Thermodynamics and Statistical Mechanics
Thermodynamics and Statistical Mechanics is roughly 10% of the GRE Physics Subject Test. It covers the laws of thermodynamics, ideal-gas processes, heat capacities, entropy, engines and refrigerators, kinetic theory, and the statistical description of distinguishable and quantum gases.
Core rules
- First law: ΔU = Q − W, with W the work done by the gas. For a quasi-static process W = ∫P dV.
- Ideal gas: PV = nRT and U = (f/2)nRT for f quadratic degrees of freedom. Equipartition gives ½kT per degree of freedom.
- Molar heat capacities: C_V = (f/2)R and C_P = C_V + R. For a monatomic gas γ = 5/3, for a diatomic gas (rotation only) γ = 7/5.
- Processes: isothermal (T constant, ΔU = 0), isobaric (W = PΔV), isochoric (W = 0), adiabatic (PV^γ constant, Q = 0, so ΔU = −W).
- Entropy: ΔS = ∫dQ_rev/T. For an ideal gas, ΔS = nC_V ln(T₂/T₁) + nR ln(V₂/V₁). Entropy of the universe never decreases.
- Carnot efficiency η = 1 − T_c/T_h; a refrigerator's COP is T_c/(T_h − T_c). No engine between the same reservoirs beats Carnot.
- Kinetic theory: v_rms = √(3kT/m), mean kinetic energy (3/2)kT per molecule.
- Statistical mechanics: P(state) ∝ e^{−E/kT}; the partition function Z = Σ e^{−E_i/kT} generates all thermodynamic averages. Fermions obey the Pauli principle; bosons may occupy the same state.
Worked example
A monatomic ideal gas at 300 K expands adiabatically and reversibly to eight times its volume. With γ = 5/3, T₂ = T₁(V₁/V₂)^{γ−1} = 300 × (1/8)^{2/3} = 300 × 0.25 = 75 K. The gas cools sharply because it does work at the expense of internal energy, and no heat enters.
Traps
Getting the sign of W backwards (work done by versus on the gas); using C_P when the volume, not the pressure, is held fixed; assuming an adiabatic process is also isothermal; using the monatomic γ for a diatomic gas at room temperature; forgetting that entropy changes are computed along a reversible path even when the actual process is irreversible; assuming Carnot efficiency is attainable in practice.