THE IDEAL GAS LAW

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.