Single Choice

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).

A$$n=\displaystyle\frac{C_P}{C_V}$$
B$$n=\displaystyle\frac{C-C_P}{C-C_V}$$
Correct Answer
C$$n=\displaystyle\frac{C_P-C}{C-C_V}$$
D$$n=\displaystyle\frac{C-C_V}{C-C_P}$$

Solution

For a polytropic process given as $$PV^n = constant$$
The specific heat is given as,

$$C = \dfrac{R}{\gamma - 1} + \dfrac{R}{1 - n}$$

$$C = C_v \dfrac{\gamma - n}{1 - n}$$

$$C = \dfrac{C_p - C_vn}{1 - n}$$

$$\therefore n = \dfrac{C - C_p}{C - C_v}$$

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