An ideal Carnot engine operates between a warm reservoir at 233 K and a colder reservoir. During each cycle, this engine extracts $15.0 \mathrm {~J}$ of heat from the warm reservoir and does $4.0 \mathrm {~J}$ of work. What is the temperature of the colder reservoir?
A) 171 K
B) 62 K
C) 47 K
D) 67 K

Question

Quizplus · Accepted Answer

From the Carnot cycle, we know that the efficiency of the engine is given by 
$$\eta = 1 - \frac{T_c}{T_h}$$
where $T_c$ and $T_h$ are the temperatures of the cold and hot reservoirs, respectively. We are given that the engine extracts $15.0 \mathrm{~J}$ of heat from the warm reservoir and does $4.0 \mathrm{~J}$ of work during each cycle. Thus, the amount of heat rejected to the cold reservoir is 
$$Q_c = Q_h - W = 15.0 \mathrm{~J} - 4.0 \mathrm{~J} = 11.0 \mathrm{~J}$$
Since the engine is ideal and operates in a Carnot cycle, we know that it is a reversible process and the amount of heat rejected to the cold reservoir is related to the amount of heat absorbed from the hot reservoir by 
$$\frac{Q_h}{T_h} = \frac{Q_c}{T_c}$$
Substituting our values and manipulating the equation we have 
$$T_c = \frac{Q_c}{Q_h}\cdot T_h = \frac{11.0 \mathrm{~J}}{15.0\mathrm{~J}}\cdot 233 \mathrm{~K} \approx 171 \mathrm{~K}$$
Therefore, the temperature of the colder reservoir is approximately $171 \mathrm{~K}$. Thus, the best choice is (A).

An Ideal Carnot Engine Operates Between a Warm Reservoir at 233