Assume that in the virial expansion [begin{equation*}frac{P v}{k T}=1-sum_{j=1}^{infty} frac{j}{j+1} beta_{j}left(frac{lambda^{3}}{v} ight)^{j} tag{10.4.22}end{equation*}] where (beta_{j}) are the

Assume that in the virial expansion

\[\begin{equation*}\frac{P v}{k T}=1-\sum_{j=1}^{\infty} \frac{j}{j+1} \beta_{j}\left(\frac{\lambda^{3}}{v}\right)^{j} \tag{10.4.22}\end{equation*}\]

where \(\beta_{j}\) are the irreducible cluster integrals of the system, only terms with \(j=1\) and \(j=2\) are appreciable in the critical region. Determine the relationship between \(\beta_{1}\) and \(\beta_{2}\) at the critical point, and show that \(k T_{c} / P_{c} v_{c}=3\).