Question

An ideal gas at initial temperature $T_{0}$ and initial volume $V_{0}$ is expanded adiabatically to a volume $2 V_{0}$. The gas is then expanded isothermally to a volume $5 V_{0}$ and there after compressed adiabatically so that the temperature of the gas becomes again $T_{0}$. If the final volume of the gas is $\alpha V_{0}$, then the value of constant $\alpha$ is

• A. 2.5
• B. 1.5
• C. 2.0
• D. 3.0

Answer 1

Answer: (A)

For adiabatic expansion,
$T_{1} V_{1}^{\gamma-1}=T_{2} V_{2}^{\gamma-1} \Rightarrow \frac{T_{2}}{T_{1}}=\left(\frac{V_{1}}{V_{2}}\right)^{\gamma-1}$
Here, $T_{1}=T_{0}$
$\frac{T_{2}}{T_{0}} =\left(\frac{V_{0}}{2 V_{0}}\right)^{\gamma-1} $
$T_{2}=\left(\frac{1}{2}\right)^{\gamma-1} T_{0} \begin{bmatrix}V_{1}=V_{0} \\ V_{2}=2 V_{0}\end{bmatrix}$
Again, gas is expanded isothermally, therefore
$p_{1} V_{1}^{\prime}=p_{2} V_{2}^{\prime}$
Here, $ \begin{bmatrix}
V_{1}^{\prime}=V_{2}=2 V_{0} \\
V_{2}^{\prime}=5 V_{0}
\end{bmatrix}$
$\frac{p_{1}}{p_{2}}=\frac{V_{2}^{\prime}}{V_{1}^{\prime}}=\frac{5 V_{0}}{2 V_{0}}$
$\frac{p_{1}}{p_{2}}=\frac{V_{2}^{\prime}}{V_{1}^{\prime}}=\frac{5}{2}$
Finally gas compressed again adiabatically, therefore, by adiabatic relation,
$T V^{\gamma-1}=$ constant
$T_{2}\left(V_{2}^{\prime}\right)^{\gamma-1}=T_{3}\left(V_{3}\right)^{\gamma-1} \Rightarrow \left(\frac{V_{3}}{V_{2}^{\prime}}\right)^{\gamma-1}=\frac{T_{2}}{T_{3}}$
$\left[\therefore T_{3}=T_{0}\right]$
$\left(\frac{V_{3}}{V_{2}^{\prime}}\right)^{\gamma-1}=\frac{\left(\frac{1}{2}\right)^{\gamma-1} T_{0}}{T_{0}} \Rightarrow V_{3}=\frac{V_{2}^{\prime}}{2}=\frac{5 V_{0}}{2}=2.5 V_{0}$
Hence, the value of constant is $2.5$.