= Solution
For a fixed-composition <simple compressible system>, $dG=-S\,dT+V\,dp$ gives $S=-(\partial G/\partial T)_p$ and $V=(\partial G/\partial p)_T$. Equality of mixed <derivatives> yields $(\partial S/\partial p)_T=-(\partial V/\partial T)_p$. Therefore
$$
dS=-\left(\frac{\partial V}{\partial T}\right)_pdp
+\left(\frac{\partial S}{\partial T}\right)_pdT.
$$
For one mole of <ideal gas>, $V_m=RT/p$, so $(\partial V_m/\partial T)_p=R/p$. Reversible <heat> at fixed <pressure> gives $(\partial S_m/\partial T)_p=C_{mp}/T$, and consequently
$$
dS_m=-\frac Rp\,dp+\frac{C_{mp}}T\,dT.
$$
With constant positive <molar heat capacity>, integrate from a reference state $(p_0,T_0)$:
$$
\boxed{S_m=S_0-R\ln\frac p{p_0}+C_{mp}\ln\frac T{T_0}.}
$$
If <heat capacity> varies, the last term must instead be $\int_{T_0}^{T}C_{mp}(T')\,dT'/T'$. The reference ratios make the logarithms dimensionless. The following <classical ideal-gas entropy limits> test where the constant-heat-capacity expression remains applicable.
Back to article page
