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 $$PV^n$$ = constant, then n is given by (Here $$C_P$$ and $$C_V$$ are molar specific heat at constant pressure and constant volume, respectively):

$$dQ = dU + dW$$

$$dQ = C dT$$, $$dU = C_V dT$$, $$dW = P dV$$

$$C dT = C_V dT + P dV$$

$$PV^n = \text{constant}$$

$$\left(\frac{RT}{V}\right)V^n = \text{constant} \implies TV^{n-1} = \text{constant}$$

Differentiating both sides with respect to $$T$$ (To rewrite $$P dV $$ in terms of $$dT$$ ):

$$V^{n-1} + T(n-1)V^{n-2} \frac{dV}{dT} = 0$$

$$V + T(n-1) \frac{dV}{dT} = 0$$

$$\frac{dV}{dT} = \frac{-V}{T(n-1)}$$

$$P dV = \left(\frac{RT}{V}\right) \left(\frac{-V}{T(n-1)}\right) dT = \frac{-R}{n-1} dT = \frac{R}{1-n} dT$$

$$C dT = C_V dT + \frac{R}{1-n} dT$$

$$C = C_V + \frac{C_P - C_V}{1-n}$$

$$C - C_V = \frac{C_P - C_V}{1-n}$$

$$1 - n = \frac{C_P - C_V}{C - C_V}$$

$$n = 1 - \frac{C_P - C_V}{C - C_V}$$

$$n = \frac{C - C_P}{C - C_V}$$