HEAT & WORK

Four roads between two states — and the tolls differ. That difference is why engines exist.

W_by = ∫P dV, signed by direction; ΔU = Q − W_by closes while one computed piston state draws the entire ledger.

Four exact isobars and isochores close ΔU and ΔS_gas to zero while Q_net = W_net = the signed rectangular P–V area.

Q_h = W + Q_c to the last joule, η = 1 − T_c/T_h — the ceiling nobody beats.

Tf is the heat-capacity–weighted average; heat lost = heat gained, audited to 1e-12.

q = ΔT/(L₁/k₁+L₂/k₂) — R-values add in series. Ohm’s law for heat, drawn.

P = εσAT⁴ and λ_max = b/T — Stefan, Wien, and the Planck curve that broke classical physics.