1. Fugacity of Non-Ideal Mixtures (SVNA 11.6 and 11.7)
In our attempt to describe the Gibbs energy of real gas and liquid mixtures, we examine two “sources” of non-ideal behaviour:
Pure component non-ideality due to PVT behaviour
concept of fugacity
Non-ideality in mixtures
partial molar properties
mixture fugacity and residual properties
We will begin our treatment of non-ideality in mixtures by considering gas behaviour.
Start with an ideal gas mixture expression.
Modify this expression for cases where pure component non-ideality is observed.
Further modify this expression for cases in which non-ideal mixing effects occur.
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2. Ideal Gas Mixtures In an ideal gas mixture
all molecules have negligible volume
interactions between molecules of any type are negligible.
Based on this model, the chemical potential of any component in a perfect gas mixture is:
where the reference state, Giig(T,P) is the pure component Gibbs energy at the given P,T.
For a pure ideal gas, we derived:
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3. Ideal Gas Mixtures Substitute to get: (11.29)
which is the chemical potential of component i in an ideal gas mixture at T,P.
The total Gibbs energy of the ideal gas mixture is the sum of the contributions from the individual components:
(11.11)
(11.30)
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4. Ideal Mixtures of Real Gases
Consider an ideal solution of real gases.
The molecules have finite volume and interact, but assume these interactions are equivalent between components
The appropriate model is that of an ideal solution:
where Gi(T,P) is the Gibbs energy of the real pure gas:
(11.31)
Our ideal solution model applied to real gases is therefore:
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5. Non-Ideal Mixtures of Real Gases
In cases where molecular interactions differ between the components (polar/non-polar mixtures) the ideal solution model does not apply
Our knowledge of pure component fugacity is of little use in predicting the mixture properties
We require experimental data or correlations pertaining to the specific mixture of interest
To cope with highly non-ideal gas mixtures, we define a mixture fugacity:
(11.47)
where fi is the fugacity of species i in solution, which replaces the product yiP in the perfect gas model, and yifi of the ideal solution model.
CHEE 311
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6. Non-Ideal Mixtures of Real Gases
To describe non-ideal gas mixtures, we define the solution fugacity:
and the fugacity coefficient for species i in solution:
(11.52)
In terms of the solution fugacity coefficient:
Notation:
fi, i - fugacity and fugacity coefficient for pure species i
fi, i - fugacity and fugacity coefficient for species i in solution
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7. Calculating iv from Compressibility Data
Consider a two-component vapour of known composition at a given pressure and temperature
If we wish to know the chemical potential of each component, we must calculate their respective fugacity coefficients
In the laboratory, we could prepare mixtures of various composition and perform PVT experiments on each.
For each mixture, the compressibility (Z) of the gas can be measured from zero pressure to the given pressure.
For each mixture, an overall fugacity coefficient can be derived at the given P,T:
How do we use this overall fugacity coefficient to derive the fugacity coefficients of each component in the mixture?
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8. Calculating iv from Compressibility Data
The mixture fugacity coefficients are partial molar properties of the residual Gibbs energy, and hence partial molar properties of the overall fugacity coefficient:
In terms of our measured compressibility:
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9. Calculating iv from the Virial EOS
We know how to use the virial equation of state to calculate the fugacity and fugacity coefficient of pure, non-polar gases at moderate pressures.
The virial equation can be generalized to describe the calculation of mixture properties.
The truncated virial equation is the simplest alternative:
where B is a function of temperature and composition according to:
(11.61)
Bij characterizes binary interactions between i and j; Bij=Bji
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10. Calculating iv from the Virial EOS
Pure component coefficients (B11≡ B1, B22≡ B2,etc) are calculated as previously and cross coefficients are found from:
(11.69b)
where,
and
( ]
Bo and B1 for the binary pairs are calculated using the standard equations 3.65 and 3.66 using Tr=T/Tcij.
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11. Calculating iv from the Virial EOS
We now have an equation of state that represents non-ideal PVT behaviour of mixtures: or
We are equipped to calculate mixture fugacity coefficients from equation 11.60
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12. Calculating iv from the Virial EOS
The result of differentiation is:
(11.64)
with the auxiliary functions defined as:
In the binary case, we have
(11.63a)
(11.63b)
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13. 6. Calculating iv from the Virial EOS
Method for calculating mixture fugacity coefficients:
1. For each component in the mixture, look up:
Tc, Pc, Vc, Zc, 
2. For each component, calculate the virial coefficient, B
3. For each pair of components, calculate:
Tcij, Pcij, Vcij, Zcij, ij
and
using Tcij, Pcij for Bo,B1
4. Calculate ik, ij and the fugacity coefficients from:
CHEE 311
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