Problem — Van der Waals isotherm below T_c

Problem

A Van der Waals isotherm with $T < T_c$ shows a horizontal stretch in the $p$-$V$ plane. Explain the physical meaning of that stretch and how its extent varies as $T$ increases.

Solution

Meaning of the horizontal stretch. On the real isotherm, the constant-pressure plateau represents the coexistence of liquid and vapour during the phase transition. Compressing the gas along the isotherm, once the saturation pressure is reached the substance starts to condense: with further compression the volume decreases (more and more liquid, less and less vapour) but the pressure does not change, because the system simply converts vapour into liquid. Hence the horizontal segment. $p = p_\text{sat}(T) = \text{constant along the whole plateau}$ The S-shaped curve predicted by the Van der Waals equation (with the positive-slope stretch, mechanically unstable) is replaced by this segment via the Maxwell construction, which fixes $p_\text{sat}$ by imposing equal areas above and below.

Variation with temperature. Left end of the plateau = volume of the saturated liquid; right end = volume of the saturated vapour. As $T$ increases the liquid expands and the vapour is compressed: the two ends approach each other, so the horizontal stretch shortens. $T\uparrow\ \Longrightarrow\ V_\text{vapour}-V_\text{liquid}\downarrow\ \Longrightarrow\ \text{shorter plateau}$ At the critical temperature $T_c$ the two volumes coincide: the plateau vanishes and shrinks to the critical point (an inflection with horizontal tangent). For $T > T_c$ there is no longer any distinction between liquid and vapour. $\ev{\text{plateau} = \text{liquid-vapour coexistence at } p_\text{sat};\ \text{shortens with } T,\ \text{vanishes at } T_c}$

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Topics: Kinetic theory of gases Concepts: Ideal gas law · Pressure Skills: Reading graphs