Question

An polytropic ideal gas undergoes a process in which $P{V}^{-a}=\text{constant}$, where $V$ is the volume occupied by the gas initially at pressure $P$. At the end of the process,$rms$ speed of gas molecules has become $a^{\frac{1}{2}}$ times of its initial value. Find the value of $C_v$ so that energy transferred by the heat to the gas is ${}^{\prime }{a}^{\prime }$ times of the initial energy.

• A. $\frac{(a+1)R}{(a-1)}$
• B. $\frac{(a^2-1)R}{(a^2+1)}$
• C. $\frac{(a-1)R}{(a+1)}$
• D. $\frac{(a^2+1)R}{(a^2-1)}$

Answer 1

The correct option is D $\frac{(a^2+1)R}{(a^2-1)}$

Formula used:$C=\frac{R}{\gamma -1}+\frac{R}{1-x}$
Given, $P{V}^{-a}=\text{constant}$
Molar heat capacity in the polytropic process
$C=\frac{R}{\gamma -1}+\frac{R}{1-x}$
$=C_V+\frac{R}{1-(-a)}=C_V+\frac{R}{1+a}$
Formula used: $\Delta Q=nC\Delta T$
At the end of process $V_{rms}$ is $a^{\frac{1}{2}}$ times of the initial value. Temperature has become $\text{'}a\text{'}$ times (as we know $V_{rms}\alpha T^{\frac{1}{2}}$)
$\Delta Q=nC\Delta T$
$\Delta Q=nCT(a-1)=nT(a-1)$
$[C_V+\frac{R}{1+a}]$

But given $\Delta Q=aPV$
$aPV-\frac{(a-1)}{(a+1)}PV=n(a-1)C_{V}T$
(Since,$PV=nRT$)
Solving $(a_2+1)PV=n(a^2-1)C_{V}T$
Or $C_V=\frac{R\left(a^2+1\right)}{(a^2-1)}$