One mole of an ideal monoatomic gas undergoes the following four reversible processes:

Step $1$ It is first compressed adiabatically from volume $V_{1}$ to $1 \;m ^{3}$.

Step $2$ Then expanded isothermally to volume $10 \;m ^{3}$.

Step $3$ Then expanded adiabatically to volume $V _{3}$.

Step $4$ Then compressed isothermally to volume $V_{1}$. If the efficiency of the above cycle is $3 / 4$, then $V_{1}$ is ............ $m^3$

Question

One mole of an ideal monoatomic gas undergoes the following four reversible processes:

Step $1$ It is first compressed adiabatically from volume $V_{1}$ to $1 \;m ^{3}$.

Step $2$ Then expanded isothermally to volume $10 \;m ^{3}$.

Step $3$ Then expanded adiabatically to volume $V _{3}$.

Step $4$ Then compressed isothermally to volume $V_{1}$. If the efficiency of the above cycle is $3 / 4$, then $V_{1}$ is ............ $m^3$

KVPY 2017, Advanced

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Solution

$(d)$ Given $p-V$ cycle is

$A B:$ Adiabatic compression,

$V_{A}=V_{1}, V_{B}=1 \,m ^{3}$

$B C:$ Isothermal expansion,

$V_{C}=V=10 \,m ^{3}$

$C D$ : Adiabatic expansion,

$V_{D}=V_{3}$

$D A$ : Isothermal compression,

$V_{A}=V_{1}$

Cycle efficiency is given, $\eta=\frac{3}{4}$

For given Carnot's cycle,

$\eta=1-\frac{T_{1}}{T_{2}}=1-\left(\frac{V_{2}}{V_{1}}\right)^{\gamma-1}$

$[\therefore$ Process $A B$ is adiabatic, $\gamma=\frac{5}{3}$ for monoatomic gas]

$\Rightarrow \quad \frac{3}{4}=1-\left(\frac{1}{V_{1}}\right)^{\frac{5}{3}-1} \Rightarrow \frac{1}{V_{1}^{2 / 3}}=\frac{1}{4}$

$\Rightarrow V_{1}=8 \,m ^{3}$

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