entropy example 1

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Tomasz Downarowicz (2007), Scholarpedia, 2(11):3901.

revision #25680 [link to/cite this article]

Curator: Tomasz Downarowicz, Institute of Mathematics Computer Science, Wroclaw University of Technology, Poland

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.

Invited by:	Dr. Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia
Action editor:	Dr. Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia

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entropy example 1

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