6. Coping with Non-Ideality SVNA 11.3

1. 6. Coping with Non-Ideality SVNA 11.3 • Up until now, we have considered only ideal mixtures that (unfortunately) do not occur often in practice. • Non-ideality can take two forms: • Deviations in pure component behaviour e.g. pure gases at high pressure are not ideal • 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 ideal gas mixtures and ideal solutions to account for non-ideal mixing effects. Lecture 13

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 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 have to solve engineering problems (separations, property calculations, …) for these non-ideal systems. Lecture 13

3. Partial Properties: Thought Experiment • The volume change of the acetone-water mixture does not equal the volume of the water droplet • 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. Lecture 13

4. Partial Molar Quantities • We prefer to think of mixtures in terms of their components: • An overall property like V or H 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 alternative way of defining molar quantities. • If nM is the total thermodynamic property of interest: • (11.7) • where is a partial molar property, also a function of (T,P, nj) • A partial molar property depends on the P,T and composition from which it is derived. • It is difficult to predict, but can be measured experimentally. Lecture 13

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: • M =  xi (11.11) • If we know the partial properties of the components of the mixture (from experimental data) we can determine its total property. Lecture 13

6. The Gibbs-Duhem Equation • An important question we need to answer is: • how do are the partial molar properties in a mixture related? • 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 (11.2) • which at constant P,T is: • d(nG) =  i dni Lecture 13

7. The Gibbs-Duhem Equation • d(nG) provided by these two relations must be equal. Therefore, • d(nG) =  i dni =  ni di +  i dni • For this to be true, •  ni di = 0 (11.14) • This is the Gibbs-Duhem equation for chemical potential. It is also true for any other partial molar property. • What does it look like for a two-component system? • What does it mean? • 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 or correlations) can be checked for consistency Lecture 13

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: • (11.7) • For binary systems: • (11.11) • therefore • The Gibbs-Duhem equation tells us that: • (11.14) • Leaving us with: Lecture 13

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: • (11.15, 11.16) • All that we need to calculate a partial molar property is an expression for the total molar property (M) as a function of composition. How would this work for volume? Lecture 13

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 11.1, 11.2, 11.3, 11.4 - Partial Molar Properties Lecture 13