Question

A certain ideal gas undergoes a polytropic process ${\text{PV}}^{\text{n}}=$ constant such that the molar specific heat during the process is negative. If the ratio of the specific heats of the gas be $\gamma$ , then the range of values of $n$ will be

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

Answer 1

Answer: (B)

Since $P{V}^n=$ constant and also $PV=RT$, taking 1 mol of the gas for simplicity, $dU=C_{v}dT$
where $C_v\to$ molar specific heat at constant volume.
Now the molar specific heat in a polytropic process $P{V}^n=$ constant is given by $C=\left(\frac{R}{\gamma -1}\right)-\left(\frac{R}{n-1}\right)=\frac{(n-\gamma )R}{(n-1)(\gamma -1)}\dots (i)$
From this equation, we see that $C$ will be negative when $n<\gamma$ and $n>1$, simultaneously, i.e., $1<n<\gamma$. Since $\gamma$ for all ideal gases is greater than 1 , if $n>\gamma$ or $n<1$, then $C_V$ will be positive.