Calculate the isothermal compressibility and Helmholtz free energy for the CarnahanStarling equation of state (10.3.25) and show that the Helmholtz free energy is given by

\[\frac{\beta A}{N}=\frac{\beta A^{\text {ideal }}}{N}+\frac{\eta(4-3 \eta)}{(1-\eta)^{2}}\]

where \(A^{\text {ideal }}\) is the Helmholtz free energy of a classical monatomic ideal gas at the same density.