Question

A certain ideal gas undergoes a polytropic process $\textit{PV}^{\text{n}}=$ 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 $P V=R T$, taking 1 mol of the gas for simplicity, $d U=C_{ v } d T$
where $C_{ v } \rightarrow$ molar specific heat at constant volume.
Now the molar specific heat in a polytropic process $P V^{ n }=$ constant is given by $C=\left(\frac{R}{\gamma-1}\right)-\left(\frac{R}{ n -1}\right)=\frac{( n -\gamma) R}{( n -1)(\gamma-1)} \ldots( i )$
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.