One mole of a monatomic ideal gas undergoes the cyclic process J → K → L→ M → J, as shown in the P-T diagram.

Question

One mole of a monatomic ideal gas undergoes the cyclic process \(J \rightarrow K \rightarrow L\rightarrow M \rightarrow J\), as shown in the P-T diagram.

Match the quantities mentioned in List-I with their values in List-II and choose the correct option. [R is the gas constant.]

	List-I		List-II
(P)	Work done in the complete cyclic process	(1)	\(RT_0 - 4RT_0\,ln2\)
(Q)	Change in the internal energy of the gas in the process JK	(2)	0
(R)	Heat given to the gas in the process KL	(3)	\(3RT_0\)
(S)	Change in the internal energy of the gas in the process MJ	(4)	\(-2 RT_0 \, ln2\)
		(5)	\(-3 RT _0 \, ln2\)

(A) P → 1; Q → 3; R → 5; S → 4 

(B) P → 4; Q → 3; R → 5; S → 2 

(C) P → 4; Q → 1; R → 2; S → 2 

(D) P → 2; Q → 5; R → 3; S → 4

Answer 1

Correct option is (B) P → 4; Q → 3; R → 5; S → 2 

\(\mathrm{J}\left(\mathrm{P}_0, \mathrm{~V}_0, \mathrm{~T}_0\right)\)

\(\mathrm{K}\left(\mathrm{P}_0, 3 \mathrm{~V}_0, 3 \mathrm{~T}_0\right) \)

\( \mathrm{M}\left(2 \mathrm{P}_0, \frac{\mathrm{V}_0}{2}, \mathrm{~T}_0\right) \)

\( \mathrm{L}\left(2 \mathrm{P}_0, \frac{3 \mathrm{~V}_0}{2}, 3 \mathrm{~T}_0\right)\)

\(\mathrm{P}_0 \mathrm{~V}_0=\mathrm{nRT}_0 \)

\(\mathrm{JK} \rightarrow \text { isobaric } \Rightarrow \mathrm{W}=\mathrm{P}_0\left(2 \mathrm{~V}_0\right)=2 \mathrm{nRT}_0 \)

\(\Delta \mathrm{U}=\frac{3}{2} \mathrm{nR}\left(2 \mathrm{~T}_0\right)=3 n R \mathrm{~T}_0\)

\(\mathrm{KL} \rightarrow \text { isothermal } \rightarrow \mathrm{W}=\mathrm{nR}(3 \mathrm{~T}) \ln \left(\frac{1}{2}\right)=-3 \mathrm{nRT}_0 \ln 2\)

\(\Delta \mathrm{U}=0 \Rightarrow \mathrm{Q}=-3 \mathrm{nRT}_0 \ln 2\)

\( \mathrm{LM} \rightarrow \text { isobaric }=2 \mathrm{P}_0\left(-\mathrm{V}_0\right)=-2 \mathrm{nRT}_0 \)

\(\mathrm{MJ} \rightarrow \text { isothermal } \Rightarrow \mathrm{nRT}_0 \ln 2 ; \Delta \mathrm{U}=0\)

\(\mathrm{WD}_{\text {net }}=-2 \mathrm{nRT}_0\ln2\)

\(\mathrm{P} \rightarrow 4, \mathrm{Q} \rightarrow 3, \mathrm{R} \rightarrow 5, \mathrm{~S} \rightarrow 2\)