A fixed amount of an ideal gas ( n mol) goes through this cycle: 1-2 reversible isothermal expansion at T h from V 1 to V 2 ; 2-3 isochoric cooling to T c ; 3-4 reversible isothermal compressi

Question

A fixed amount of an ideal gas ( n mol) goes through this cycle: 1-2 reversible isothermal expansion at T h ​ from V 1 ​ to V 2 ​ ; 2-3 isochoric cooling to T c ​ ; 3-4 reversible isothermal compression at T c ​ from V 2 ​ to V 1 ​ ; 4-1 isochoric heating to T h ​ . No heat regenerator is used. Select the correct statement(s).

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Work in 1-2 = n R T h ​ ln ( V 2 ​ / V 1 ​ ) ; in 3-4 = − n R T c ​ ln ( V 2 ​ / V 1 ​ ) ; isochoric steps do no work. Net = n R ( T h ​ − T c ​ ) ln ( V 2 ​ / V 1 ​ ) : (A) correct. Step 4-1 is at constant volume: q = n C V ​ ( T h ​ − T c ​ ) : (B) correct. U is a state function, so Δ U cyc l e ​ = 0 : (C) correct. Heat taken in = n R T h ​ ln ( V 2 ​ / V 1 ​ ) + n C V ​ ( T h ​ − T c ​ ) , which is more than n R T h ​ ln ( V 2 ​ / V 1 ​ ) . So η < 1 − T c ​ / T h ​ (the Carnot value): (D) wrong. Answer: A, B, C.

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