The difference between isothermal compressibility and adiabatic compressibility for an ideal gas is

Solution(By Examveda Team)

By $$T-Ds$$  Equations at constant entropy
$${C_p}dT = T{\frac{{\partial V}}{{\partial T}}_P}dP$$     and  $${C_v} = - T{\left( {\frac{{\partial P}}{{\partial T}}} \right)_P}{\left( {\frac{{\partial V}}{{\partial T}}} \right)_S}$$
$$ \Rightarrow \frac{{{C_P}}}{{{C_V}}} = \frac{{\left( {\frac{{\partial P}}{{\partial V}}} \right)S}}{{\left( {\frac{{\partial P}}{{\partial V}}} \right)T}}$$
Since, $${C_P}$$ is always greater than $${C_V}$$ the ratio of isothermal compressibility and isentropic (reversible adiabatic) process is always greater than $$1 \Rightarrow $$ the difference is greater than zero.