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Question: A Carnot engine operates with a source at \(500\;{\text{K}}\) and sinks at \(375\;{\text{K}}\). If t...

A Carnot engine operates with a source at 500  K500\;{\text{K}} and sinks at 375  K375\;{\text{K}}. If the engine takes 600  kcal600\;{\text{kcal}} of heat in one cycle, the heat rejected to sink per cycle is
A. 250  kcal250\;{\text{kcal}}
B. 350  kcal350\;{\text{kcal}}
C. 450  kcal450\;{\text{kcal}}
D. 550  kcal550\;{\text{kcal}}

Explanation

Solution

The above problem is based on the Carnot cycle. The efficiency of the Carnot cycle is equal to the heat output to heat input. The efficiency can be expressed in terms of temperature also. The heat rejected by the sink will be equal to the difference between heat input of the cycle and heat used in the cycle.

Complete step by step answer:
Given: The temperature of the source is T1=500  K{T_1} = 500\;{\text{K}}
The temperature of the sink is T2=375  K{T_2} = 375\;{\text{K}}
The heat taken by the engine is Q1=600  kcal{Q_1} = 600\;{\text{kcal}}
The expression to calculate the efficiency of the Carnot engine is,
η=1−T2T1\eta = 1 - \dfrac{{{T_2}}}{{{T_1}}}
Substitute 500  K500\;{\text{K}}for T1{T_1} and 375  K375\;{\text{K}}for T2{T_2} in the above expression to find the efficiency of the Carnot engine.
η=1−375  K500  K\eta = 1 - \dfrac{{375\;{\text{K}}}}{{500\;{\text{K}}}}
⇒η=0.25\Rightarrow\eta = 0.25
The expression to calculate the heat rejected to sink per cycle is,
Q=(1−η)Q1Q = \left( {1 - \eta } \right){Q_1}
Substitute 0.25 for η\eta and 600  kcal600\;{\text{kcal}}for Q1{Q_1} to find the heat rejected to sink per cycle.
Q=(1−0.25)(600  kcal)Q = \left( {1 - 0.25} \right)\left( {600\;{\text{kcal}}} \right)
∴Q=450  kcal\therefore Q = 450\;{\text{kcal}}

Thus, the heat rejected to sink per cycle is 450  kcal450\;{\text{kcal}} and the option (C) is the correct answer.

Additional Information:
The Carnot cycle is the heat cycle that consists of two isothermal processes and two isobaric processes. This heat cycle is the standard cycle for the thermodynamic processes. All thermodynamic devices are designed on the basis of the Carnot efficiency of the process.

Note: Always remember the both formula to find the efficiency of the Carnot cycle. One method to find the efficiency is by the ratio of heat output to heat input and another method is on the basis of the temperature of the source and sink.