
aP m
m
2
2mm V
a
bV
RTP
(8.2)
van der Waals equation
Ideal Gas Law/Perfect Gas Equation:
PV = nRT (1.18)*
RTPVm
Real Gases
Physical Chemistry
Van der Waals Equation of State
RTbVV
aP m
m
))((2
RTPVm
Real Gases
Physical Chemistry
: to correct the effect of intermolecular attractive forces on the gas pressure
2m/a V
b: the volume excluded by intermolecular repulsive forces
Virial Equation of State
32
)()()(1
mmmm V
TD
V
TC
V
TBRTPV (8.4)
Redlich-Kwong Equation
2/1)( TbVV
a
bV
RTP
mmm
(8.3)
Real Gases
Physical Chemistry
Real Gas Equations of State
The limited accuracy of the data allows evaluation of only B(T) and sometimes C(T).
Virial Equation of State
])(')(')('1[ 32 PTDPTCPTBRTPVm (8.5)222 )''(,' TRCBCRTBB (8.6)
BP
RTVm (8.7)low P
mmmm
mm
RTV
a
VbRTV
a
bV
VZ
RT
PV
/1
1vdW gas
2mm V
a
bV
RTP
(8.2)
RT
Vm
Real Gases
Physical Chemistry
32
)()()(1
mmmm V
TD
V
TC
V
TBRTPV (8.4)
Power series in 1/Vm
Power series in P
Gas Mixtures
2211222
2/121211
21 )(2 bxbxbandaxaaxxaxa (8.10)
For a mixture of two gases, 1 and 2, use a two-parameter equation,
(8.11)tot
m n
VV mean molar volume
Real Gases
Physical Chemistry
x1 and x2: the mole fractions of the components
b: a weighted average of b1 and b2
a: related to intermolecular attractions
(a1a2)1/2: intermolecular interaction between gases 1 and 2
Isotherms of H2O
P
Vm
400 oC
U
RJ N
Y
374 oC300 oC200 oC
H2O
L + VL
VL
G
H
T S
K
MW
Condensation
T < 374 oC
gas condenses to liquid when PT = 300 oC
R(vapor)S(saturated vapor), P, V
S(saturated vapor)W(saturated liquid), P, V W(saturated liquid)Y(liquid), P , V
Real Gases
Physical Chemistry
t/℃
A
DC
0.00611
0.01
solid
gas
liquid
O
P /
10 5
Pa
374.2
218 atm
H2O phase diagram: P — T
99.974
1 atm
0.0024
I
R
S
Y
Tf TbT3
Real Gases
Physical Chemistry
400 oC
Condensation
T 374 oC
No amount of compression will cause the separation out of a liquid phase in equil. with the gas.
T = 374 oCCritical temperature TcCritical pressure PcCritical volume Vm,cCritical constants Isotherms of H2O
P
Vm
U
RJ N
Y
374 oC300 oC200 oC
H2O
L + VL
VL
G
H
T S
K
MW
Real Gases
Physical Chemistry
Fig. 8.3
Critical constantsCritical T (Tc), Tc(CO2)=304.2 K
Critical P (Pc), Pc(CO2)=7.38 MPa
Critical molar V (Vm,c), Vm,c(CO2)=94×10-6 m3·mol-1
Isotherms of CO2{P}
{Vm,c}
T3
c
Tc
gb a
l
T1 T2
Real Gases
Physical Chemistry
Table 8.1 Critical Constants
Species Tc / K Pc / atm Vm,c / cm3·mol-1
Ar 150.7 48.3 74.6
Ne 44.4 27.2 41.7
N2 126.2 33.5 89.5
H2O 647.1 217.8 56.0
D2O 643.9 213.9 56.2
H2S 373.2 88.2 98.5
CO2 304.2 72.88 94.0
HCl 324.6 82.0 81.0
CH3OH 512.5 80.8 117
Real Gases
Physical Chemistry
FluidThere is a continuity between the gaseous and the liquid states. In recognition of this continuity, the term fluid is used to mean either a liquid or a gas.
An ordinary liquid can be viewed as a very dense gas. Only when both phases are present in the system is there a clear-cut distinction between liquid and gaseous states.
For a single-phase liquid system it is customary to define as a liquid a fluid whose temperature is below Tc and whose molar volume is less than Vm,c.
If these two conditions are not met, the liquid is called a gas. So a further distinction between gas and vapor can be made, but these two words are used interchangeably in this book.
Real Gases
Physical Chemistry
Supercritical fluid
A supercritical fluid is one whose T and P satisfy
A supercritical fiquid usually has liquidlike density but its viscosity is much lower than typical for a liquid and diffusion coefficients in it are much higher than in liquids.
T > Tc and P > Pc
Real Gases
Physical Chemistry
Supercritical fluid
-60 -40 -20 0 20 40 60 80 100
B
5
10
15
20
25
30
35
-56.6℃ tc=31.06℃ t/℃
o
C
gas
liquid
solid
100
200
300
400
500
600
700
800
900100011001200p/MPa
P c=7.38MPa
A
0.518MPa
Supercritical CO2 is used commercially as a solvent to decaffeinate coffee.
Real Gases
Physical Chemistry CO2
Critical data and equations of state
Differentiating the van der Waals equation (8.2)
32
2
)( mmTm V
a
bV
RT
V
P
and
0
TmV
P0
2
2
TmV
P
At the critical point:
(8.12)
2mm V
a
bV
RTP
432
2 6
)(
2
mmTm V
a
bV
RT
V
P
Application of the conditions (8.12) gives
3,
2,
2
)( cmcm
c
V
a
bV
RT
and 4,
3,
3
)( cmcm
c
V
a
bV
RT
(8.13)
Real Gases
Physical Chemistry
Critical data and equations of state
Division of the first equation in (8.13) by the second yields
2,, cmcm
cc V
a
bV
RTP
From van der Waals equation:
(8.14)
4,
3,
3,
2,
3
2
)(
)(
cm
cm
cm
c
cm
c
Va
Va
bVRT
bVRT
Use of (8.15) in the first equation in (8.13) gives
3
2 ,,
cmcm
VbV
and32 27
2
4 b
a
b
RTc (8.16)
bV cm 3, (8.15)
Rb
aTc 27
8
Real Gases
Physical Chemistry
Critical data and equations of state
2,, cmcm
cc V
a
bV
RTP
Substitution of (8.15) and (8.16) into (8.14)
(8.14)
gives
22 2792
27/8
b
a
b
a
b
baPc
(8.16)
bV cm 3, (8.15)
Rb
aTc 27
8
(8.17)
Real Gases
Physical Chemistry
Critical data and equations of state
Substitution of (8.15) and (8.16) into (8.14)
Three equations for two parameters, a and b
227b
aPc
(8.16)
bV cm 3, (8.15)
Rb
aTc 27
8
(8.17)
c
c
P
TRa
64
27 22
(8.18)c
c
P
RTb
8 vdW gas
Real Gases
Physical Chemistry
Critical data and equations of state
Combination of (8.15) to (8.17)
227b
aPc
(8.16)
bV cm 3, (8.15)
Rb
aTc 27
8
(8.17)
375.08
3, c
cmcc RT
VPZ (8.19)
Real Gases
Physical Chemistry
Predicts the compressibility factor at the critical point
Van der waals equation
Critical data and equations of state
375.08
3, c
cmcc RT
VPZ (8.19)
1, c
cmc
RT
VPideal gas
c
c
c
c
P
RT
P
RTb 08664.0
3
)12( 3/1
(8.20)c
c
c
c
P
TR
P
TRa
2/52
3/1
2/52
42748.0)12(9
(8.21)
333.03
1, c
cmcc RT
VPZ (8.22)
R-K equation
Real Gases
Physical Chemistry
Van der waals equation
Selected equations of state
Equation Critical constants
Perfect gas
van der Waals
Berthelot
mV
RTP
2mm V
a
bV
RTP
2mm TV
a
bV
RTP
227b
a
2/1
33
2
12
1
b
aR
bR
a
27
8
2/1
3
2
3
2
bR
a
b3
b3
cV cTcP
Real Gases
Physical Chemistry
Selected equations of state
Equation Critical constants
Perfect gas
R-K
virial
mV
RTP
2/1)( TbVV
a
bV
RTP
mmm
2
)()(1
mmm V
TC
V
TB
V
RTP
cV cTcP
Real Gases
Physical Chemistry
The law of corresponding states
The critical constants are characteristic properties of gases
The reduced variables of a gas by dividing the actual variable by the corresponding constant.
The observation that the real gases at the same reduced volume and reduced temperature exert the same reduced pressure is called the law (principle) of corresponding states.
,c
r P
PP ,
,cm
mr V
VV ,
cr T
TT (8.27)
reduced pressure
reduced volume
reduced temperature
),( rrr TPfV (8.28)
Real Gases
Physical Chemistry
