Thermodynamics and Statistical Mechanics
Core concepts
Know the laws of thermodynamics, the distinction between state functions and path functions, the ideal gas law, heat capacities at constant volume and pressure, entropy, Carnot efficiency, and the canonical ensemble. Understand that temperature is ∂S/∂E, that entropy counts microstates (S = k ln Ω), and that the Boltzmann factor e^(−E/kT) sets populations of energy levels.
Traps
- Work and heat are path-dependent; internal energy is a state function. Q = ΔU + W with sign conventions must be stated.
- Carnot efficiency η = 1 − T_c/T_h uses absolute temperatures; using Celsius gives nonsense.
- Entropy of the universe never decreases, but a subsystem's entropy can fall if work is done on it.
- C_p − C_v = nR for an ideal gas; mixing them up changes adiabatic results.
- Equipartition counts ½kT per quadratic degree of freedom; a diatomic gas has more modes than a monatomic one.
Worked example
A Carnot engine operates between 500 K and 300 K. Efficiency is 1 − 300/500 = 0.40, so 40% of absorbed heat becomes work. Trap: converting to Celsius (227 °C and 27 °C) and computing 1 − 27/227 gives a wrong, meaningless number.
Rule of thumb
For any thermal process, first identify whether it is isothermal, adiabatic, isobaric, or isochoric; each fixes one variable and selects the right equation.