Question

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

Answer 1

Hint: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 ${T_1} = 500\;{\text{K}}$
The temperature of the sink is ${T_2} = 375\;{\text{K}}$
The heat taken by the engine is ${Q_1} = 600\;{\text{kcal}}$
The expression to calculate the efficiency of the Carnot engine is,
$\eta = 1 - \dfrac{{{T_2}}}{{{T_1}}}$
Substitute $500\;{\text{K}}$for ${T_1}$ and $375\;{\text{K}}$for ${T_2}$ in the above expression to find the efficiency of the Carnot engine.
$\eta = 1 - \dfrac{{375\;{\text{K}}}}{{500\;{\text{K}}}}$
$\Rightarrow\eta = 0.25$
The expression to calculate the heat rejected to sink per cycle is,
$Q = \left( {1 - \eta } \right){Q_1}$
Substitute 0.25 for $\eta $ and $600\;{\text{kcal}}$for ${Q_1}$ to find the heat rejected to sink per cycle.
$Q = \left( {1 - 0.25} \right)\left( {600\;{\text{kcal}}} \right)$
$\therefore Q = 450\;{\text{kcal}}$

Thus, the heat rejected to sink per cycle is $450\;{\text{kcal}}$ 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.