Deduce first law of thermodynamics for : (a) isothermal process (b) isochoric process

Answer: The first law is \(\Delta U = Q + W\).

(a) Isothermal process (temperature constant): for an ideal gas the internal energy depends only on temperature, so \(\Delta U = 0\). Hence \(0 = Q + W\), i.e. \(Q = -W\). The heat absorbed by the system is completely used to do work on the surroundings (or work done on the system is released as heat).

(b) Isochoric process (volume constant): \(\Delta V = 0\), so \(W = -P_{ext}\Delta V = 0\). Hence \(\Delta U = Q_v\). The heat absorbed at constant volume entirely increases the internal energy of the system.