Question

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)

• A. $\begin{array}{c}n=\frac{C_p}{C_v}\end{array}$
• B. $\begin{array}{c}n=\frac{C-C_p}{C-C_v}\end{array}$
• C. $\begin{array}{c}n=\frac{C_p-C}{C-C_v}\end{array}$
• D. $\begin{array}{c}\\ n=\frac{C-C_v}{C-C_p}\end{array}$

Answer 1

The correct option is B $\begin{array}{c}n=\frac{C-C_p}{C-C_v}\end{array}$

For a polytropic process given as $P{V}^n$ = constant.
The specific heat is given as,
$C=C_v+\frac{R}{1-n}$
$⟹1-n=\frac{R}{C-C_v}$
$n=1-\frac{R}{C-C_v}$

$=\frac{C-(C_v+R)}{C-C_v}$

$=\frac{C-C_p}{C-C_v}$