= Low-temperature liquid volume of a van der Waals gas
{title2=$V_l\sim b+RTb^2/a$}
For one mole, liquid–vapour <phase coexistence> requires equal <pressure> and <chemical potential>. At $\tau=RTb/a\ll1$, the saturation pressure is exponentially small. Neglecting it in the dense-phase <equation of state> gives $RTV_l^2-aV_l+ab=0$. The root near the excluded volume is
$$
V_l\simeq\frac{a}{2RT}\left(1-\sqrt{1-4RTb/a}\right)
=\frac{2b}{1+\sqrt{1-4\tau}}
=b(1+\tau+2\tau^2+\cdots).
$$
This is an asymptotic low-temperature result, not an exact coexistence volume at every subcritical temperature. Away from that limit, solve equal pressure together with the <Maxwell equal-area construction>.
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