Chapter 6. The EoS/GE mixing rules for cubic equations of
state
Problem 1. Infinite Pressure Limit
Table 6.2 presents the excess Gibbs free energy expression for the van der Waals
(vdW) Equation of State (EoS).
(i) Following a procedure similar to that shown in Appendix 6.A for the SRK
equation of state, show that the excess Gibbs free energy equation at infinite
pressure obtained from the van der Waals equation is given by the following
equation:


g E    xi  i   
 i

6.28
a

b
(ii) Which activity coefficient model can be derived from equation 6.28?
(iii) Consider a binary mixture. Derive the expression for the activity coefficient of
compound 1 based on equation 6.28 in the following cases - types of mixing
rules:
6.29
   xi  i
i
and
   xi x j  ij
i
 ij   i  j
6.30
j
Comment on the results and mention what type of mixtures these models can
potentially be applied to. Give some examples.
(iv) Derive for the vdW EoS at infinite pressure the mixing rule for the energy
parameter, similar to that for SRK shown in equation 6.27 (and Table 6.2).
Problem 2. A special version of the Huron-Vidal mixing rule
Huron and Vidal, Fluid Phase Equilibria, 1979, 3, 255 showed that using a special
version of the NRTL equation (see Chapter 5, Appendix 5.B) and under certain
assumptions, the classical van der Waals one fluid mixing rule can be recovered for
the energy parameter( a   ai a j ). Show that this is indeed the case for the
i
j
following values of the NRTL coefficients:
a12  0
g i  (ln 2)
g ij  2
ai
bi
bi b j
bi  b j
6.31
g i g j 1  k ij 
1
Problem 3. Zero Pressure Limit via constant packing fraction
An assumption that approximates the zero pressure limit is the so-called “constant
volume packing fraction” for pure compounds and for the mixture expressed as ui =
Vi/bi = u = V/b.
(i) Derive the expression of the excess Gibbs free energy from the SRK Equation of
State at the (near) zero pressure limit using the constant volume packing fraction
assumption.
(ii) Derive based on (i) the expression for the mixing rule of the energy parameter and
compare it to that for the PSRK and MHV1 mixing rules (Table 6.4). Typical
packing fraction (u-) values are between 1.1 and 1.3 (see Table 6.7).
Problem 4. The zero-pressure limit: the exact and the MHV1/MHV2
approximations
For the SRK EoS, the fugacity coefficient (for a pure compound or for a mixture) is
given by the expression:
a
 P(V  b)  PV
V b 
ln    ln 

1
ln 


bRT  V 
 RT  RT
6.32
1. Show that at zero (reference) pressure and using reduced variables, SRK can be
written as:
1


0
6.33
u 0  1 u 0 (u 0  1)
where u0 is the value of V/b at zero pressure and the reduced energy parameter is
a
defined as:  
.
bRT
2. Show that the liquid volume at zero pressure is given by the equation:
u0 
  1   2  6  1
6.34
2
which has a liquid-like solution for  lim  3  2 2 .
3. Show that at zero reference pressure the fugacity is expressed by the following
equation via the so-called q-function:
 u 1 
 f 
6.35
q    1  ln  u0  1   ln  0   ln  0   ln b
 RT 
 u0 
This q-expression represents the so-called “exact” q-equation.
and that an exact EoS/GE mixing rule at zero pressure is given by equation 6.3:
2
 gE

 RT
*
b

   xi ln 
i
0
 bi

  q e     xi qie  i 

6.36
4. The q-function can be approximated by a linear or a quadratic equation of the
energy parameter, i.e. (see also left column of Table 6.4):
6.37
q( )  qo  q1
q( )  qo  q1  q 2 2
6.38
Then show that the MHV1 and MHV2 mixing rules can be respectively derived for
the energy parameter and they are expressed by the equations (see also Table 6.4):
MHV1:
E ,*
 b 
1 g
 
  xi ln    xi  i
q1  RT
i
 bi   i
6.39
MHV2:
b



  g E ,* 
   xi ln 
q1    xi i   q2  2   xi i2   
i
i



  RT  i
 bi



6.40
Problem 5. Activity coefficient expressions from cubic EoS using various mixing
rules
The fugacity coefficient of SRK is:
Pure compound or mixture:
a
 P(V  b)  PV
V b 
ln    ln 

1
ln 


bRT  V 
 RT  RT
6.41
Compound i in a mixture:
ai
 P(V  b)  bi PV
V  b 
ln ˆ i   ln 

(

1
)

ln


bi RT  V 
 RT  b RT
6.42
where the composition derivatives of the co-volume and energy parameters depend on
the mixing rules used and they are defined as:
bi 
 (nb)
ni
6.43
T , P ,n j  i
ai
  na 



bi RT ni  bRT 
6.44
T , P ,n j  i
3
1. Show that the general expression for the activity coefficient from SRK is given by
the equation:
V  bi 
V  bi 
ai
ai
V  bi 
V  b 
ln  i  ln  i
1  i
ln  i
ln 




 V b 
 V  b  bi RT  Vi  bi RT  V 
a bVi  Vbi 

bRT V (V  b)
6.45
2. Consider a binary mixture.
i. Assuming the validity of the van der Waals one fluid mixing rules and the linear
mixing rule for the co-volume parameter, show that:
V  b 
a
a
V  b 
V  b 
 V1 b1 
ln  1  ln  1 1   1   1 1   1 ln  1 1  
  
 V b 
 V  b  b1 RT  V1  RT (V  b)  V b 
6.46
 2 x j a ji

b1  V  b 
a  j
  ln 
bRT 
a
b
V 

 


iii. Assume the validity of the van der Waals one fluid mixing rules, the linear mixing
rule for the co-volume parameter and the geometric mean rule for the cross-energy
parameter (with kij=0) [ a ij  a i a j ]. Show that at infinite dilution conditions:
ln  1
 b1 
1  
V1 
 V1  b1 
 V1  b1 
a1
a
  1  
 
 ln 
ln 
 2
 V2  b2 
 V2  b2  b1 RT 1  b2  b2 RT
 V 
2 

 V1b2  V2 b1 
 2

 V b V 
2 2 
 2
6.47
2

b1
RT
 a2
a   b 
 1  ln 1  2 

b1   V2 
 b2
iv. Assuming the validity of the linear mixing rule for the co-volume parameter and
a
a
the so-called a/b rule i.e.   xi i , show that:
b
bi
i
ln  1
 b1 
1  
V1 
 V1  b1 
 V1  b1 
a1
a
  1  
 
 ln 
ln 
 2
 V2  b2 
 V2  b2  b1 RT 1  b2  b2 RT
 V 
2 

4
 V1b2  V2 b1 
 2

 V b V 
2 2 
 2
6.48
What do you observe upon comparing equations 6.47 and 6.48? Explain why equation
6.48 represents a measure of the combinatorial-free volume (or non-residual or nonenergetic) term originating from the SRK EoS. How does this term compare to the
combinatorial or combinatorial-free volume terms of well-known models such as the
Flory-Huggins and Entropic-FV (discussed in Chapters 4 and 5)?
How should equation 6.47 be written assuming again the validity of the van der Waals
one fluid mixing rules but without making the assumption of the linear mixing rule for
the co-volume parameter ?
3.
i. Show that in the case of the Huron-Vidal mixing rule, then the activity coefficient is
given by equation 6.45 with:
ai
ln  iM
a

 i
bi RT
C
bi RT
6.49
where C=-ln2 for SRK and M indicates the external activity coefficient model
associated with the mixing rule e.g. NRTL or UNIFAC.
Assume that only the residual term of UNIFAC is used (as typically done for the
infinite pressure Huron-Vidal mixing rule). In this case, how is equation 6.49 written
for alkane mixtures ? What do you observe ?
ii. Show that in the case of the MHV1 mixing rule, then the activity coefficient is
given by equation 6.45 with:


ai
a
1

ln  iM  ln  iFH  i
bi RT q1
bi RT
ln 
FH
i
6.50
b 
b 
 ln  i   1   i 
b
b
M indicates the external activity coefficient model associated with the mixing rule e.g.
NRTL or UNIFAC.
iii. Show that in the case of the LCVM mixing rule, then the activity coefficient is
given by equation 6.45 with:

1  
ai
a
1   1   
 ln  iM  
 ln  iFH   i
 

bi RT q1  AV
AM 
 AM 
 bi RT
b 
b 
ln  iFH  ln  i   1   i 
b
b
6.51
M indicates the external activity coefficient model associated with the mixing rule e.g.
NRTL or UNIFAC.
5
4. Show that the general expression for the infinite dilution activity coefficient which
can be used for EoS/GE mixing rules, assuming the linear mixing rule for the covolume parameter is given as:
V b 
V b 
V  b 
a
a  V b  V2 b1 

ln  1  ln  1 1   1   1 1   1 ln  1 1   2  1 22

V

b
V

b
b
RT
V
b
RT
V

b
V
2 
2 
1
1
 2
 2

 2
2 2 
 2
6.52
 V2  b2 
ai


ln 
bi RT  V2 
Then, prove the following expressions for some of the well-known EoS/GE mixing
rules (in some cases some approximations are made which should be explained):
MHV1:
V b
ln  1  ln  1 1
 V 2  b2

V b
  1   1 1

 V 2  b2
 b1 
1  
V1 
a1
a

ln 
 2
b1 RT  b2  b2 RT
1  
 V2 

  ln  1M ,comb,  ln  1FH ,



 V1b2  V2 b1 
 2

V

b
V
2 2 
 2
6.53
 ln  1M ,res,
Huron-Vidal (assuming that an external activity coefficient model with only a residual
term is used):
ln  1
 b1 
1  
V1 
 V1  b1 
 V1  b1 
a1
a
  1  
 
 ln 
ln 
 2
 V 2  b2 
 V2  b2  b1 RT 1  b2  b2 RT
 V 
2 

 V1b2  V2 b1 
 2

 V b V 
2 2 
 2
6.54
 ln  1M ,res,
LCVM:
V b
ln  1  ln  1 1
 V 2  b2

V b
  1   1 1

 V 2  b2
 b1 
1  
V1 
a1
a

ln 
 2
b1 RT  b2  b2 RT
1  
 V2 

   1   
1   
 ln  1M ,comb,  
  
 ln  1FH , 

AM 

  AV
 AM 
 V1b2  V2 b1 
 2

 V b V 
2 2 
 2
6.55
  1  
 ln  1M ,res,
 

AM 
 AV
6
CHV (or  -MHV1):
V b
ln  1  ln  1 1
 V 2  b2

V b
  1   1 1

 V 2  b2
 b1 
1  
V1 
a1
a

ln 
 2
b1 RT  b2  b2 RT
1  
 V2 

  ln  1M ,comb,  1    ln  1FH ,



 V1b2  V2 b1 
 2

V

b
V
2 2 
 2
6.56
 ln  1M ,res,
How could the above expressions be divided into combinatorial-Free Volume and
energetic (residual) contributions?
Problem 6. EoS/GE mixing rules for asymmetric mixtures
1. Show that the MHV1 mixing rule shown in Table 6.4 can be equivalently
written in the form of equation 6.7.
2. Show that the LCVM mixing rule which was originally defined as shown in
equations 6.12-6.14 or 6.15 can be equivalently written in the form shown in
Table 6.6.
3. Table 6.11 below gives the van der Waals volume (r) and critical properties of
ethane, octane, decane, eicosane and CO2.
Calculate for the mixtures of ethane with octane, decane and eicosane as well as for
CO2 with octane, decane and eicosane the “combinatorial terms’ difference” in
activity coefficients (at infinite dilution) for the models: MHV1,  -MHV1, LCVM
and CHV (see expressions in Table 6.6). Compare the results between the four mixing
rules, especially as the size-asymmetry increases. What do you observe? Discuss the
results.
Hint. The “combinatorial difference” is defined as the difference between the
combinatorial term of the external activity coefficient model used minus the FloryHuggins term originated from the equation of state e.g. for MHV1 it is:
ln  1M ,  ln  1FH , (binary mixture and at infinite dilution).
Table 6.11. Some pure component parameters for selected compounds
Compound
Ethane
Octane
Decane
Eicosane
CO2
r
1.800
5.850
7.200
13.94
1.300
Tc (K)
305.4
568.8
617.6
769.0
304.2
7
Pc (bar)
48.8
24.8
21.1
11.6
73.8
Problem 7. Choice of thermodynamic models - 1
Phase diagrams (experimental data and calculations) have been presented for the
binary systems CO2/ethane, chloroform/acetone, propanol/water, methanol/benzene,
acetone/water and ethanol/water (figures 1.3, 3.2, 3.3, 3.5, 6.2, 6.4).
i. How well do you expect that cubic equations of state will perform using the van
der Waals one fluid (quadratic) mixing rules and zero interaction parameters? For
which mixtures do you expect that use of a single kij would result to large
improvements? Do you expect that SRK and PR would provide markedly
different phase equilibrium results?
ii. Mention at least three thermodynamic models which are likely to provide better
representation of the VLE for these mixtures compared to cubic EoS with the
vdW1f mixing rules. Justify your answer.
Problem 8. Choice of thermodynamic models - 2
Many methanol-alkane mixtures e.g. methanol-heptane (see figures 1.4 and 3.11)
exhibit both VLE and LLE (at low temperatures).
i. Assume that SRK or PR are used with the vdW1f mixing rules and a single kij
fitted to the VLE data. How well do you think LLE will be represented with these
cubic equations of state?
ii. Which of the classical thermodynamic models presented in Chapters 3-6 are
expected to provide reasonably good representation of both VLE and LLE?
8