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Chapter 8: Free Energies

This chapter examines the crucial concepts of Helmholtz free energy (F), Gibbs free energy (G), and enthalpy (H) within thermodynamic systems. It builds on prior discussions by exploring equilibrium conditions characterized by minimum F at constant temperature, volume, and particle number. The relationships among these thermodynamic functions are clarified, including the extremum principle for equilibrium in varied conditions. Practical examples highlight their application in real-world scenarios, such as calorimetry and adiabatic processes, enhancing the understanding of thermal energy exchanges.

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Chapter 8: Free Energies

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  1. Chapter 8: Free Energies Helmholtz Free Energy, F Gibbs Free Energy, G Enthalpy, H Table 8.1

  2. I. Equilibrium – Ch. 7 • In Ch. 7, we started with differences in T, p or μ between two subsystems. • The total system was isolated from the surroundings but U, V or N were exchanged (the variables) between the subsystems. • As a result, equilibrium was associated with maximum S(U, V, N).

  3. Equilibrium – Ch. 8 • Now consider a sealed test tube in which V and N are constant. This test tube is in a constant water bath, so heat (= energy) is exchanged to maintain constant T. (Confirm using First Law) • When T, V and N are the variables, then equilibrium is associated with minimum F(T, V, N): dF ≤ 0. Eqns 8.1 – 8.6

  4. Natural Variables • There is a distinction between “variables” and “natural variables” of thermodynamic functions (TF). • When a TF is expressed in terms of its natural variables, then the TF can be used to find equilibrium. I.e. the Extremum Principle is justified. • Max S(U, V, N) but not S(T, V, N) • Min G(T, p, N) but not G(U, V, N)

  5. Thermodynamic Function, F • F(T, V, N) = U – TS is called the Helmhotz Free Energy. • Then dF = dU – T dS ≤ 0 and equilibrium is associated with min U and max S at constant T. • dF = (δF/δT)V,NdT + (δF/δV)T,NdV + Σ(δF/δNj)V,T,Ni dNj = - S dT – p dV + Σμj dNj • Examples 8.1, 8.2

  6. Thermodynamic Function, H • H(S, p, N) = U + pV is called Enthalpy. • dH = T dS + V dp + Σμj dNj Eqn 8.22 • ΔH values are experimentally accessible (calorimetry).

  7. Thermodynamic Function, G • G(T, p, N) = H –TS is called the Gibbs Free Energy. • dG = -S dT + V dp + Σμj dNjEqn 8.25 • Then dG = dH – T dS ≤ 0 and equilibrium is associated with min H and max S at constant T. • Note Table 8.1

  8. II. More Applications • Ch 7 Tools • In an adiabatic process, w  ΔU (dU = δw) • Using the First Law, q  ΔS (δq = dS/T) (Carnot Cycle) • Ch 8 Tools • Heat Capacity • Cycles

  9. Heat Capacity  ΔU, ΔS  ΔF • q = heat αΔT and the proportionality constant is specific heat (J/g-K) or heat capacity (C, J/mol-K) • Note C may = f(T); Table 8.2 • When ΔV = 0 (bomb calorimeter), dU = δq. Then CV = (δq/dT)V = (∂U/∂T)V = T(∂S/∂T)V • dU = CV dT and dS = CV dT /T Eqn 8.31 • Ex 8.5, 8.6

  10. Heat Capacity  ΔH, ΔS  ΔG • When Δp = 0, then dH = δq. Then Cp = (δq/dT)p = (∂H/∂T)p = T(∂S/∂T)p • dH = Cp dT and dS = Cp dT /T Eqn 8.34

  11. Cycles (ΔTF = 0)  non-measurables • Determine ΔH for a phase change not at equilibrium. • Internal combustion engine • Adiabatic gas expansion • Otto Cycle (note compression ratio)

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