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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)
$n = \frac{{C - {C_V}}}{{C - {C_P}}}$
$n = \frac{{{C_P}}}{{{C_V}}}$
$n = \frac{{C - {C_P}}}{{C - {C_V}}}$
$n = \frac{{{C_P} - C}}{{C - {C_V}}}$
Solution
Specific heat $C=\frac{R}{1-n}+C_{V}$ for polytropic process
$\therefore \frac{\mathrm{R}}{1-\mathrm{n}}+\mathrm{C}_{\mathrm{V}}=\mathrm{C}$
$\frac{\mathrm{R}}{1-\mathrm{n}}=\mathrm{C}-\mathrm{C}_{\mathrm{v}} \Rightarrow \frac{\mathrm{R}}{\mathrm{C}-\mathrm{C}_{\mathrm{V}}}=1-\mathrm{n}$
(Where $\left.\mathrm{R}=\mathrm{C}_{\mathrm{P}}-\mathrm{C}_{\mathrm{V}}\right)$
$\Rightarrow \mathrm{n}=\frac{\mathrm{C}-\mathrm{C}_{\mathrm{p}}}{\mathrm{C}-\mathrm{C}_{\mathrm{V}}}$
