Consider a heat engine placed between two finite bodies at temperature 900 K and 400 Kand heat capacity 4200 J/K. What is the final temperature obtained by the two bodies for maximum possible work extraction and the maximum work respectively?

650 K, 2100 kJ

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500 K, 2100 kJ

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600 K, 420 kJ

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700 K, 420 kJ

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Solution

The correct option is C 600 K, 420 kJ
Given: $T_{1} = 900 K, T_{2} = 400 K, C_{p} = 4200 J / K$

Maximum work is obtained when the heat engine is working reversibly between the two bodies.

$Δ S_{u n i v} = 0$

$Δ S_{1} + Δ S_{2} = 0$

$C_{p} l n \frac{T_{f}}{T_{1}} + C_{p} l n \frac{T_{f}}{T_{2}} = 0$

$C_{p} l n \frac{T_{f}^{2}}{T_{1} T_{2}} = 0$
$T_{f} = \sqrt{T_{1} T_{2}}$ = 600 K

$∴$ $W_{m a x} = C_{p} (\sqrt{T_{1}} - \sqrt{T_{2}})^{2}$

$W_{m a x} = \frac{4200}{1000} (30 - 20)^{2}$

$W_{m a x} = 420 k J$

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