The ratio of adiabatic bulk modulus and isothermal bulk modulus of a gas is $\left(\gamma =C_p/C_v\right)$

For adiabatic process :

${\mathrm{PV}}^{\mathrm{\gamma }}=\mathrm{constant}$

Differentiating the above equation

$dP.V^{\gamma }=-P\gamma V^{\gamma -1}dV$

$\Rightarrow -\frac{dV}{V}=\frac{1}{\gamma P}$

For isothermal process :

PV = Constant

Differentiating the above equation

$dP.V=-PdV$

$\Rightarrow -\frac{dV}{V}=\frac{1}{P}$

The ratio of adiabatic bulk modulus and isothermal bulk modulus of a gas is 

${\left(\frac{PV}{dV}\right)}_{adiabatic}:{\left(\frac{PV}{dV}\right)}_{isothermal}$

$=\gamma :1$