4.6 Entropy

The second law of thermodynamics is best expressed in terms of a change in the thermodynamic variable known as entropy    , which is represented by the symbol S . Entropy, like internal energy, is a state function. This means that when a system makes a transition from one state into another, the change in entropy $\text{Δ}S$ is independent of path and depends only on the thermodynamic variables of the two states.

We first consider $\text{Δ}S$ for a system undergoing a reversible process at a constant temperature. In this case, the change in entropy of the system is given by

where Q is the heat exchanged by the system kept at a temperature T (in kelvin). If the system absorbs heat—that is, with $Q>0$ —the entropy of the system increases. As an example, suppose a gas is kept at a constant temperature of 300 K while it absorbs 10 J of heat in a reversible process. Then from [link] , the entropy change of the gas is

Similarly, if the gas loses 5.0 J of heat; that is, $Q=\mathrm{-5.0}\text{J}$ , at temperature $T=200\text{K}$ , we have the entropy change of the system given by

The change in entropy of a system for an arbitrary, reversible transition for which the temperature is not necessarily constant is defined by modifying $\text{Δ}S=Q\text{/}T$ . Imagine a system making a transition from state A to B in small, discrete steps. The temperatures associated with these states are $T_A$ and $T_B,$ respectively. During each step of the transition, the system exchanges heat $\text{Δ}{Q}_i$ reversibly at a temperature $T_i.$ This can be accomplished experimentally by placing the system in thermal contact with a large number of heat reservoirs of varying temperatures $T_i$ , as illustrated in [link] . The change in entropy for each step is $\text{Δ}{S}_i=Q_i\text{/}{T}_i.$ The net change in entropy of the system for the transition is

We now take the limit as $\text{Δ}{Q}_i\to 0$ , and the number of steps approaches infinity. Then, replacing the summation by an integral, we obtain

where the integral is taken between the initial state A and the final state B . This equation is valid only if the transition from A to B is reversible.