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 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]

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 = ∞

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)

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.