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 Incoming links: Virial criterion for Joule-Thomson cooling

Past exam of the mathematics course of the University of Cambridge / 2015 / ii / Paper 3 / 33C / b / iii / Solution Created 2026-09-24 Updated 2026-10-06

Write

n = N / V

and

B = B_{2} (T)

. Differentiating the virial expansion

p = k_{B} T (n + B n^{2})

at fixed pressure gives

0 = (n + B n^{2} + T B^{'} n^{2}) d T + T (1 + 2 B n) d n .

Since

d V / V = - d n / n

,

\frac{T}{V} (\frac{\partial V}{\partial T})_{p} - 1 = \frac{n ( T B ^{'} - B )}{1 + 2 B n} = n (T B^{'} - B) + O (n^{2}) .

Thus the virial criterion for Joule-Thomson cooling gives, for the displayed truncated equation of state,

μ = \frac{N}{C _{p}} \frac{T B _{2}^{'} - B _{2}}{1 + 2 B _{2} n} = \frac{N}{C _{p}} (T B_{2}^{'} - B_{2}) [1 - 2 B_{2} n + O (n^{2})] .

In particular the first density correction to the ideal-gas thermal-expansion factor gives the leading result

μ = N (T B_{2}^{'} - B_{2}) / C_{p}

. The heat capacity here is the actual

C_{p}

at the state considered; replacing it by an ideal-gas value is a further leading-order approximation. On the low-density mechanically stable branch with

C_{p} > 0

and

1 + 2 B_{2} n > 0

, positive cooling coefficient requires

T B_{2}^{'} - B_{2} > 0 ⟺ \frac{d}{d T} (\frac{B _{2} ( T )}{T}) > 0.

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