A heat engine operates with an efficiency of \(0.5 .\) What can the temperatures of the high-temperature and lowtemperature reservoirs be? a) \(T_{\mathrm{H}}=600 \mathrm{~K}\) and \(T_{\mathrm{L}}=100 \mathrm{~K}\) b) \(T_{\mathrm{H}}=600 \mathrm{~K}\) and \(T_{\mathrm{L}}=200 \mathrm{~K}\) c) \(T_{\mathrm{H}}=500 \mathrm{~K}\) and \(T_{\mathrm{L}}=200 \mathrm{~K}\) d) \(T_{\mathrm{H}}=500 \mathrm{~K}\) and \(T_{\mathrm{L}}=300 \mathrm{~K}\) e) \(T_{\mathrm{H}}=600 \mathrm{~K}\) and \(T_{\mathrm{L}}=300 \mathrm{~K}\)

Write down the formula for efficiency and plug in the given efficiency value: $$ 0.5 = 1 - \frac{T_L}{T_H} $$

Plug in the temperatures mentioned in each option and check if the equation is satisfied. (a) \(T_H = 600 \mathrm{~K}\) and \(T_L = 100 \mathrm{~K}\): $$ 0.5 = 1 - \frac{100}{600} = 1 - \frac{1}{6} = \frac{5}{6} $$ Answer (a) not correct. (b) \(T_H = 600 \mathrm{~K}\) and \(T_L = 200 \mathrm{~K}\): $$ 0.5 = 1 - \frac{200}{600} = 1 - \frac{1}{3} $$ Answer (b) is correct. (c) \(T_H = 500 \mathrm{~K}\) and \(T_L = 200 \mathrm{~K}\): $$ 0.5 = 1 - \frac{200}{500} = 1 - \frac{2}{5} = \frac{3}{5} $$ Answer (c) not correct. (d) \(T_H = 500 \mathrm{~K}\) and \(T_L = 300 \mathrm{~K}\): $$ 0.5 = 1 - \frac{300}{500} = 1 - \frac{3}{5} = \frac{2}{5} $$ Answer (d) not correct. (e) \(T_H = 600 \mathrm{~K}\) and \(T_L = 300 \mathrm{~K}\): $$ 0.5 = 1 - \frac{300}{600} = 1 - \frac{1}{2} $$ Answer (e) not correct.

The heat engine operating with an efficiency of \(0.5\) would have the high-temperature and low-temperature reservoirs as \(T_H = 600 \mathrm{~K}\) and \(T_L = 200 \mathrm{~K}\) (Option b).