6. Coping with Non-Ideality SVNA 10.3

1. 6. Coping with Non-Ideality SVNA 10.3 • Up until now, we have considered only ideal mixtures that (unfortunately) do not represent many cases experienced in practice. • Non-ideality can take two forms: • Deviations in pure component behaviour e.g. pure gases at high pressure • Deviations in mixture behaviour e.g. V   xi Vi • So far, our treatment of non-ideality has involved: • the development of a method for describing non-ideal, single-component, gas behaviour • the extension of this treatment to the description of pure liquids • What remains is to revise our treatment of perfect gas mixtures and ideal solutions to account for non-ideal mixing effects. J.S. Parent

2. nw moles H2O na moles Acetone Partial Properties: Thought Experiment • Suppose we add a drop of water • to pure acetone. • What change in volume • would result? • If the resulting water/acetone mixture is ideal (recall definition of ideal) the volume increase is simply that of the volume of the water droplet. • If the mixture behaves non-ideally, the volume increase will not equal the volume of the water droplet. The effect may, in fact, be quite different. • Non-ideality in mixtures results from complex intermolecular interactions that we cannot predict. • We still have to solve engineering problems (separations, property calculations, …) using these non-ideal systems. J.S. Parent

3. Partial Properties: Thought Experiment • If the volume change of the acetone-water mixture does not equal the volume of the water droplet, then the properties of pure water are irrelevant. • We like to assign values or “contributions” to each component in non-ideal mixtures to account for the variation of a property with respect to composition. • This leads us to define partial molar properties, which in our thought experiment gives us the partial molar volume for water in an acetone-water solution. • This quantity represents change in solution volume as the number of moles of water is varied at a given P,T, and nacetone. J.S. Parent

4. Partial Molar Quantities • We prefer to think of mixtures in terms of their components: • Overall property has a contribution from each component in the mixture. • In non-ideal systems, the properties of the pure components have little meaning, forcing us to find an alternate way of defining molar quantities. • If nM represents the total thermodynamic property of interest: • (10.7) • where is a partial molar property, also a function of (T,P, nj) • A partial molar property is specific to the P,T and composition from which it is derived by equation 10.7 • It is difficult to predict, but can be measured experimentally. J.S. Parent

5. Total Properties of Non-Ideal Mixtures • Ideal mixtures result from a lack of molecular interactions (ideal gas) or equivalent molecular interactions (ideal solution). In these cases, a total thermodynamic property (nM) for a mixture is: • nM =  ni Mi where Mi represents the pure • component property of i. • Non-ideal systems do not obey this simple formula, as cross-component molecular interactions differ from pure component interactions. • nM =  ni where represents the partial molar • property of component i. • In terms of mole fractions (dividing by n): • M =  xi (10.11) • If we know the partial properties of the components of the mixture (from experimental data) we can derive its total property. • This is summability relation, which is opposite to 10.7 that defines a partial property J.S. Parent

6. The Gibbs-Duhem Equation • An important question we need to answer is: • how do the partial molar properties of a mixture relate? • Start with the definition of total Gibbs Energy at T,P: • nG =  ni Gi =  ni i • If we change the composition of the system at constant T,P, the Gibbs energy responds accordingly: • dnG =  d(ni i) • =  ni di +  i dni • But we know that the total change in Gibbs energy is defined by: • d(nG) = nV dP - nS dT +  i dni (10.3) • which at constant P,T (dP=0, dT=0), is: • d(nG) =  i dni J.S. Parent

7. The Gibbs-Duhem Equation • d(nG) provided by these two relations must be equal. Therefore, • d(nG) =  ni di +  i dni • =  i dni • For this to be true, •  ni di = 0 (10.14) • This is the Gibbs-Duhem equation applied to chemical potential • Why is it useful? • It states that partial molar properties cannot change independently, if one partial property increases, others must decrease • Estimates of partial molar properties (from experimental data, correlations…) can be checked for consistency J.S. Parent

8. Calculating Partial Molar Props. - Binary Systems • Partial molar properties (Mi) can always be derived from an equation for the solution property (M) as a function of composition: • (10.7) • Comparatively simple relations exist for binary systems: • (10.11) • therefore • The Gibbs-Duhem equation tells us that: • (10.14) • Leaving us with: J.S. Parent

9. Calculating Partial Molar Props. - Binary Systems • We left off with: • Since x1+x2=1, dx1 = - dx2 and • which we can write as: • Substituting the total solution property: • (10.15, 10.16) • What we need to calculate a partial molar property is an expression for the total molar property (M) as a function of composition. J.S. Parent

10. Notation for the Course • Our superscripts and subscripts are propagating rapidly, so let’s revisit our definitions: • S = entropy of one mole of the mixture • Si = entropy of one mole of pure i • Si = entropy of one mole of pure i in the mixture • = partial molar entropy • Suggested Readings • Examples 10.1, 10.2, 10.3, 10.4 - Partial Molar Properties J.S. Parent

11. 6. Non-Ideal Mixtures • 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. J.S. Parent

12. Perfect Gas Mixtures • We examined perfect gas mixtures in lecture 9. 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: J.S. Parent

13. Perfect Gas Mixtures • Substituting for our new reference state yields: • (10.28) • 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: • (10.11) • (10.29) J.S. Parent

14. 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: • (10.30) • Our ideal solution model applied to real gases is therefore: J.S. Parent

15. 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: • (10.42) • 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. J.S. Parent

16. 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: • (10.47) • 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 J.S. Parent

17. 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? J.S. Parent

18. 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: J.S. Parent

19. 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: • (3.31) • where B is a function of temperature and composition according to: • (10.65) • Bij characterizes binary interactions between i and j; Bij=Bji J.S. Parent