Question

Write the defining equation for isothermal compressibility

Answer 1

The isothermal compressibility is related to the density fluctuations of a system, and these in turn are given by the integral of the pair density.

The isothermal compressibility eqation is :

=> χT = −1V(∂V¯∂p)T

=> N=ρ−1(∂ρ∂p)T

=> ρ−1(∂ρ∂μ)T(∂μ∂p)T = βVρ2(∂<N>∂βμ)T.

The second equality follows because at equilibrium an intensive variable is a function of only two other intensive variables, so that (∂ρ/∂V)p,T = 0, and it doesn't matter whether N or V is held fixed. The final line utilises the Gibbs–Duhem relation.

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