Use [ begin{aligned} {left[kappa_{T}(n, T) ight]^{-1} } & =nleft(frac{partial p}{partial n} ight)_{T} P(n, T)

Use

\[
\begin{aligned}
{\left[\kappa_{T}(n, T) \right]^{-1} } & =n\left(\frac{\partial p}{\partial n} \right)_{T} \\
P(n, T) & =n^{2}\left(\frac{\partial f}{\partial n} \right)_{T}
\end{aligned}
\]

where \(f=A / N\) is the Helmholtz free energy per particle. Then

\[
\begin{aligned}
P(n, T) & =p\left(n_{0}, T \right)+\int_{n_{0}}^{n} \frac{d n^{\prime}}{n^{\prime} \kappa\left(n^{\prime}, T \right)}, \\
f(n, T) & =f\left(n_{0}, T \right)+\int_{n_{0}}^{n} \frac{P\left(n^{\prime}, T \right)}{\left(n^{\prime} \right)^{2}} d n^{\prime} .
\end{aligned}
\]