Notice: Undefined offset: 148 in /var/www/scholarpedia.org/mediawiki/includes/parser/Parser.php on line 5961
Entropy/entropy example 1 - Scholarpedia

# 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.