chapter8 - Chapter 8 Excess Mixture Properties and Activity...

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Chapter 8 Excess Mixture Properties and Activity Coe cients Thebehaviorofmostliquidandso lidm ixturescannotberepresentedbythecubicequat ionsof state. For this reason, it is necessary to de f ne another quantity, called the activity coe cient , to express fugacities of components in liquid and solid mixtures. In the literature, various activity coe cient models are used in phase equilibrium calculations. The purpose of this chapter is to introduce such models and to show how to evaluate the parameters appearing in these models. 8.1 PROPERTY CHANGES ON MIXING FOR AN IDEAL MIXTURE Property change on mixing per mole, e ϕ mix ,isde f ned by Eq. (6.3-4), i.e., e ϕ mix = k X i =1 x i ( ϕ i e ϕ i ) (8.1-1) For an ideal mixture, Eq. (8.1-1) takes the form e ϕ IM mix = k P i =1 x i ( ϕ i e ϕ i ) (8.1-2) The properties of an ideal mixture are given in Section 7.4 as V i = e V i and H i = e H i (8.1-3) The use of Eq. (8.1-3) in Eq. (8.1-2) gives e V mix =0 and e H mix (8.1-4) indicating that volume and enthalpy changes on mixing are both zero. For an ideal mixture this is an expected result because not only the size of the molecules are equal to each other, but also the interactions between unlike molecules are equal to those between like molecules. To obtain an expression for the Gibbs energy change on mixing for an ideal mixture, it is necessary to relate G i to e G i . The partial molar Gibbs energy of component i in an ideal mixture is given by G i = λ i ( T )+ RT ln b f i = λ i ( T RT ln( x i f i ) (8.1-5) The molar Gibbs energy of pure i is expressed by Eq. (5.2-6), i.e., e G i = λ i ( T RT ln f i (8.1-6) 203
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Subtraction of Eq. (8.1-6) from Eq. (8.1-5) leads to G IM i e G i = RT ln x i (8.1-7) Thus, the use of Eq. (8.1-7) in Eq. (8.1-2) results in e G mix = RT k X i =1 x i ln x i (8.1-8) For a binary mixture of components 1 and 2, a representative plot of e G mix versus x 1 is shown in Figure 8.1. Since x i < 1 , it follows from Eq. (8.1-8) that e G mix < 0 . In other words, the Gibbs energy of an ideal mixture is always less than the summation of the Gibbs energies of the unmixed pure components. Thus, upon mixing at a speci f ed temperature and pressure, components 1 and 2 form a stable 1 ideal mixture. IM mix G ~ Δ 1 x 0 IM mix S ~ Δ Figure 8.1 Representative plots of e G mix and e S mix as a function of composition. To obtain an expression for the entropy change on mixing for an ideal mixture, note that e G mix = e H mix | {z } 0 T e S mix (8.1-9) Substitution of Eq. (8.1-8) into Eq. (8.1-9) yields e S mix = R k X i =1 x i ln x i (8.1-10) For a binary mixture of components 1 and 2, a representative plot of e S mix versus x 1 is also shown in Figure 8.1. Mixing increases disorder and thus e S mix > 0 . 8.2 EXCESS PROPERTIES In the literature, it is customary to split the property change on mixing into two terms as ϕ mix = ϕ mix + ϕ ex (8.2-1) where ϕ ex is called the excess property . The use of Eq. (8.2-1) in Eq. (6.3-2) leads to ϕ mix = k X i =1 n i e ϕ i + ϕ mix | {z } ϕ mix + ϕ ex (8.2-2) 1 In other words, components 1 and 2 are completely miscible in each other. Miscibility of components will
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chapter8 - Chapter 8 Excess Mixture Properties and Activity...

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