Isothermal compressibility of an ideal gas is

Compressibility (τ):\(\tau = - \frac{1}{v}\frac{{\partial v}}{{\partial P}} = \frac{1}{\RHO }\frac{{\partial \rho }}{{\partial P}}\)Compressibility is thus INVERSE of bulk modulus. Hence compressibility can be defined as the incurred volummetric strain for unit change in pressure.Isentropic compressibility: \(\tau = - \frac{1}{v}{\LEFT( {\frac{{\partial v}}{{\partial P}}} \right)_{s = constant}} = \frac{1}{\rho }{\left( {\frac{{\partial \rho }}{{\partial P}}} \right)_{s = constant}}\)Isothermal compressibility: \(\tau = - \frac{1}{v}{\left( {\frac{{\partial v}}{{\partial P}}} \right)_{T = constant}} = \frac{1}{\rho }{\left( {\frac{{\partial \rho }}{{\partial P}}} \right)_{T = constant}}\)SINCE: PV=NRT\({\left( {\frac{{\partial v}}{{\partial P}}} \right)_{T = constant}} = - \frac{{nRT}}{{{P^2}}} = - \frac{{nRT}}{P}.\frac{1}{P} = - \frac{v}{P}\)Hence, isothermal compressibility is\(\tau = \frac{{nRT}}{{V{P^2}}} = \frac{1}{P}\)