Question

Question:(I) A heat engine exhausts 8200 J of heat while performing 2600 J of useful work. What is the efficiency of this engine?

Short answer

The efficiency of the heat engine is \(24.07\% \).

Step-by-step solution

Understanding the principle of the heat engine

The heat engine can be used to convert an amount of heat into an amount of mechanical energy.

This engine receives some amount of heat from a high-temperature source. It utilizes some part of the heat for work. Thereafter, it rejects the remaining (non-utilized) part of the energy into the low-temperature source.

Identification of given data

The given data can be listed below as:

• The heat rejected by the heat engine is\({Q_{\rm{L}}} = 8200{\rm{ J}}\).
• The work done by the heat engine is \(W = 2600{\rm{ J}}\).

Determination of the efficiency of the heat engine

The work done by the heat engine can be expressed as:

\(\begin{aligned}{c}W &= {Q_{\rm{H}}} - {Q_{\rm{L}}}\\{Q_{\rm{H}}} &= W + {Q_{\rm{L}}}\end{aligned}\)

Here,\({Q_{\rm{H}}}\)is the heat added or supplied to the heat engine.

The efficiency of the heat engine can be expressed as:

\(\begin{aligned}{c}e &= \left( {\frac{W}{{{Q_{\rm{H}}}}}} \right)\\e &= \left( {\frac{W}{{W + {Q_{\rm{L}}}}}} \right)\end{aligned}\)

Substitute the values in the above equation.

\(\begin{aligned}{c}e &= \left( {\frac{{2600{\rm{ J}}}}{{2600{\rm{ J}} + 8200{\rm{ J}}}}} \right)\\e &= 0.2407\\e &= 0.2407 \times 100\% \\e &= 24.07\% \end{aligned}\)

Thus, the efficiency of the heat engine is \(24.07\% \).