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(B) For a polytropic process $PV^n = 	ext{constant}$, the molar heat capacity $C$ is given by the formula:$C = C_V + \frac{R}{1-n} = \frac{R}{\gamma-

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$-1$

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(B) For a polytropic process $PV^n = 	ext{constant}$, the molar heat capacity $C$ is given by the formula:$C = C_V + \frac{R}{1-n} = \frac{R}{\gamma-1} + \frac{R}{1-n}$According to the problem, the molar heat capacity $C$ is the arithmetic mean of $C_P$ and $C_V$:$C = \frac{C_P + C_V}{2}$Since $C_P = C_V + R$, we can write:$C = \frac{(C_V + R) + C_V}{2} = \frac{2C_V + R}{2} = C_V + \frac{R}{2}$Equating the two expressions for $C$:$C_V + \frac{R}{1-n} = C_V + \frac{R}{2}$Subtracting $C_V$ from both sides:$\frac{R}{1-n} = \frac{R}{2}$This implies:$1 - n = 2$$n = 1 - 2 = -1$Therefore, the value of $n$ is $-1$.

An ideal gas undergoes a polytropic process given by the equation $PV^n = \text{constant}$. If the molar heat capacity of the gas during this process is the arithmetic mean of its molar heat capacity at constant pressure $(C_P)$ and constant volume $(C_V)$, then the value of $n$ is ..............

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