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. CHEE 311 Lecture 11

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: CHEE 311 Lecture 11

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) CHEE 311 Lecture 11

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: CHEE 311 Lecture 11

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 Lecture 11

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 CHEE 311 Lecture 11

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? CHEE 311 Lecture 11

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: CHEE 311 Lecture 11

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 CHEE 311 Lecture 11

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 using Tr=T/Tcij. CHEE 311 Lecture 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 CHEE 311 Lecture 11

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) CHEE 311 Lecture 11

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 Lecture 11