b

is the excluded volume per particle caused by short-range repulsion. The positive constant

a

measures cohesive attraction, which lowers the pressure through the Boltzmann factor.

ii

Solution

At low density,

\frac{p}{k _{B} T} = \frac{1}{v - b} exp (- \frac{a}{k _{B} T v}) = \frac{1}{v} + \frac{b - a / ( k _{B} T )}{v ^{2}} + O (v^{- 3}) .

B_{2} (T) = b - \frac{a}{k _{B} T} .

(\partial p / \partial v)_{T} = (\partial^{2} p / \partial v^{2})_{T} = 0

. The first logarithmic derivative gives

\frac{a}{k _{B} T v ^{2}} = \frac{1}{v - b} .

\frac{1}{( v - b ) ^{2}} - \frac{2 a}{k _{B} T v ^{3}} = 0.

Combining the equations gives

v_{c} = 2 b

, and substitution then gives

T_{c} = \frac{a}{4 b k _{B}} .

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