= Solution
For a simple system of fixed composition, the <first law of thermodynamics> and
$$
G=E-TS+PV
$$
give
$$
dG=-S\,dT+V\,dP.
$$
Thus
$$
S=-\left(\frac{\partial G}{\partial T}\right)_P,
\qquad
V=\left(\frac{\partial G}{\partial P}\right)_T.
$$
Equality of mixed derivatives gives the <Gibbs free-energy Maxwell relation>
$$
\boxed{
\left(\frac{\partial S}{\partial P}\right)_T
=-\left(\frac{\partial V}{\partial T}\right)_P
}.
$$
The <heat capacity at constant volume> and <heat capacity at constant pressure> are
$$
C_V=\left(\frac{\partial E}{\partial T}\right)_V
=T\left(\frac{\partial S}{\partial T}\right)_V,
$$
$$
C_P=\left(\frac{\partial H}{\partial T}\right)_P
=T\left(\frac{\partial S}{\partial T}\right)_P,
\qquad H=E+PV.
$$
Regarding $S$ as a function of $T,V$ and differentiating along a constant-pressure curve gives
$$
\left(\frac{\partial S}{\partial T}\right)_P
=\left(\frac{\partial S}{\partial T}\right)_V
+\left(\frac{\partial S}{\partial V}\right)_T
\left(\frac{\partial V}{\partial T}\right)_P.
$$
The Maxwell relation from the Helmholtz free energy is
$$
\left(\frac{\partial S}{\partial V}\right)_T
=\left(\frac{\partial P}{\partial T}\right)_V.
$$
Therefore the <difference between constant-pressure and constant-volume heat capacities> is
$$
\boxed{
C_P-C_V
=T\left(\frac{\partial V}{\partial T}\right)_P
\left(\frac{\partial P}{\partial T}\right)_V
}.
$$
Along the liquid-gas <phase coexistence curve>, the molar Gibbs free energies agree. Differentiating $G_{\rm l}=G_{\rm g}$ along the curve gives
$$
-S_{\rm l}\,dT+V_{\rm l}\,dP
=-S_{\rm g}\,dT+V_{\rm g}\,dP.
$$
Hence the <Clausius-Clapeyron relation> is
$$
\boxed{
\frac{dP}{dT}
=\frac{S_{\rm g}-S_{\rm l}}{V_{\rm g}-V_{\rm l}}
=\frac{L}{T(V_{\rm g}-V_{\rm l})}
},
$$
where $L=T(S_{\rm g}-S_{\rm l})$ is the molar <latent heat>.
If $V_{\rm g}\gg V_{\rm l}$ and the gas is ideal, then $V_{\rm g}=RT/P$, so
$$
\frac{dP}{dT}\simeq\frac{LP}{RT^2}.
$$
For constant $L$, integration yields
$$
\boxed{
\log P=C-\frac{L}{RT},
\qquad
P(T)=P_0e^{-L/(RT)}
},
$$
where $P_0$ is constant over the range in which the approximations apply.
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