The coefficient of volume expansion \(\alpha\) and the isothermal bulk modulus \(B\) of a gas are given by the empirical expressions

\[\alpha=\frac{1}{T}\left(1+\frac{3 a^{\prime}}{v T^{2}}\right) \quad \text { and } \quad B=P\left(1+\frac{a^{\prime}}{v T^{2}}\right)^{-1}\]

where \(a^{\prime}\) is a constant parameter. Show that these expressions are mutually compatible. Also derive the equation of state of this gas.