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Understanding Maxwell's Relations and Mixtures in Thermodynamics

This article explores Maxwell's relations and their implications in thermodynamics, highlighting their role in predicting nonmeasurable thermodynamic quantities from measurable ones. It covers fundamental concepts, including degrees of freedom, conjugate driving forces, and thermodynamic properties as functions of natural variables. The article also delves into partial molar properties in mixtures, showcasing how component amounts influence total properties at constant temperature and pressure. Applications, examples, and mathematical tools such as the Legendre Transform are discussed, providing a comprehensive overview for students and professionals.

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Understanding Maxwell's Relations and Mixtures in Thermodynamics

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  1. Ch 9: Maxwell’s Relations and Mixtures

  2. I. General Thermodynamic Eqns • Given expt conditions, we want an eqn for a fundamental thermodynamic property as a function of its natural variables and then finding the associated extremum eqn. • Define degree of freedom (= X = extensive variable) and conjugate driving force (= P). • E.g. dU = Σ Pj dXj where Pj = (∂U/∂Xj)Xi≠j • Use the math tool, Legendre Transform, to do this.

  3. Legendre Transforms (Ch 8) • Transform y = f(x) to a function expressed in terms of c(x) = dy/dx = slopes and b(x) = intercepts. • Start with y(x) = f(x) = c(x) x + b(x) and dy(x) = (dy/dx)dx = c(x)dx • Transform to b(x) = y(x) – c(x) x and db(x) = dy(x) – c(x) dx - x dc(x) = -x dc(x) • Ex. 9.1 and 9.2; Prob 8.1

  4. II. Maxwell’s Relations • Handout • These relations or eqns allow us to “…predict nonmeasurable thermodynamic quantities from measurable ones” p 155 • More Applications – Chapter 9 Tools are presented in the next 3 slides as MR (Maxwell’s Relations) 1, 2, 3

  5. MR 1: (∂U/∂V)T • How does U change with V at constant T?: Use dU = T dS - p dV (IIa) • Then (∂U/∂V)T = πT = T(∂S/∂V)T – p = T(∂p/∂T)V – p Note: LHS = nonmeas. But RHS = meas. • Try Ideal G. L., van der Waals G. L. • Ex 9.5

  6. MR 2: (∂S/∂ℓ)T,p • Example 9.3

  7. MR 3: Measurable α and к • α = (1/V) (∂V/∂T)p = thermal expansion coefficient; units = K-1; Fig 9.2 • к = - (1/V) (∂V/∂p)T = isothermal compressibility; units = atm-1 ; Fig 9.4 • Unusual behavior of water: Fig 9.5 (low α) and Fig 9.6 (low and negative к) due to H-bonds. Prob 9.2 • Ex 9.4 S = f(p) and Ex 9.5 U = f(V)

  8. III. Mixtures • Now consider a system with more than one component. How do we determine how the amount of each component affects a particular total property at constant T and p? • Partial molar [Property] = (∂Prop/∂nj)T, p, ni • This partial describes how Prop changes with moles of the jth component, all other variables kept constant.

  9. Partial Molar Volume • Partial molar volume = vj = (∂V/∂nj)T, p, ni This tells us how the total volume V changes as nj moles of component j are added. • Then V = Σ vjnj • dV = Σ(∂V/∂nj)T, p, ni dnj = Σ vjdnj

  10. Calculating Total Volume • A mixture of one mol of water (v = 18.0 mL) and one mol of ethanol (v = 58.3 mL) are mixed. What are the partial volumes of water and alcohol? What is the total volume of this mixture?

  11. Chemical Potential • Eqn 9.24 presents four expressions for chemical potential. Each one describes how the thermodynamic function varies with composition. • But the partial free function is the one at constant T and p. • Partial molar free energy = (∂G/∂Nj)T, p, Ni = μj = (∂H/∂Nj)T, p, Ni -T (∂S/∂Nj)T, p, Ni

  12. More Applications – Ch 9 Tools • Gibbs Duhem Eqn tells us that partial molar quantities (from homogenous thermo fnts) are related and how they are linked. • Σ njdvj = 0 at const p, T Eqn 9.33 • Also Σ Njdμj = 0 at const p, T Eqn 9.37

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