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Chapter 12 Temperature and Heat Capacity

Chapter 12 Temperature and Heat Capacity. Schottky Two-State Model and the Meaning of Temperature. Ex. 10.5, p 185-6 N distinguishable particles distributed in two non-degenerate states: 0 energy (N–n particles) and ε 0 energy (n particles). n = U/ ε 0 and dn/dU = 1/ ε 0

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Chapter 12 Temperature and Heat Capacity

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  1. Chapter 12 Temperature and Heat Capacity

  2. Schottky Two-State Model and the Meaning of Temperature • Ex. 10.5, p 185-6 • N distinguishable particles distributed in two non-degenerate states: 0 energy (N–n particles) and ε0 energy (n particles). • n = U/ε0 and dn/dU = 1/ε0 • S = k ℓn W = k ℓn N!/[n! (N-n)!] = - kn ℓn (n/N) – k(N-n) ℓn [(N-n)/N]

  3. Temperature • 1/T = (∂S/∂U)V,N = k (∂ℓn W/∂n)V,N (dn/dU) = (k/ε0) ℓn (fgrd/fexc) = (k/ε0) ℓn R • fexc = fraction in ε0 = (n/N) = 1 - fgrd • Fig 12.2 (S/k vs U), Eqn 12.4, Fig 6.3 • Low U means R > 1: LHS of figure, system absorbs energy to max S, 1/T and T > 0 • High T means R < 1 (pop inv): RHS; system gives up energy to max S, 1/T and T < 0 • R = 1: equilibrium or max S, T = ∞

  4. Heat Capacity • Fig 12.3 compares IGL (infinite energy ladder) and Schottky model (2 energy states) at constant V and N. • Start with S vs U and end with CV vs T. • IGL: S = 3Nk/2 ℓn U (Eqn p 225; confirm) • 2-State: S = k ℓn W (Eqn 12.2; Fig 12.2)

  5. T and CV • 1/T = driving force for taking up energy • IGL surrounded by heat bath reaches equilibrium when Tbath = TIGL (thermal equilibrium). CV = change in U per change in T (fluctuations) = constant) • 2-State system reaches equilibrium when fgrd/fexc = 1.

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