Phase Transitions derivations

On the coexistence curve, $g_{liq} = g_{gas}$ . Moving along it:

d g_{liq} = - s_{liq} d T + v_{liq} d p = d g_{gas} = - s_{gas} d T + v_{gas} d p

Rearranging:

\frac{d p}{d T} = \frac{s_{gas} - s_{liq}}{v_{gas} - v_{liq}} = \frac{L}{T (v_{gas} - v_{liq})}

where $L = T (s_{gas} - s_{liq})$ is the specific latent heat.

p = \frac{k_{B} T}{v - b} - \frac{a}{v^{2}}

Set $d p / d v = 0$ and $d^{2} p / d v^{2} = 0$ :

\frac{d p}{d v} = - \frac{k_{B} T}{(v - b)^{2}} + \frac{2 a}{v^{3}} = 0 ⟹ k_{B} T = \frac{2 a (v - b)^{2}}{v^{3}}

\frac{d^{2} p}{d v^{2}} = \frac{2 k_{B} T}{(v - b)^{3}} - \frac{6 a}{v^{4}} = 0 ⟹ k_{B} T = \frac{3 a (v - b)^{3}}{v^{4}}

Dividing:

\frac{3 (v - b)}{2 v} = 1 ⟹ v_{c} = 3 b

Substituting back:

k_{B} T_{c} = \frac{8 a}{27 b}

Verification at $(v_{c}, T_{c})$ : substituting $v_{c} = 3 b$ , $v_{c} - b = 2 b$ , $k_{B} T_{c} = 8 a / 27 b$ :

$d p / d v = - \frac{2 a}{27 b^{3}} + \frac{2 a}{27 b^{3}} = 0$

$d^{2} p / d v^{2} = \frac{2 a}{27 b^{4}} - \frac{2 a}{27 b^{4}} = 0$