Find out an expression for the total work done per cycle by a carnot's engine

To derive an expression for the total work done per cycle by a Carnot engine, we first need to understand the basic principles behind this idealized heat engine. A Carnot engine operates between two heat reservoirs: a hot reservoir at temperature \(T_H\) and a cold reservoir at temperature \(T_C\). The efficiency of a Carnot engine is determined by the temperatures of these reservoirs.

Efficiency of a Carnot Engine

The efficiency (\( \eta \)) of a Carnot engine is given by the formula:

η = 1 - \frac{T_C}{T_H}

Where:

• η is the efficiency of the engine.
• T_C is the absolute temperature of the cold reservoir (in Kelvin).
• T_H is the absolute temperature of the hot reservoir (in Kelvin).

Work Done by the Engine

The work done (\(W\)) by the engine during one complete cycle can be expressed in terms of the heat absorbed from the hot reservoir (\(Q_H\)). The relationship between work done and heat absorbed is given by:

W = Q_H - Q_C

Here, \(Q_C\) is the heat rejected to the cold reservoir. According to the first law of thermodynamics, the work done by the engine is equal to the heat absorbed minus the heat rejected.

Heat Absorbed and Rejected

For a Carnot engine, the heat absorbed from the hot reservoir can be related to the temperatures and the efficiency:

Q_H = \frac{W}{η}

Substituting the expression for efficiency, we have:

Q_H = \frac{W}{1 - \frac{T_C}{T_H}}

Now, we can express \(Q_C\) in terms of \(Q_H\) and the efficiency:

Q_C = Q_H - W

By substituting \(Q_H\) into this equation, we can find the total work done per cycle.

Final Expression for Work Done

Now, let's combine everything to find the total work done per cycle:

W = Q_H - Q_C = Q_H - (Q_H - W) = η Q_H

Substituting \(Q_H\) in terms of \(T_H\) and \(T_C\), we can express the work done as:

W = Q_H \cdot \left(1 - \frac{T_C}{T_H}\right)

Conclusion

In summary, the total work done per cycle by a Carnot engine can be expressed as:

W = Q_H \cdot \left(1 - \frac{T_C}{T_H}\right)

This formula highlights the relationship between the heat absorbed from the hot reservoir and the temperatures of the two reservoirs, illustrating how the efficiency of the Carnot engine directly influences the work output. Understanding this concept is crucial for grasping the principles of thermodynamics and the operation of heat engines.