Question

A Carnot engine takes $300{\text{cal}}$ of heat from a reservoir at $500{\text{K}}$ and rejects $150{\text{cal}}$ to a sink. Find the temperature of the sink
a) $1000{\text{K}}$
b) $750{\text{K}}$
c) $250{\text{K}}$
d) $125{\text{K}}$

Answer 1

Hint: Apply the Carnot’s theorem to find the temperature of the sink

Formula Used: Carnot’s theorem gives us $\dfrac{{{Q_1}}}{{{T_1}}} = \dfrac{{{Q_2}}}{{{T_2}}}$ where ${Q_1}$ and ${Q_2}$ represents the heat taken from the reservoir (source) and ${T_1}$ its temperature, ${Q_2}$ represents the heat rejected to the sink and ${T_2}$ represents the sink’s temperature.

Complete step by step answer:
Step 1: List the information provided in the question
     The heat taken from the source is ${Q_1} = 300{\text{cal}}$
     The temperature of the source or reservoir is ${T_1} = 500{\text{K}}$
     The heat taken from the sink is ${Q_2} = 150{\text{cal}}$
     The temperature of the sink ${T_2}$ is unknown
Step 2: State the Carnot’s theorem

Carnot’s theorem states that every Carnot heat engine between a pair of reservoirs is equally efficient, regardless of the working conditions.

This suggests that $\dfrac{{{Q_1}}}{{{T_1}}} = \dfrac{{{Q_2}}}{{{T_2}}}$, where ${Q_1}$ represents the heat taken from the reservoir (source) and ${T_1}$ its temperature, ${Q_2}$ represents the heat rejected to the sink and ${T_2}$ represents the sink’s temperature.

Step 3: Express the Carnot’s relation and substitute the values of ${Q_1} = 300{\text{cal}}$ , ${T_1} = 500{\text{K}}$ , ${Q_2} = 150{\text{cal}}$ to find ${T_2}$
     From the Carnot’s theorem we have, $\dfrac{{{Q_1}}}{{{T_1}}} = \dfrac{{{Q_2}}}{{{T_2}}}$

On substituting we get, $\dfrac{{300{\text{cal}}}}{{500{\text{K}}}} = \dfrac{{150{\text{cal}}}}{{{T_2}}}$

Rearrange the above equation to get, ${T_2} = \dfrac{{150 \times 500}}{{300}}$

     Hence, we get, ${T_2} = 250{\text{K}}$
     Therefore, the correct option is c) $250{\text{K}}$

Additional Information: Knowing the values for ${Q_1}$ and ${Q_2}$ , we can easily find the work done by the engine. The work done by the engine is given by, $W = {Q_1} - {Q_2}$
On substituting we get it as, $W = 300 - 150 = 150{\text{cal}}$

Note: The heat from the reservoir is used to do work by the working substance and some of it gets rejected to the sink. Make sure that all the quantities are expressed in their respective S. I. units. If not, necessary conversions should be made.