An engine runs between a reservoir at temperature $200 \,K$ and a hot body which is initially at temperature of $600 \,K$. If the hot body cools down to a temperature of $400 \,K$ in the process, then the maximum amount of work that the engine can do (while working in a cycle) is (the heat capacity of the hot body is $1 \,J / K )$
KVPY 2020, Advanced
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$(b)$ The efficiency of engine, $\eta=\frac{W}{Q_{\text {in }}}$ or $W=\eta Q_{\text {in }}$
Also, $Q=\int C d t$
For maximum work, efficiency should be maximum, i.e. for Carnot engine,
$\eta =1-\frac{T_{2}}{T_{1}}=1-\frac{200}{T}$
$\therefore \quad W =\int \eta Q_{\text {in }}$
$=-\int \limits_{600}^{400}\left(1-\frac{200}{T}\right) C d T$
$\Rightarrow \quad W =-C[T-200 \ln T]_{600}^{400}$
$=-C\left[-200+\ln \left(\frac{3}{2}\right) 200\right] J$
Here, $C=1$ (given)
$\therefore \quad W=200-200 \ln \left(\frac{3}{2}\right) \,J$
or $\quad W=200\left[1-\ln \left(\frac{3}{2}\right)\right] \,J$
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