Chapter 7: Properties of Real Gases

Winter 2013
Chem 254: Introductory Thermodynamics
Chapter 7: Properties of Real Gases ............................................................................................. 83
Properties of Real Gases ........................................................................................................... 83
Compression Factor Z ............................................................................................................... 86
Law of Corresponding State ...................................................................................................... 86
Fugacities (pure substances) ..................................................................................................... 87
Chapter 7: Properties of Real Gases
Properties of Real Gases
Ideal gases:
- molecules have no intrinsic volume
- molecules do not interact
This works pretty well at :
- Low pressures
- Low densities
- High temperature (kinetic energy dominates)
n RT PV for each component in gas
i
i
n RT P V
total
P x P
i i total
total
established experimentally
Today we derive this from statistical mechanics (particle in
box)
Very complicated to solve QM + Stats mech
experimentalist uses empirical laws
Van der Waals:
nRT n
P a
V nb V
2
2
Chapter 7: Properties of Real Gases 83

Winter 2013
Chem 254: Introductory Thermodynamics
Redlich-Kwong:
Virial equation of State (systematic)
2
nRT n a 1
P V nb T V nb
1 B( T) C( T)
P RT ....
V
2
m V
m V
m
BT ( ): power series in T
V
m
V
n
for a range of volumes (condensation), pressure is
the same
no way to fit against analytical function
Ideal gas: cannot capture gas liquid transition
Van der Waals
nRT n
P a
V nb V
2
2
2 2 2
P V nb V nRTV an V nb
3 2 2 2 2
PV PnbV nRTV an V an b 0
solve cubic equation for V :
x
x x 1 2 0 x 0,1, 2
2
x x 1 0 x 0,
i;
2
i
1
From VDW cubic solution construct the
flat region of graph
Chapter 7: Properties of Real Gases 84

Winter 2013
Chem 254: Introductory Thermodynamics
T c
at which
P
P
V
Tc
2
0
2
V
Tc
Do the calculation:
8a
Tc
ab , are VDW parameters
27Rb
a
Pc
2
27b
V b
mc ,
3
What if T
T
c
Increase density, but all in one supercritical
phase (no phase transition)
Chapter 7: Properties of Real Gases 85

Winter 2013
Chem 254: Introductory Thermodynamics
Compression Factor Z
V PV
Z
V RT
actual
m
ideal
m
V
Vm
n
ideal RT
Vm
P
Z : tabulated for many gases
m
Low T , low P , Z 1,
V
m
V
High T , (very) high P , Z 1,
ideal
m
V
m
attraction dominates
V
ideal
m
finite volume term dominates
Law of Corresponding State
-reduced, dimensionless variables
T
r
T
, Pr
T
c
P
, V
,
P
c
mr
V
V
m
mc ,
T , P , V
c c m,
c
= parameters for specific gas
Van der Waals has a unique form:
T
c
8T
3
Pr
3V
1
r
2
mr ,
Vmr
,
same for every gas
8a
a
, Pc
, V
2 mc ,
3b
ab , = VDW parameters
27Rb
27b
actual
Vm
For any gas: Plot Z vs T ,
ideal r
P
r
V
c
c
m
virtually the same plot for any gas (not exactly)
know T , P for gas, use numerical curve to get V
mr ,
(figure 7.8 in book)
Example:
1.00 kg of CH4
at T 200 K, P 68 , V
Chapter 7: Properties of Real Gases 86

Winter 2013
Chem 254: Introductory Thermodynamics
1 kg
16.09 10 kg mol
n
3 1
ideal nRT
Vm
17.52 L
P
Table 7.2 in book CH
4
Tc
190.56 K , Pc
45.99 bar
T
r
T
P
1.21 Pr
1.48
T
P
c
look at figure 7.8
c
Z T , P 0.63
ideal
V ZV m
0.6317.52 L 11.04 L
r
r
Two gases with the same T , P have the same V
,
r
r
mr
Fugacities (pure substances)
o
For ideal gases
P
T, P T RT
ln , where
Po
substance at pressure P
o
= 1bar. This is not really correct
o
is for the pure
Real gases
Pf
Pi
d V dP (constant T )
m
ideal RT
Vm
P
RT
d dP
P
P P
d
RT dP RT ln
Po
Po
V
m
ZV
ideal
m
ideal
V Z 1 V
m
ideal
m
RT RT
Z
1
P P
dP
RT
d VmdP RT RT Z 1
dP
P
P
Pf
P P P
Pi Po Po Po
P
RT ln RT
ln
Po
= fugacity coefficient (calculated from integral)
Chapter 7: Properties of Real Gases 87

Winter 2013
Chem 254: Introductory Thermodynamics
Fugacity: f P
f is like pressure, correction for non ideal gas behavior
P
T, P T, Po
RT
ln
Po
f
T, Po
RT
ln
Po
Back to chemical equilibrium
G
TP ,
T,
P
i
i
o
f
i
rGT
RT ln
Po
o
0 G RT ln K
r T f
o
f
i
rGT
RT ln
Po
o
P
i
rGT
RT ln
Po
i
K
f
i
i
K
P
i
o
P
i
o
P
i
i
rGT RT ln rGT
RT ln
RT ln
Po
Po
i
o
P
rGT RT ln RT ln RT ln xi
Po
i
i
K
x
K
K
i
f
Kp
i
ngas
i
P
f
Kx
i
Po
i
o
RT ln K G Where
f r T
G is found from G f
r
o
T
Chapter 7: Properties of Real Gases 88