Question

The efficiency of the Carnot engine is $\dfrac{1}{5}$ . Now the temperature of the source is increased by ${100^ \circ }\,C$ then efficiency becomes doubled. The initial temperature of sink and source are respectively.
(1) $240\,K,\,360\,K$
(2) $240\,K,\,300\,K$
(3) $200\,K,\,250\,K$
(4) $300\,K,\,240\,K$

Answer 1

Hint:Use the below formula of the efficiency, and substitute the parameters and the conditions given in the questions. Based on the two conditions, form two equations from and solve the two equations to find the value of the sink and the source temperature.

Useful formula:
The efficiency of the Carnot’s engine is given by

$\eta = \dfrac{{{T_1} - {T_2}}}{{{T_1}}}$

Where $\eta $ is the efficiency of the Carnot’s engine, ${T_1}$ is the temperature of the sink and ${T_2}$ is the temperature of the source.

Complete step by step solution:
It is given that the efficiency of the Carnot’s engine, $\eta = \dfrac{1}{5}$
The temperature of the source increased by ${100^ \circ }\,C$ , then the efficiency became doubled.
Using the formula of the efficiency,

$\eta = \dfrac{{{T_1} - {T_2}}}{{{T_1}}}$

Substituting the initial conditions,

$\dfrac{1}{5} = \dfrac{{{T_1} - {T_2}}}{{{T_1}}}$ --------------(1)

Substituting the final condition in the formula, we get

$\dfrac{2}{5} = \dfrac{{\left( {{T_1} + 100} \right) - {T_2}}}{{\left( {{T_1} + 100} \right)}}$ ------------(2)
Solving equation (1) and (2), we get

$\dfrac{{{T_1} - {T_2}}}{{{T_1}}} = \dfrac{{\left( {{T_1} + 100} \right) - {T_2}}}{{2\left( {{T_1} + 100} \right)}}$

By simplifying the above equation, we get

\[2\left( {{T_1} + 100} \right){\left( {{T_1} - T} \right)_2} = {T_1}\left( {{T_1} + 100} \right) - {T_2}\]
${T_1} = 240\,K$

Substituting the value of the source temperature in the equation (1), we get

$\dfrac{1}{5} = \dfrac{{240 - {T_2}}}{{240}}$

By simplifying the above equation, we get

${T_2} = 300\,K$

Hence the temperature of the sink is obtained as $300\,K$.

Thus the option (2) is correct.

Note:Normally the sink temperature represents the cold environment and the source represents the hot environment. The practical examples of this are the heat engine is the source and the natural environment represents the sink that cools the heat.