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Thermodynamics: Free Energies, Helmholtz & Gibbs, and Enthalpy Explained

Chapter 8 delves into the concepts of Helmholtz Free Energy (F), Gibbs Free Energy (G), and Enthalpy (H) within thermodynamics. It explains equilibrium in isolated systems and how constant temperature, volume, and particle number affect free energy. The chapter outlines the relationship between natural variables and thermodynamic functions, emphasizing extremum principles for equilibrium conditions. Additionally, it discusses heat capacity, adiabatic processes, and practical applications, providing essential equations and examples for better understanding.

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Thermodynamics: Free Energies, Helmholtz & Gibbs, and Enthalpy Explained

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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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