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 Incoming links: Ideal gas free energy

u (r)

, so that in the dilute limit only one interacting pair need be retained at a time. The leading Mayer f-function expansion is

e^{- U / T} - 1 ≃ \sum_{1 \leq i < j \leq N} (e^{- u (∣ x_{i} - x_{j} ∣) / T} - 1) .

For each pair, translation of its center coordinate supplies

V

and the other

N - 2

particles supply

V^{N - 2}

. Neglecting boundary corrections,

\frac{1}{V ^{N}} \int (e^{- U / T} - 1) d^{3 N} x ≃ \frac{N ( N - 1 )}{2 V} \int_{R^{3}} (e^{- u (r) / T} - 1) d^{3} r .

lo g (1 + x) = x + O (x^{2})

and

N (N - 1) ≃ N^{2}

F = F_{ideal} + \frac{N ^{2} T}{V} B (T), B (T) = \frac{1}{2} \int_{R^{3}} (1 - e^{- u (r) / T}) d^{3} r = 2 π \int_{0}^{\infty} (1 - e^{- u (r) / T}) r^{2} d r .

Using

P = - (\partial F / \partial V)_{T, N}

P = \frac{NT}{V} + \frac{N ^{2} T}{V ^{2}} B (T) .

n = N / V

, this is

P / (n T) = 1 + B (T) n + O (n^{2})

, so

B (T)

- N k_{B} T lo g V

. With

V = 4 π r^{3} /3

4 π t r^{2}

, the radius-dependent free energy is therefore

F (r) = - N k_{B} T lo g V + 4 π t r^{2} + constant .

For equilibrium against external pressure

P_{0}

, minimize

F + P_{0} V