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An adiabatic thermalization between $n$ bodies is an irreversible process,leading to a rise in the total entropy of the bodies and yields a final commontemperature $T_F$. We express the Clausius formula that computes the entropychange between the initial non-equilibrium state and the final equilibriumstate, using another equilibrium state of the $n$ bodies for the given initialentropy, that corresponds to a temperature $T_f$. The second law inequalityfollows from the fact $T_f < T_F$, under the assumption of positive heatcapacities of the bodies. We derive this inequality for the discrete case of$n$ bodies as well as the continuum case of an unequally heated rod. As anexample, we illustrate our results for the case of temperature-independent heatcapacity.

Clausius formula and the Second law in the process of thermalization