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 $1m^3$.
Step 2 Then expanded isothermally to volume $10m^3$.
Step 3 Then expanded adiabatically to volume $V_3$.
Step 4 Then compressed isothermally to volume $V_1$.

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

Answer 1

Answer: (D)

Given $p-V$ cycle is

$AB$ : Adiabatic compression,
$V_A=V_1,V_B=1m^3$
$BC$ : Isothermal expansion,
$V_C=V=10m^3$
$CD$ : 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 $AB$ is adiabatic, $\gamma =\frac{5}{3}$ for monoatomic gas]
$\Rightarrow \frac{3}{4}=1-{\left(\frac{1}{V_1}\right)}^{\frac{5}{3}-1}$
$\Rightarrow \frac{1}{V_1^{2/3}}=\frac{1}{4}$
$\Rightarrow V_1=8m^3$