One mole of a monatomic ideal gas undergoes the cyclic process , as shown in the diagram. Match the quantities mentioned in List-I with their values in List-II and choose the correct option. [R is the gas constant] [2024]

Question

One mole of a monatomic ideal gas undergoes the cyclic process $J \to K \to L \to M \to J$ , 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] [2024]

	List - I		List - II
(P)	Work done in the complete cyclic process	(1)	$R T_{0} - 4 R T_{0} \ln 2$
(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)	$3 R T_{0}$
(S)	Change in the internal energy of the gas in the process MJ	(4)	$- 2 R T_{0} \ln 2$
		(5)	$- 3 R T_{0} \ln 2$

Smart Achievers · Accepted Answer

(2)

$From given P - T diagram,$

$For process J K : Isobaric$ $W = n R (3 T_{0} - T_{0}) = 2 R T_{0}$

$and Δ u = n c_{v} Δ T = n \times \frac{3 R}{2} \times 2 T_{0} = 3 R T_{0}$

$3 R T_{0}$ $W = - n R (3 T_{0}) \ln 2$

$W = - 3 R T_{0} \ln 2$

$Δ u = 0 and Q = - 3 R T_{0} \ln 2$

$For process L M : Isobaric$

$W = n R (T_{0} - 3 T_{0}) = - 2 R T_{0}$

$Δ u = n c_{v} Δ T = n \times \frac{3 R}{2} \times (- 2 T_{0}) = - 3 R T_{0}$

$For process M J : Isothermal$

$W = n R T \ln 2 = R T_{0} \ln 2$

$Δ u = 0$

$And work done in the complete cyclic process,$

$W_{net} = R T_{0} \ln 2 + 2 R T_{0} - 3 R T_{0} \ln 2 - 2 R T_{0}$

$= - 2 R T_{0} \ln 2$

$So correct option is (2)$

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