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(B) For a heat engine operating between a variable temperature source $T$ and a fixed temperature sink $T_0 = 200 \,K$,the maximum work is obtained wh

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$200(1-\ln(3/2)) \,J$

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(B) For a heat engine operating between a variable temperature source $T$ and a fixed temperature sink $T_0 = 200 \,K$,the maximum work is obtained when the engine operates as a Carnot engine at every instant.The efficiency of a Carnot engine is given by $\eta = 1 - \frac{T_0}{T}$.The infinitesimal work done $dW$ for an infinitesimal heat $dQ_{in} = -C dT$ extracted from the hot body is $dW = \eta dQ_{in} = (1 - \frac{T_0}{T})(-C dT)$.Integrating from the initial temperature $T_i = 600 \,K$ to the final temperature $T_f = 400 \,K$:$W = \int_{600}^{400} -(1 - \frac{200}{T}) C dT = C \int_{400}^{600} (1 - \frac{200}{T}) dT$.Given $C = 1 \,J/K$:$W = [T - 200 \ln T]_{400}^{600} = (600 - 400) - 200(\ln 600 - \ln 400)$.$W = 200 - 200 \ln(600/400) = 200 - 200 \ln(3/2) = 200(1 - \ln(3/2)) \,J$.

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