Question 88: Thermodynamics & KTG

Question

In the given $P-V$ diagram, a monoatomic gas $\left(\gamma=\frac{5}{3}\right)$ is first compressed adiabatically from state $A$ to state $B$. Then it expands isothermally from state $B$ to state $C$. [Given: $\left(\frac{1}{3}\right)^{0.6} \simeq 0.5, \ln 2 \simeq 0.7$ ].

Which of the following statement(s) is(are) correct?

Accepted Answer

For adiabatic process $(\mathrm{A} ightarrow \mathrm{B})$
$$
\begin{aligned}
& \mathrm{P}_{\mathrm{A}} \mathrm{V}_{\mathrm{A}}^\gamma=\mathrm{P}_{\mathrm{B}} \mathrm{V}_{\mathrm{B}}^\gamma \\
& 10^5 	imes(0.8)^{\frac{5}{3}}=3 	imes 10^5\left(\mathrm{~V}_{\mathrm{B}}ight)^{\frac{5}{3}} \\
& \Rightarrow \mathrm{V}_{\mathrm{B}}=0.8 	imes\left(\frac{1}{3}ight)^{0.6}=0.4
\end{aligned}
$$
Work done in process $\mathrm{A} ightarrow \mathrm{B}$
$$
\begin{aligned}
& \mathrm{W}_{\mathrm{AB}}=\frac{\mathrm{P}_{\mathrm{A}} \mathrm{V}_{\mathrm{A}}-\mathrm{P}_{\mathrm{B}} \mathrm{V}_{\mathrm{B}}}{\gamma-1} \\
& \Rightarrow \mathrm{W}_{\mathrm{AB}}=\frac{10^5 	imes 0.8-3 	imes 10^5 	imes 0.4}{\frac{5}{3}-1} \\
& \Rightarrow \mathrm{W}_{\mathrm{AB}}=-60 \mathrm{~kJ}=\Rightarrow\left|\mathrm{W}_{\mathrm{AB}}ight|=60 \mathrm{~kJ}
\end{aligned}
$$
Work done in process $\mathrm{B} ightarrow \mathrm{C}$ (Isothermal process)
$$
\begin{aligned}
& \mathrm{W}_{\mathrm{BC}}=\mathrm{nRT} \ell \mathrm{n} \frac{\mathrm{V}_{\mathrm{C}}}{\mathrm{V}_{\mathrm{B}}}=\mathrm{P}_{\mathrm{B}} \mathrm{V}_{\mathrm{B}} \ell \mathrm{n} \frac{\mathrm{V}_{\mathrm{C}}}{\mathrm{V}_{\mathrm{B}}} \\
& \Rightarrow \mathrm{W}_{\mathrm{BC}}=3 	imes 10^5 	imes 0.4 \ell \mathrm{n} \frac{0.8}{0.4} \\
& \Rightarrow \mathrm{W}_{\mathrm{BC}}=84 \mathrm{~kJ}
\end{aligned}
$$
Work done in process $\mathrm{C} ightarrow \mathrm{A}$
$$
\mathrm{W}_{\mathrm{CA}}=\mathrm{P} \Delta \mathrm{V}=0 \quad(\because \Delta \mathrm{V}=0)
$$
So total work done in the process $\mathrm{A} ightarrow \mathrm{B} ightarrow \mathrm{C}$
$$
\begin{aligned}
& \mathrm{W}_{\mathrm{ABC}}=\mathrm{W}_{\mathrm{AB}}+\mathrm{W}_{\mathrm{BC}}+\mathrm{W}_{\mathrm{CA}}=-60+84+0 \\
& \mathrm{~W}_{\mathrm{ABC}}=24 \mathrm{~kJ}
\end{aligned}
$$
So correct options are (B,C,D)

Answer

The magnitude of the total work done in the process $A \rightarrow B \rightarrow C$ is $144 \mathrm{~kJ}$.

Answer

The magnitude of the work done in the process $B \rightarrow C$ is $84 \mathrm{~kJ}$.

Answer

The magnitude of the work done in the process $A \rightarrow B$ is $60 \mathrm{~kJ}$.

Answer

The magnitude of the work done in the process $C \rightarrow A$ is zero.

Thermodynamics & KTG

Choose the correct answer: