Two ideal Carnot heat engines have the same efficiency.One operates between 5.0 × 10² K and 3.0 × 10² K,and the other between 4.0 × 10² K and some lower temperature.What is the lower temperature?

Question

Quizplus · Accepted Answer

Efficiency of a Carnot engine is given by:
η = (T1 - T2)/T1
where η is the efficiency, T1 is the hot reservoir temperature, and T2 is the cold reservoir temperature.

Since both engines have the same efficiency,
(T1 - T2)/T1 = (T3 - T2)/T3

where T3 is the temperature of the cold reservoir for the second engine.

Simplifying and solving for T3, we get:
T3 = 2T2 - T1

Substituting the given values for T1 and T2, we get:
T3 = 2(4.0 × 10^3) - 5.0 × 10^3
T3 = 3.0 × 10^2 K

Therefore, the lower temperature is 240 K.

Two Ideal Carnot Heat Engines Have the Same Efficiency