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Question: A Carnot engine takes \(300{\text{cal}}\) of heat from a reservoir at \(500{\text{K}}\) and rejects ...

A Carnot engine takes 300cal300{\text{cal}} of heat from a reservoir at 500K500{\text{K}} and rejects 150cal150{\text{cal}} to a sink. Find the temperature of the sink
a) 1000K1000{\text{K}}
b) 750K750{\text{K}}
c) 250K250{\text{K}}
d) 125K125{\text{K}}

Explanation

Solution

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

Formula Used: Carnot’s theorem gives us Q1T1=Q2T2\dfrac{{{Q_1}}}{{{T_1}}} = \dfrac{{{Q_2}}}{{{T_2}}} where Q1{Q_1} and Q2{Q_2} represents the heat taken from the reservoir (source) and T1{T_1} its temperature, Q2{Q_2} represents the heat rejected to the sink and T2{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 Q1=300cal{Q_1} = 300{\text{cal}}
The temperature of the source or reservoir is T1=500K{T_1} = 500{\text{K}}
The heat taken from the sink is Q2=150cal{Q_2} = 150{\text{cal}}
The temperature of the sink T2{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 Q1T1=Q2T2\dfrac{{{Q_1}}}{{{T_1}}} = \dfrac{{{Q_2}}}{{{T_2}}}, where Q1{Q_1} represents the heat taken from the reservoir (source) and T1{T_1} its temperature, Q2{Q_2} represents the heat rejected to the sink and T2{T_2} represents the sink’s temperature.

Step 3: Express the Carnot’s relation and substitute the values of Q1=300cal{Q_1} = 300{\text{cal}} , T1=500K{T_1} = 500{\text{K}} , Q2=150cal{Q_2} = 150{\text{cal}} to find T2{T_2}
From the Carnot’s theorem we have, Q1T1=Q2T2\dfrac{{{Q_1}}}{{{T_1}}} = \dfrac{{{Q_2}}}{{{T_2}}}

On substituting we get, 300cal500K=150calT2\dfrac{{300{\text{cal}}}}{{500{\text{K}}}} = \dfrac{{150{\text{cal}}}}{{{T_2}}}

Rearrange the above equation to get, T2=150×500300{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 Q1{Q_1} and Q2{Q_2} , we can easily find the work done by the engine. The work done by the engine is given by, W=Q1−Q2W = {Q_1} - {Q_2}
On substituting we get it as, W=300−150=150calW = 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.