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 $V_{1}$.

• A. $2 \,m^3$
• B. $4 \,m^3$
• C. $6 \,m^3$
• D. $8 \,m^3$

Answer 1

Answer: (D)

Given $p-V$ cycle is

$AB$ : Adiabatic compression,
$V_A=V_1,V_B=1\, m^3$
$BC$ : Isothermal expansion,
$V_C = V = 10 \,m^3$
$CD$ : Adiabatic expansion,
$V_D = V_3$
$DA$ : Isothermal compression,
$V_A = V_1$
Cycle efficiency is given,$\eta=\frac{3}{4}$
For given Carnot’s cycle,y-
$\eta=1-\frac{T_{1}}{T_{2}}=1-\left(\frac{V_{2}}{V_{1}}\right)^{\gamma-1}$
$[\therefore $ Process $A B$ is adiabatic, $\gamma=\frac{5}{3}$ for monoatomic gas]
$\Rightarrow \frac{3}{4}=1-\left(\frac{1}{V_{1}}\right)^{\frac{5}{3}-1}$
$ \Rightarrow \frac{1}{V_{1}^{2 / 3}}=\frac{1}{4}$
$\Rightarrow V_{1}=8 \,m ^{3}$