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Fugacity of Non-Ideal Mixtures (SVNA 11.6 and 11.7)

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- 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
- concept of fugacity

- Non-ideality in mixtures
- partial molar properties
- mixture fugacity and residual properties

- Pure component non-ideality
- We will begin our treatment of non-ideality in mixtures by considering gas behaviour.
- Start with the perfect gas mixture model derived earlier.
- 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.

Lecture 11

- We examined perfect gas mixtures in a previous lecture. The assumptions made in developing an expression for the chemical potential of species i in a perfect gas mixture were:
- 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.
- We can choose a more convenient reference pressure that is standard for all fluids, that is P=unit pressure (1 bar,1 psi,etc)
- In this case the pure component Gibbs energy becomes:

Lecture 11

- Substituting for our new reference state yields:
- (11.29)
- which is the chemical potential of component i in a perfect gas mixture at T,P.
- The total Gibbs energy of the perfect gas mixture is provided by the summability relation:
- (11.11)
- (11.30)

Lecture 11

- One source of mixture non-ideality resides within the pure components. Consider an ideal solution that is composed of real gases.
- In this case, we acknowledge that 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:

Lecture 11

- 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 solution 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.

Lecture 11

- 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

Lecture 11

- 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?

Lecture 11

- It can be shown that 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 compressiblity:

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- We have used the virial equation of state to calculate the fugacity and fugacity coefficient of pure, non-polar gases at moderate pressures.
- Under these conditions, it represents non-ideal PVT behaviour of pure gases quite accurately
- 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

Lecture 11

- Pure component coefficients (B11≡ B1, B22≡ B2,etc) are calculated as previously and cross coefficients are found from:
- (11.69b)
- where,
- and
- (11.70-73]
- Bo and B1 for the binary pairs are calculated using the standard equations 3.65 and 3.66 at Tr=T/Tcij.

Lecture 11

- 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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- The result of differentiation is:
- (11.64)
- with the auxilliary functions defined as:
- In the binary case, we have
- (11.63a)
- (11.63b)

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- 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:

Lecture 11