= Solution
An <equation of state> relates equilibrium pressure, <temperature> and <density>. The ideal-gas form $p=nk_BT$ neglects intermolecular <forces> and excluded <volume>. A low-density real-gas expansion is
$$
\frac p{nk_BT}=1+B_2(T)n+B_3(T)n^2+\cdots,
$$
where the virial coefficients encode interaction effects and generally depend on <temperature>.
A simple phenomenological example is the molar <Van der Waals equation>
$$
\boxed{p=\frac{RT}{v-b}-\frac a{v^2}.}
$$
Here v is molar <volume>, b represents excluded <volume> and the attractive interaction lowers pressure through a. Its dilute expansion gives $pv/(RT)=1+[b-a/(RT)]/v+\cdots$. It recovers the ideal-gas limit but is not a quantitatively universal equation for all <real gases>.
Below its <critical temperature>, the raw isotherm contains a region with $(\partial p/\partial v)_T>0$, corresponding to negative <isothermal compressibility> and mechanical instability. Equilibrium <phase> separation replaces that segment by a coexistence pressure satisfying equal <chemical potentials>, equivalently the <Maxwell equal-area construction>. The <critical point> follows from $p_v=p_{vv}=0$:
$$
\boxed{v_c=3b,\qquad T_c=\frac{8a}{27Rb},\qquad p_c=\frac{a}{27b^2}.}
$$
The distinction between an unstable homogeneous branch and a stable two-phase state is essential when interpreting a real-gas <equation of state>.
\Image[/past-exam-of-the-thermal-and-statistical-physics-course-of-the-university-of-cambridge/2013/paper-1-tsp2013-real-gas.png]
{title=Original reduced van der Waals isotherms and a calculated Maxwell coexistence segment; these are model curves, not copied experimental data}
Back to article page
