  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 common
temperature $T_F$. We express the Clausius formula that computes the entropy
change between the initial non-equilibrium state and the final equilibrium
state, using another equilibrium state of the $n$ bodies for the given initial
entropy, that corresponds to a temperature $T_f$. The second law inequality
follows from the fact $T_f < T_F$, under the assumption of positive heat
capacities 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 an
example, we illustrate our results for the case of temperature-independent heat
capacity.
