Question

A polytropic process for an ideal gas is represented by equation $=P{V}^n$ constant. If $\gamma$ is ratio of specific heats $\left(\frac{C_p}{C_v}\right)$ . then value of n for which molar heat capacity of the process is negative, is given as

• A. $\gamma >n$
• B. $\gamma >n>1$
• C. $n>\gamma$
• D. none as it is not possible

Answer 1

The correct option is B $\gamma >n>1$

$P{V}^n=const$

$C=\frac{fR}{2}+\frac{R}{1-n}<0$

$\Rightarrow CV+\frac{R}{n-1}<0$

$\Rightarrow CV<\frac{R}{n-1}$ $$

$\because CV>0$

$\therefore n-1>0$

$\Rightarrow n>1$

$\Rightarrow \left(n-1\right)<\frac{R}{CV}$

$\Rightarrow n<\frac{R+CV}{CV}$

$\Rightarrow n<\frac{CP}{CV}$

$\Rightarrow n<\gamma$

$\therefore \gamma >n>1$

Hence,

option $\left(B\right)$ is correct answer.