Discussion
A heat engine extracts heat (\(Q_{H}\)) from a thermal reservoir at a temperature of 1000 K and rejects heat (\(Q_{L}\)) to a thermal reservoir at a temperature of 100 K, while producing work (\(W\)). Which one of the combinations of [\(Q_{H}\), \(Q_{L}\) and \(W\)] given is allowed?
Answer: B
The allowed combination is \(Q_{H}\) = 2000 J, \(Q_{L}\) = 750 J, \(W\) = 1250 J.
A heat engine must satisfy the first law, \(Q_{H} = Q_{L} + W\), and the second law, which caps its efficiency at the Carnot value:
\[ \begin{aligned} \eta_{max} &= 1 - \frac{T_{L}}{T_{H}} \\ &= 1 - \frac{100}{1000} = 0.9 \end{aligned} \]
- 2000, 750, 1250: \(750 + 1250 = 2000\), and \(\eta = 1250/2000 = 0.625 \lt 0.9\). Allowed.
- 2000, 500, 1000: \(500 + 1000 = 1500 \ne 2000\); energy is not conserved.
- 6000, 500, 5500: energy balances, but \(\eta = 5500/6000 \approx 0.917 \gt 0.9\), better than a Carnot engine, which the second law forbids.
- 6000, 600, 5500: \(600 + 5500 = 6100 \ne 6000\); more energy leaves than enters.
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