Question

For which combination of working temperatures the efficiency of Carnot's engine is highest. (A) \(80 \mathrm{~K}, 60 \mathrm{~K}\) (B) \(100 \mathrm{~K}, 80 \mathrm{~K}\) (C) \(60 \mathrm{~K}, 40 \mathrm{~K}\) (D) \(40 \mathrm{~K}, 20 \mathrm{~K}\)

Short answer

The combination of working temperatures that gives the highest efficiency for a Carnot engine is \(40 \mathrm{~K}\) and \(20 \mathrm{~K}\) (option D), with an efficiency of \(0.5\).

Step-by-step solution

Calculate the efficiency for option A

For option (A), we have \(T_h = 80 \mathrm{~K}\) and \(T_c = 60 \mathrm{~K}\). Using the efficiency formula: \[ \eta_A = 1 - \frac{60}{80} = 1 - \frac{3}{4} = \frac{1}{4} = 0.25 \]

Calculate the efficiency for option B

For option (B), we have \(T_h = 100 \mathrm{~K}\) and \(T_c = 80 \mathrm{~K}\). Using the efficiency formula: \[ \eta_B = 1 - \frac{80}{100} = 1 - \frac{4}{5} = \frac{1}{5} = 0.2 \]

Calculate the efficiency for option C

For option (C), we have \(T_h = 60 \mathrm{~K}\) and \(T_c = 40 \mathrm{~K}\). Using the efficiency formula: \[ \eta_C = 1 - \frac{40}{60} = 1 - \frac{2}{3} = \frac{1}{3} \approx 0.333 \]

Calculate the efficiency for option D

For option (D), we have \(T_h = 40 \mathrm{~K}\) and \(T_c = 20 \mathrm{~K}\). Using the efficiency formula: \[ \eta_D = 1 - \frac{20}{40} = 1 - \frac{1}{2} = \frac{1}{2} = 0.5 \]

Compare the efficiencies and find the highest one

Now that we have calculated the efficiencies for each combination of temperatures, we can compare them to see which one gives the highest efficiency. Efficiencies: (A) \(\eta_A = 0.25\) (B) \(\eta_B = 0.2\) (C) \(\eta_C \approx 0.333\) (D) \(\eta_D = 0.5\) Comparing the efficiencies, we find that option (D) has the highest efficiency of \(0.5\). So the combination of working temperatures of \(40 \mathrm{~K}\) and \(20 \mathrm{~K}\) gives the highest efficiency for a Carnot engine. The correct answer is (D).