Download Real Gases

The IDE, PV = nRT is accurate for gases at low densities. At high pressures and
low temperatures the finite molecular volume and inter-particle interactions can
no longer be omitted. Molecules are not entities with point masses. Sufficiently
brought close together, the closer molecules do exert attractive and repulsive
forces (potentials) on one another.
Real Gases
A major limitation of the ideal gas law is that it does not predict that a gas can
be liquefied under appropriate conditions.
Chapter 7
Properties
To account for such non-idealities for each specific gas, a different P-V-T
relationship (Real gas equation ) is required.
Real gas equation (high density, low temperature)  ideal gas equation (low
density, high temperature).
2
Van der Waals constants

n2 a 
 P  2  V  nb   nRT
V 

b (L/mol)
17.71
0.1065
Acetic anhydride
20.158
0.1263
Acetone
16.02
0.1124
Acetonitrile
17.81
0.1168
Acetylene
4.516
0.0522
Ammonia
4.170
0.0371
Argon
1.355
0.03201
Benzene
18.24
0.1154
Bromobenzene
28.94
0.1539
Butane
14.66
0.1226
Carbon dioxide
3.640
0.04267
11.77
0.07685
1.505
0.03985
Carbon disulfide
Carbon monoxide
Carbon tetrachloride
https://en.wikipedia.org/wiki/Van_der_Waals_constants_(data_page)
2
a (L bar/mol )
Acetic acid
19.7483
0.1281
Chlorine
6.579
0.05622
Chlorobenzene
25.77
0.1453
Van der Waals (VDW) equation of state:

n2 a 
 P  2  V  nb   nRT
V 

Redlich-Kwong (RW) equation of state:
Parameters a and b in the equations of state shown above have different
values for a given gas in the two equations.
Beattie-Bridgeman equation of state
IDE
Dots - accurate data
from the NIST Chemistry
webbook
VDW
Redlich-Kwong equation is
a better fit of data.
Virial equation of state
virial coefficients
RWE
First, second, third, … virial coefficients. Virial equation gives the best fit.
PV  nRT

n2 a 
 P  2  V  nb   nRT
V 

The van der Waals equation fails to
reproduce the gas-phase properties
in the region of the phase diagram in
which liquid and vapor coexist.
The oscillations that are predicted by
the calculated isotherms are unreal
because they predict that the volume of
the system will decrease as the
pressure decreases.
In these two-phase regions it is usual to replace the calculated isotherm by
a horizontal line in what is called the Maxwell construction.
CO2 VDW Isotherms Maxwell Construction
VDW Isotherms
Note: the oscillation region, unreal.
critical temperature Tc
g
l
inflection
 2 P 
 P 
 V    V 2   0

TC 
TC
lco-exist g
a
real
real
Compressibility Factor, z: z  Vmideal  PVm
Vm
z measures the error in P-V curves
from the ideal gas law, w.r.t. the
VDW or RK equations.
The P-V plots would depend on
temperature.
Ideal gas, z = 1 for all P and Vm.
z > 1, the real gas exerts a greater
pressure than the ideal gas
for the same values of T and Vm…..
RT
b
c
d
Dots - accurate data from the NIST Chemistry
webbook
N2
z>1
z<1
Ideal gas law obeyed
if P is sufficiently small.
VDW and RW equations generate
satisfactory P-V isotherms for real gases
only in the single-phase gas region (T > Tc)
and for densities well below the critical
density, c = M/Vmc.
The dots calculated Vm from the NIST Chemistry Workbook.
z - the two equations (VDW and RW) are not in quantitative agreement with
accurate results the trends of dependence of z on P for different T values are
correct.
@ T = 200 K, z initially decreases with pressure. The compression factor only
becomes greater than the ideal gas value of one for pressures in excess of
200 bar. For T = 400. K, z increases linearly with T. This functional
dependence is predicted by the Redlich-Kwong equation. The van der Waals
equation predicts that the initial slope is zero, and slow increase with P.
For both temperatures, z1 as P  0, i.e. ideal gas law is obeyed if pressure is
sufficiently small.
It can be shown that for VDW equation the initial slope of the z vs. P curve is
1
 z 
lim 
  RT
P  0 P

T
a

 b  RT


;

a
~ attraction
RT
b ~ repulsion
Boyle temperature is the temperature at which a non ideal gas behaves
mostly like an ideal gas.
and will be zero (i.e. z = 1 for all P ), if b = a/RT.
At the Boyle temperature, the compressibility factor z = 1 and,
The corresponding temperature is termed
the Boyle temperature TB.
a
~ attraction
RT
 b ~ repulsion
Reminder: a and b are substance
dependent. TB is different for each gas.
1
 z 
lim 
  RT
P 0 P

T
a

 b  RT


;

a
~ attraction
RT
z > 1 repulsive forces dominate
z < 1 attractive forces dominate
 z 
T above TB 
  0 as P  0
 P T
 z 
T below TB 
  0 as P  0
 P T
 z 
T around TB 
 ~ 0 as P  0
 P T
b ~ repulsion
VDW (solid lines): T = 400K
110
r>a
400
a>r
735
Law of Corresponding States
Calculation of z requires different parameters for each gas. It is useful to derive
an equation of state for real gases that does not explicitly contain material
dependent parameter and universally valid. Real gases differ from ideal gas
because of the differences in their molecular volumes and the strength of the
attractive potential.
The critical temperature is a measure of the strength of the attractive potential.
The critical volume is a measure of the molecular volume. The technique is to
use express the equation of state in terms of critical values as dimensionless
reduced variables.
Pr 
Example Problem 7.1
V
P
; Vmr  m ;...
Pc
Vmc
The law of corresponding states is obeyed by many gases for
reduced temperatures Tr > 1.
Using z, the error in assuming that
the pressure can be calculated
using the ideal gas law can be
defined as,
It is not exactly universal because the material-dependent quantities a and b
enter through values of Pc, Tc, and Vc.
Error 
z 1
 100
z
Error 
Vactual  Videal
z 1
 100 
 100
Vactual
z
Implementation of Law of Corresponding States:
The compression factor at the critical point for the Van der Waals equation of
state is independent of a and b and same for all gases.
For a given gas at specific P and T values the Pr and Tr is calculated.
The value for z can then be read off the curves of z vs. Pr .
The z for specific values of Pr and Tr is used to estimate the error in using
the ideal gas law.
Error 
= 0.375
z  1 V  Videal

 V  zVideal
z
V
A comparison of this prediction of zc with experiment is in qualitative
agreement. For RW equation zc = 0.333.
Pr
Tr
Fugacity, f ~ P
The pressure exerted by a real gas can be greater or less than that for an ideal
gas. The fugacity is the effective pressure a real gas exerts.
Equilibrium Constant for Real Gases
For ideal gases: G  G 0  nRT ln
   0  RT ln
P
P0
P
P0
P (ideal gas pressure)
and GR0   RT ln K
   0  RT ln
f
f0
Starting from dG,
ideal
m
dG
V
real
m
ideal
m
V
z
V
ideal
m
dP
PVmreal

RT
dGmreal  dGmideal
RT
dP :note per mole (n =1)
P
dGmideal  RT ln P
dGmreal  RT ln f
f
f0
:How is f related to P?
dGmreal  Vmreal dP
dGmideal  Vmideal dP
 zRT RT 
d (Gmreal  Gmideal )  

 dP
P 
 P
dP
d (Gmreal  Gmideal )  ( z  1) RT
P
at P  0, Gmreal  Gmideal
: keeping the same mathematical form
d (Gmreal  Gmideal )
f
 d ln  d ln 
RT
P
  fugacity coefficient
density
V real PVmreal
z  mideal 
Vm
RT
d (Gmreal  Gmideal )  (Vmreal  Vmideal )dP
dGm  Vm dP at pressure P and constant T
dGmideal 
   0  RT ln
Again starting from dGm  Vm dP
dG   SdT  VdP  dGm   S m dT  Vm dP
dGmreal  Vmreal dP
f  P as P  0
‘
‘
In terms of fugacity (real gases)
the chemical potential;
For densities with the attractive IMF dominate; Grealm< Gidealm and f < P,
for densities with the repulsive IMF; Grealm> Gidealm and f < P.
For an ideal gas f = P
d (Gmreal  Gmideal ) ( z  1)

dP
RT
P
For a real gas z should be accounted
d (Gmreal  Gmideal )
f
 d ln  d ln  :previous slide
RT
P
d (Gmreal  Gmideal ) ( z  1)

dP
RT
P
f ( z  1)
d ln 
dP
P
P
P
P
f
 z 1 
 z 1 
ln   
 dP  ln    
 dP
P 0 P 
P 
0
f P
Tr
The fugacity coefficients for H2, N2, and
NH3 are plotted as a function of the partial
pressure of the gases for T = 700K.
The fugacity coefficient is plotted as a function of the reduced
pressure for the indicated values of the reduced temperature, Tr.
Fugacity and the Equilibrium Constant for Real Gases
For a given gas at specific P and T values the Pr and Tr is calculated.
The value for  can then be read off the curves of  vs. Pr . The  for specific
values of Pr and Tr can be used to estimate f of any gaseous species.
Once the I for all gaseous species are found, KP can be written in terms of Kf.
‘
Kf 
f C ' f D  C ' D PC ' PD

f A fC  A  C PA PC
Kf 
 C ' D
 P 
K P and K x   0 
 A  C
P 
ngas
KP
P  Total pressure
Tr