= Solution
An <equation of state> relates equilibrium state variables such as <pressure>, <volume (thermodynamics)>, <temperature> and material amount. At fixed composition, it reduces the number of independent thermodynamic variables. It does not by itself specify all caloric properties: information such as <heat capacity> or <internal energy> is also needed.
At a liquid-gas first-order transition, an equilibrium <isotherm> can contain a constant-pressure interval in which changing total <volume (thermodynamics)> changes the phase fractions. A <metastable> single-phase extrapolation can instead show a region with $(\partial p/\partial V)_T>0$; that region has negative <isothermal compressibility> and is mechanically unstable. <Thermodynamic phase> coexistence requires equal <temperature>, <pressure> and <chemical potential>, and a consistent free-energy description replaces the unstable region by coexistence. For an appropriate analytic <fluid> <equation of state>, a <liquid-gas critical point> occurs where $(\partial p/\partial V)_T=(\partial^2p/\partial V^2)_T=0$.
These tests diagnose the phase behaviour represented by that <equation of state>. They do not guarantee detection of every possible transition involving an additional <order parameter> which the mechanical equation omits.
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