Chemical Potential in Mixtures

1. Chemical Potential in Mixtures • Purpose of this lecture: • To derive expressions for total Enthalpy, Entropy, and Gibbs energy of mixtures of ideal gas mixtures • Highlights • Gibbs Theorem • Enthalpy (T, yi) of an ideal gas mixture is independent of P, but entropy and Gibbs energy depend on P • Chemical potential of a species in a mixture is a function of T, P and composition. • Reading assignment: Sections 11.4 from textbook (eds. 7 or 6) Lecture 6

2. Chemical Potential in Mixtures • When we add dn moles of a component to n moles of itself, Gibbs energy increases by: • where mi,T,P represents the chemical potential of the pure component. • In a mixture, how will the total Gibbs energy change if we add dn moles of one component. • Will the Gibbs energy change follow the above formula? • Why or why not? Lecture 6

3. Chemical Potential in Mixtures • Because the Gibbs energy is a function of composition, we expect chemical potential to also depend on composition. • We are looking to derive mi as a function of T,P and yi • This requires an expression for the total Gibbs energy of a mixture: • to which the defining equation for mi is applied: 11.1 • In lecture 3 we derived expressions for H, S and G on T & P for a single component system. • What are the corresponding relationships for mixtures? Lecture 6

4. Chemical Potential in Ideal Gas Mixtures • The molecular conditions that produce a ideal gas mixture are the same as those of an ideal gas. • The gas must consist of freely moving particles of negligible volume and having negligible forces of interaction • “…every gas is a vacuum to every other gas (in the mixture)” • If n moles of an ideal gas mixture occupy V at T, the pressure is: • If nk moles of component k in this mixture were to occupy the same volume V at T: • Which leads to the concept of partial pressure: Lecture 6

5. Properties of Ideal Gas Mixtures • Gibbs Theorem: • A total thermodynamic property of an ideal gas mixture is the sum of the total properties of the individual species, each at the mixture temperature, but at its own partial pressure, pi. • Any thermodynamic property M varies according to: • On a molar basis: • = yA * + yB * Gig(T,P) GAig(T,pA) GBig(T,pB) Lecture 6

6. Enthalpy of an Ideal Gas Mixture • For any fluid in a closed system: • 6.20 • Therefore, the molar enthalpy Hig(T,P) of a pure ideal gas is independent of pressure: • Hig(T,pi) = Hig(T,P) • The total molar enthalpy of an ideal mixture of ideal gases is: • Hig = S yi Hiig(T,pi) • or • Hig = S yi Hiig(T,P) 11.25 0 Lecture 6

7. Entropy of an Ideal Gas Mixture • The entropy of a pure ideal gas is a function of T and P according to: • 6.24 • Therefore, the entropy change when pure ideal is expanded from P to pi at a constant T: • The entropy of an ideal gas mixture is therefore, • 11.26 Lecture 6

8. Gibbs Energy of an Ideal Gas Mixture • In a closed system, Gibbs Energy depends on pressure and temperature: • 6.6 or 6.10 • For a pure ideal gas, i, at a given temperature, this reduces to: • (constant T) • The change in Gibbs energy associated with expanding an ideal gas from P to pi at constant T is: Lecture 6

9. Gibbs Energy of an Ideal Gas Mixture • In terms of the total Gibbs energy of a system of n moles: • The total Gibbs energy is the sum of two terms: • Pure component Gibbs energies at T,P • Term to account for component mixing Look back at the enthalpy and entropy equations. Let’s draw a picture to make sense of this. What about U? Lecture 6

10. Chemical Potential in an Ideal Gas Mixture • By definition: • which applied to our expression for nGig gives: • 11.27 • Let’s talk about the  symbol. Lecture 6