Question

A certain ideal gas undergoes a polytropic process represented by equation $P{V}^n=\text{constant}$. If Molar heat capacity for this process is negative and $\gamma$ is a standard constant which denotes the ratio of the molar heat capacities of a gas, then the range of values of $n$ will be

• A. $0<n<\gamma$
• B. $1<n<\gamma$
• C. $n=\gamma$
• D. $n>\gamma$

Answer 1

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

Molar heat capacity for a polytropic process [$P{V}^n=\text{constant}$] is given by,
$C=\left(\frac{R}{\gamma -1}\right)-\left(\frac{R}{n-1}\right)$
$\Rightarrow C=\frac{(n-\gamma )R}{(n-1)(\gamma -1)}....\left(i\right)$
For $C$ (molar heat capacity) to be $-ve$,
Since $\gamma >1$ always, thus $C$ can be negative in two case.
Case 1:
$(n-\gamma )<0$ and $(n-1)>0$
This gives $n<\gamma$ and $n>1$

Case 2:
$(n-\gamma )>0$ and $(n-1)<0$
This gives $n>\gamma$ and $n<1$
But if $\gamma >1$ and $n>\gamma$, this means $n>1$. Hence this case is not possible.

Therefore for $C$ to be negative, $(n<\gamma )$ and $(n>1)$ or $(1<n<\gamma )$