CHEE 311,Lecture 18,1,van Laar Correlation,Another two-parameter excess Gibbs energy model was developed from an expansion of (RTx1x2)/GE instead of GE/RTx1x2. The end results are: (12.16) for the excess Gibbs energy and: (12.17a) (12.17b) for the activity coefficients. Note that: as x10, ln1 A12 and as x2 0, ln2 A21,CHEE 311,Lecture 18,2,Example 2 (Problem 5, Practice Problem Set #4),P5. Vapour liquid data for the system 1,4 dioxane(1)/ethylbenzene(2) at 85 oC are provided below. From these data obtain estimates of the van Laar coefficients (estimates based on smoothly drawn curves on the enclosed graph paper are sufficient). Estimate values of P-x1-y1 for x1=0.5 based on these parameter values. Table 2: VLE data for the system 1,4 dioxane/ethylbenzene at 85 oC,CHEE 311,Lecture 18,3,Local Composition Models,Unfortunately, the previous approach cannot be extended to systems of 3 or more components. For these cases, local composition models are used to represent multi-component systems. Wilsons Theory Non-Random-Two-Liquid Theory (NRTL) Universal Quasichemical Theory (Uniquac) While more complex, these models have two advantages: the model parameters are temperature dependent the activity coefficients of species in multi-component liquids can be calculated using information from binary data. A,B,C A,B A,C B,C tertiary mixture binary binary binary,CHEE 311,Lecture 18,4,Wilsons Equations for Binary Solution Activity,A versatile and reasonably accurate model of excess Gibbs Energy was developed by Wilson in 1964. For a binary system, GE is provided by: (12.18) where (12.24) Vi is the molar volume at T of the pure component i. aij is determined from experimental data. The notation varies greatly between publications. This includes, a12 = (12 - 11), a21 = (12 - 22) that you will encounter in Holmes, M.J. and M.V. Winkle (1970) Ind. Eng. Chem. 62, 21-21.,CHEE 311,Lecture 18,5,Wilsons Equations for Binary Solution Activity,Activity coefficients are derived from the excess Gibbs energy using the definition of a partial molar property: When applied to equation 11.16, we obtain: (12.19a) (12.19b),CHEE 311,Lecture 18,6,Wilsons Equations for Multi-Component Mixtures,The strength of Wilsons approach resides in its ability to describe multi-component (3+) mixtures using binary data. Experimental data of the mixture of interest (ie. acetone, ethanol, benzene) is not required We only need data (or parameters) for acetone-ethanol, acetone-benzene and ethanol-benzene mixtures The excess Gibbs energy for multicomponent mixtures is written: (12.22) and the activity coefficients become: (12.23) where ij = 1 for i=j. Summations are over all species.,CHEE 311,Lecture 18,7,Wilsons Equations for 3-Component Mixtures,For three component systems, activity coefficients can be calculated from the following relationship: Model coefficients are defined as (ij = 1 for i=j):,CHEE 311,Lecture 18,8,Comparison of Liquid Solution Models,Activity coefficients of 2-methyl-2-butene + n-methylpyrollidone. Comparison of experimental values with those obtained from several equations whose parameters are found from the infinite-dilution activity coefficients. (1) Experimental data. (2) Margules equation. (3) van Laar equation. (4) Scatchard-Hamer equation. (5) Wilson equation.,CHEE 311,Lecture 18,9,Non-Ideal VLE to Moderate Pressures: Overview SVNA 14.1,We now have the tools required to describe and calculate vapour-liquid equilibrium conditions for even the most non-ideal systems. 1. Equilibrium Criteria: In terms of chemical potential In terms of mixture fugacity 2. Fugacity of a component in a non-ideal gas mixture: 3. Fugacity of a component in a non-ideal liquid mixture:,CHEE 311,Lecture 18,10,g, f Formulation of VLE Problems,To this point, Raoults Law was only description we had for VLE behaviour: We know that calculations based on Raoults Law do not predict actual phase behaviour due to over-simplifying assumptions. Accurate treatment of non-ideal phase equilibrium requires the use of mixture fugacities. At equilibrium, the fugacity of each component is the same in all phases. Therefore, or, determines the VLE behaviour of non-ideal systems where Raoults Law fails.,CHEE 311,Lecture 18,11,Non-Ideal VLE to Moderate Pressures,A simpler expression for non-ideal VLE is created upon defining a lumped parameter, F. The final expression becomes, (i = 1,2,3,N) 14.1 To perform VLE calculations we therefore require vapour pressure data (Pisat), vapour mixture and pure component fugacity correlations (i) and liquid phase activity coefficients (i).,CHEE 311,Lecture 18,12,Non-Ideal VLE to Moderate Pressures,Sources of Data: 1. Vapour pressure: Antoines Equation (or similar correlations) 14.3 2. Vapour Fugacity Coefficients: Viral EOS (or others) 14.6 3. Liquid Activity Coefficients Binary Systems - Margules,van Laar, Wilson, NRTL, Uniquac Ternary (or higher) Systems - Wilson, NRTL, Uniquac,