An ideal gas undergoes a quasi static, reversible process in which its molar heat capacity $$C$$ remains constant. If during this process the relation of pressure $$P$$ and volume $$V$$ is given by $$P{V^n}$$  = constant, then $$n$$ is given by (Here $${C_P}$$ and $${C_V}$$ are molar specific heat at constant pressure and constant volume, respectively):

Solution :
For a polytropic process
$$\eqalign{ & C = {C_V} + \frac{R}{{1 - n}} \cr & \therefore \,\,C - {C_V} = \frac{R}{{1 - n}} \cr & \therefore \,\,1 - n = \frac{R}{{C - {C_V}}} \cr & \therefore \,\,1 - \frac{R}{{C - {C_V}}} = n \cr & \therefore \,\,n = \frac{{C - {C_V} - R}}{{C - {C_V}}} \cr & = \frac{{C - {C_V} - {C_P} + {C_V}}}{{C - {C_V}}} \cr & = \frac{{C - {C_P}}}{{C - {C_V}}}\,\left( {\because \,{C_P} - {C_{V = R}}} \right) \cr} $$