Entropy/entropy example 1

Consider a system consisting of two identical and homogeneous solid bodies, of temperatures \(T_1\) and \(T_2\), respectively (state \(A\)). For our purposes, we take the states to be parameterized completely by \(T_1\) and \(T_2\); thus, the state space is two-dimensional. Assuming that temperature depends linearly on the heat content, the heat contained in the solids amounts to \(Q_1=cT_1\) and \(Q_2=cT_2\), respectively. All states with \(Q_1+Q_2 = {const}\) have the same energy. Let \(B\) denote the state where both solids contain the same amount of heat, \(Q_0 = \frac {Q_1+Q_2}2\).

The change of entropy as the system passes from state \(A\) to \(B\) equals

\[ \Delta S = \int_{Q_1}^{Q_0} \frac cQ\,dQ + \int_{Q_2}^{Q_0} \frac cQ\,dQ. \]

By an elementary calculus,

\[ \Delta S = c(\log Q_0 - \log Q_1) + c(\log Q_0 - \log Q_2) = 2c\left[\log\left(\frac{Q_1+Q_2}2\right) - \frac{\log Q_1+\log Q_2}2\right]. \]

Since the logarithmic function is strictly concave, this expression is positive, which means that the state \(B\) has entropy larger then \(A\). Thus \(B\) has the largest entropy among all states with the same level of energy and so it is the equilibrium state.