= Solution
An <equation of state> relates thermodynamic variables within a material model; a <phase diagram> records which equilibrium phase or phases are stable under specified controls. For a pure substance at fixed <temperature> and <pressure>, stability means minimizing <Gibbs free energy>. Two phases coexist when their molar <chemical potentials> agree. The equation of state and the free-energy stability condition must therefore be used together: an algebraic relation between $p,V,T$ alone need not identify the stable branch.
A pressure-temperature <phase diagram> contains single-phase regions separated by <phase coexistence curves>. For a one-component system the <Gibbs phase rule> gives $F=3-P$: a single phase has two independent controls, two-phase coexistence lies on a curve, and three phases coexist at a <triple point>. The liquid-gas coexistence curve ends at a <liquid-gas critical point>; beyond it a continuous path connects liquidlike and gaslike states without crossing a first-order coexistence boundary. A fixed $p,T$ point on a coexistence curve does not specify the proportions of the two phases: an additional extensive constraint, such as total volume, fixes the phase fraction.
At a <first-order phase transition>, first derivatives of <Gibbs free energy>, such as molar entropy or volume, are discontinuous. The conversion has <latent heat> $L=T\Delta s$. At a continuous transition the first derivatives match, while response functions associated with higher derivatives can be singular or discontinuous. At an ordinary liquid-gas critical endpoint, the two coexisting densities approach one another and the latent heat tends to zero; the material does not retain a sharp liquid-gas boundary above it.
The coexistence slope follows directly from equality of the two molar <chemical potentials>. Since $d\mu=-s\,dT+v\,dp$, differentiating $\mu_\alpha=\mu_\beta$ along the boundary gives
$$
-\Delta s\,dT+\Delta v\,dp=0,\qquad \boxed{\frac{dp}{dT}=\frac{\Delta s}{\Delta v}=\frac{L}{T\Delta v}.}
$$
This is the exact <Clapeyron equation>, also commonly called the <Clausius-Clapeyron relation>. For vaporization, both $L$ and $\Delta v$ are positive, so the saturation pressure rises with <temperature>. For ordinary ice melting into liquid water, $L>0$ but the molar volume decreases, giving the familiar negative melting-curve slope. The sign follows from the volume change, not from a different law.
For a dilute ideal vapour with $v_g\gg v_l$, $\Delta v\simeq RT/p$. The <ideal-vapor Clausius-Clapeyron approximation> becomes
$$
\frac{d\ln p_{\rm sat}}{dT}\simeq\frac{L}{RT^2}.
$$
If molar latent heat is approximately constant over the interval,
$$
\boxed{\ln\frac{p_{\rm sat}(T)}{p_{\rm sat}(T_0)}=-\frac LR\left(\frac1T-\frac1{T_0}\right).}
$$
These approximations deteriorate close to the critical point, where liquid and gas volumes become comparable, or over a range in which latent heat changes appreciably.<Statistical Physics by David Tong>[§5.1.2, coexistence slope and ideal-vapour approximation; surrounding discussion of the liquid-gas critical point and solid-liquid boundary]{ref}
The <ideal gas law> $pV=nRT$ is an equation of state for a dilute gas, but it has no liquid branch and cannot predict condensation. A simple model including repulsion and attraction is the <Van der Waals molar equation>,
$$
p=\frac{RT}{v-b}-\frac a{v^2},
$$
where $b>0$ represents excluded molar volume and $a>0$ represents attraction. At its critical point, $\partial p/\partial v=\partial^2p/\partial v^2=0$, giving
$$
\boxed{v_c=3b,\qquad p_c=\frac a{27b^2},\qquad T_c=\frac{8a}{27Rb}.}
$$
Below $T_c$, the homogeneous isotherm has an interval with $\partial p/\partial v>0$. That would have negative isothermal compressibility and is mechanically unstable. Equilibrium liquid-gas coexistence replaces the loop by a constant-pressure segment. Equality of chemical potentials gives the <Maxwell construction>:
$$
\boxed{p(v_l,T)=p(v_g,T)=p_*,\qquad\int_{v_l}^{v_g}[p(v,T)-p_*],dv=0.}
$$
The equal-area condition enforces the common tangent to the appropriate free energy; it is a stability construction rather than permission to follow the unstable loop physically. The average molar volume between $v_l$ and $v_g$ varies by changing the phase fraction.
\Image[/past-exam-of-the-physics-b-course-of-the-university-of-cambridge/2016/paper-2-fluid-phase-equilibrium.png]
{title=Original reduced Van der Waals isotherms and equal-area liquid-gas coexistence construction}
The model captures a liquid-gas critical point and qualitative coexistence, but its simple constants are not a complete quantitative description of real fluids and it has no solid phase. A full phase diagram needs additional solid free energies and can contain several crystal phases. Conversely, an experimentally determined coexistence curve and latent heat constrain equations of state through the <Clapeyron equation>. This is the useful connection: local thermodynamic response, global free-energy stability and the geometry of phase boundaries describe the same material at different levels.
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