The entropy change accompanying heating

Equation 2.1 refers to the transfer of heat to a system at a temperature T. In general, the temperature changes as we heat a system, so we carmot use eqn 2.1 directly. Suppose, however, that we transfer only an infinitesimal energy as heat, dq, to the system then there is only an infinitesimal change in temperature and we introduce negligible error if we keep the temperature in the denominator of eqn 2.1 equal to T during that transfer. As a result, the entropy increases by an infinitesimal amount dS given by 

To calculate dq, we recall from Section 1.4 that the heat capacity C = q/AT, where AT is macroscopic change in temperature. For an infinitesimal change dr brought about by an infinitesimal transfer of heat dq we write C = dq/dT and therefore dq = CdT, so we can write dq,.gy= CdT and therefore 

The total change in entropy, AS, when the temperature changes from Ti to Tf is the sum (integral) of all such infinitesimal terms  

For many substances and for small temperature ranges we may take C to be constant. (This is strictly true only for a monatomic perfect gas.) Then C may be taken outside the integral and the latter evaluated as follows  

Self-test 2.1 j Calculate the change in molar entropy when water vapor is heated from 160°Cto 170°C at constant volume. (Cy n = 26.92 J K mol .) 

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