Question

For an ideal heat engine, the temperature of the source is \[{127^ \circ }C\]. In order to have 60% efficiency the temperature of the sink should be ………. C. (Round off to the nearest integer)

Answer 1

Hint: An engine which is used to convert heat energy into mechanical energy is known as a heat engine. And efficiency is the capacity of the heat engine to convert energies.

Formula used:
The formula for the efficiency of the heat engine can be written as,
\[\eta = 1 - \dfrac{{{T_2}}}{{{T_1}}}\]
Where, $\eta$ is the efficiency, $T_2$ is the temperature of the sink and $T_1$ is the temperature of the source.

Complete step by step solution:
The efficiency of a heat engine is defined as the difference between the temperature of the source and the sink. It depends on the ratio of absolute temperatures. Mathematically, the formula for the efficiency of the heat engine can be written by the formula:
\[\eta = 1 - \dfrac{{{T_2}}}{{{T_1}}}\]……(i)
Efficiency is denoted by the symbol eta \[(\eta )\].
It is given that the efficiency is \[\eta = 60\% \]
Or it can be written that \[\eta = \dfrac{{60}}{{100}} = \dfrac{3}{5}\]
Given that the temperature of the source is \[{127^ \circ }C\].
\[{T_1} = 127 + 273 = 400K\]
\[\Rightarrow {T_2} = ?\]

Substituting all the values given in the question and solving for the temperature of the sink we get
\[60\% = 1 - \dfrac{{{T_2}}}{{400}}\]
\[\Rightarrow \dfrac{3}{5} = 1 - \dfrac{{{T_2}}}{{400}}\]
\[\Rightarrow \dfrac{{{T_2}}}{{400}} = 1 - \dfrac{3}{5}\]
\[\Rightarrow \dfrac{{{T_2}}}{{400}} = \dfrac{{5 - 3}}{5}\]
\[\Rightarrow {T_2} = 400 \times \dfrac{2}{5}\]
\[\Rightarrow {T_2} = 160K\]
Converting from Kelvin to Degree Celsius, we get
\[{T_2} = 160 - 273\]
\[\therefore {T_2} = - {113^ \circ }C\]

The temperature of the sink for an ideal heat engine will be \[{113^ \circ }C\].

Note: The heat engine is also known as the Carnot engine because it was invented by French physicist Carnot in 1824. A heat engine can never have 100% efficiency. This is because when heat energy is converted to mechanical energy, some energy is lost. Also, the efficiency of a heat engine is unitless. Heat engines use different principles to convert heat energy into mechanical energy, so there are many types of heat engines. When the heat burns outside, the heat engine is known as an external combustion engine and when the heat or the fuel burns inside the cylinder, that is known as an internal combustion engine.