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Chapter 9. The Cubic-Plus-Association (CPA) Equation of State
Problem 1. Expressions for the mole fraction of not-bonded molecules and for the
association term
Association equations of state like CPA and SAFT usually model alcohols using the twosite association scheme i.e. the alcohol molecules are assumed to form linear oligomers.
More specifically, these theories employ the so-called “2B” scheme (notation following
Huang and Radosz10), which implies two identical association sites with equal mole
fractions of not bonded molecules: XA=XB and AA  BB  0, AB  0 . Similarly, water
molecules are sometimes modeled using the three-site "3B" scheme, but most often using
the four-site “4C” scheme.
i. Starting from equation 9.2, prove that the mole fraction of not-bonded molecules is
given for alcohols (2B) by the equation:
1  1  4 
2 
for water using the 3B scheme by the equation:
2B:
XA 
3B:
X 
A
 1    
1   
2
 4 
4 
and for the water using the 4C scheme by the equation:
1  1  8
4C:
XA 
4 
9.78
9.79
9.80
ii. Show that the monomer fraction of a pure alcohol molecule, described using the 2Bscheme, is given as:
X monomer 
2
1  2  1  4
9.81
iii. Show that the association compressibility factor of pure alcohols (2B) is given with
CPA by the equation:
 1  1  4       ' 
Z assoc  
9.82
 1  1  4     



where ' is the derivative of the association strength with respect to density and
similarly for pure water (4C) by the equation:
 1  1  8    2(   ' ) 
Z assoc  

 1  1  8   




9.83
1
Derive equations 9.82 and 9.83 using both the “original” and the “most recent”
expressions for the association term (equation 9.1 in this chapter and equation 9.84
below for the original one) and illustrate that both approaches yield the same result.
What do you observe? Which are the advantages in employing the simplifying
equation? The “original” expression for the association contribution to the
compressibility factor is given by:
A
 1
1  X j 
assoc
Z
   xi  x j   A j  
9.84

2   i 
i
j
A  X
Problem 2. Similarities between the chemical and the perturbation theories
i. The association term of pure alcohols is given from several equations of state based
on the "chemical" theory like APACT by the equation:
 1  1  4  KRT
Z assoc  
 1  1  4  KRT




9.85
where K, the key property in the chemical theory, is the equilibrium constant. K is
related to the enthalpy ( H ) and the entropy of hydrogen bonding ( S ) via:
ln K  
H  S

RT
R
9.86
How does equation 9.85 compare to the equivalent expression from the CPA model
(equation 9.82), which is based on perturbation theory? How are the "association
energy" and "association volume" parameters of CPA ,  related to the enthalpy and
entropy of hydrogen bonding?
ii. According to Prausnitz et al.46, for the chemical theory, the following set of
combining rules are generally satisfactory for the cross enthalpy and entropy of
hydrogen bonding:
H ij 
S ij 
H i  H j
2
S i  S j
9.87
2
Show that, when the set of equations 9.87 are used, then the cross equilibrium
constant is given by the simple geometric mean combining rule:
K ij  K i K j
9.88
2
iii. Based on equations 9.87 and the analogy between the perturbation and chemical
theories (   KRT ), which combining rules do you suggest using for the association
parameters ,  of the CPA equation of state?
Problem 3. The Elliott rule
Derive an expression for the mole fraction of not-bonded molecules of an alcohol X Ai in
a binary alcohol-alcohol system. Assume that alcohols follow the two-site (2B) scheme
outlined in problem 1. Furthermore:
i. Show that the general form of the equation for the mole fraction of not-bonded
molecules X Ai is given by a 4th degree equation.
ii. Show that, when using the Elliott combining rule, this fourth-degree equation is
reduced to a third-degree one, which results to simple analytical solutions for the
mole fraction of not-bonded molecules.
iii. Assume that the two alcohols have identical association parameters. Show that the
Elliott rule gives the same result as that for a pure alcohol. (This is considered a test
for checking if a combining rule is “sensible”).
iv. Repeat questions (i) and (ii) using the 4C – 2B cross-association e.g. as used for
water-alcohol mixtures.
Problem 4. The second virial expression from the CPA EoS
Show that for the CPA equation of state, the second virial coefficient is given by the
following equations, when the 2B and 4C association schemes are used:
AB
a
B b
 e / RT  1  ABb
2B:
9.89
RT
AB
a
B b
 4 e / RT  1  ABb
4C:
9.90
RT




Problem 5. Parameters from CPA – the alternative parameterization
Using the alternative parameterization of CPA presented in Appendix 9.B and CPA
parameters shown in Table A (book’s website), calculate the “monomer” critical
temperatures, pressures and m parameters of CPA for pentane, decane, eicosane,
methanol, octanol, MEG, acetic acid and methylamine. Compare the values to the
experimental critical properties. What do you observe?
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