Discussion
Consider two Carnot heat engines A and B operating in series. Engine A receives heat from a reservoir at 1750 K and rejects heat to another reservoir at temperature T. Engine B receives an amount of energy same as that rejected by Engine A from the reservoir at temperature T. Engine B then rejects heat to another reservoir at 320 K. In both cases, the engines produce some amount of work. If the thermal efficiencies of both the engines are the same, then the temperature T is approximately
Answer: B
T is approximately 748 K.
Carnot efficiencies depend only on the reservoir temperatures. Setting them equal:
\[ \begin{aligned} 1 - \frac{T}{1750} &= 1 - \frac{320}{T} \\ T^{2} &= 1750 \times 320 = 560000 \\ T &\approx 748.3\ \text{K} \end{aligned} \]
So the intermediate temperature is the geometric mean of the extremes, not their arithmetic mean (which would be 1035 K). The other options are 100 K steps away and give unequal efficiencies: at 848 K, for example, engine A runs at 51.5% and engine B at 62.3%.
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