The isothermal compression of an ideal gas is analysed using a relativistic thermodynamics formalism based on the principle of inertia of energy (Einstein's equation) and the asynchronous formulation (Cavalleri and Salgarelli 1969 "Nuovo Cimento" 42 722-54), which is similar to the formalism developed by van Kampen (1968 "Phys. Rev." 173 295-301) and Hamity (1969 "Phys. Rev." 187 1745-52). In this 4-vector Minkowski formalism mechanical and thermodynamical processes are described by the first law of thermodynamics expressed as [delta]U[superscript [mu]] = W[superscript [mu]] + Q[superscript [mu]], in a Lorentz covariant way. This exercise is considered useful for undergraduate physics students interested in foundations of physics, with the only prerequisites in first courses in thermodynamics and special relativity. (Contains 9 figures.)