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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 concept of fugacity Non-ideality in mixtures partial molar properties

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

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Fugacity of non ideal mixtures svna 11 6 and 11 7 l.jpg

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

      • 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 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


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Perfect Gas Mixtures

  • 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


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Perfect Gas Mixtures

  • 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


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Ideal Mixtures of Real Gases

  • 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


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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 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


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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

Lecture 11


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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?

Lecture 11


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Calculating iv from Compressibility Data

  • 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:

Lecture 11


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Calculating iv from the Virial EOS

  • 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


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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

  • (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


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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

Lecture 11


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Calculating iv from the Virial EOS

  • The result of differentiation is:

  • (11.64)

  • with the auxilliary functions defined as:

  • In the binary case, we have

  • (11.63a)

  • (11.63b)

Lecture 11


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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:

Lecture 11


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