Derive the formula for efficiency of the Carnot engine.

The efficiency of the Carnot engine is a fundamental concept in thermodynamics and can be derived using the principles of the Carnot cycle. The Carnot engine is an idealized heat engine that operates between two temperature reservoirs, a hot reservoir at temperature T_H and a cold reservoir at temperature T_C. The efficiency of the Carnot engine is the maximum possible efficiency that any heat engine can achieve when operating between these two temperatures.

The efficiency (η) of the Carnot engine is given by the formula:

η = 1 - (T_C / T_H)

Where:
η is the efficiency of the Carnot engine,
T_C is the absolute temperature (in Kelvin) of the cold reservoir, and
T_H is the absolute temperature (in Kelvin) of the hot reservoir.

Now, let's derive this formula:

Start with the first law of thermodynamics, which states that the change in internal energy (ΔU) of a system is equal to the heat (Q) added to the system minus the work (W) done by the system:
ΔU = Q - W

For a heat engine, we are interested in the work done by the engine, so we can rearrange the equation as follows:
W = Q - ΔU

In the Carnot cycle, the engine operates in a reversible manner, meaning that there are no irreversibilities such as friction or heat losses. Therefore, the heat added to the engine in the isothermal expansion process (Q_H) is equal to the heat rejected in the isothermal compression process (Q_C).
Q_H = Q_C

The work done by the engine in one complete cycle is the difference between the heat added (Q_H) and the heat rejected (Q_C):
W = Q_H - Q_C

Now, we can express the efficiency (η) of the Carnot engine as the ratio of the work done by the engine to the heat added from the hot reservoir (Q_H):
η = W / Q_H

Substitute the expression for work (W) from step 4:
η = (Q_H - Q_C) / Q_H

Since Q_H is equal to Q_C (as stated in step 3), we can simplify the equation:
η = (Q_H - Q_H) / Q_H
η = 0 / Q_H

The efficiency of the Carnot engine, η, is therefore equal to 0.

To find the maximum possible efficiency, we need to consider the case where the engine is the most efficient, which occurs when there are no irreversibilities. In this case, the work done by the engine is equal to the heat added from the hot reservoir (Q_H):

W = Q_H

Therefore, the maximum possible efficiency (η_max) is:
η_max = Q_H / Q_H = 1

Finally, express this maximum efficiency in terms of temperatures:
η_max = 1 - (T_C / T_H)

This is the formula for the maximum efficiency of the Carnot engine, where T_C is the absolute temperature of the cold reservoir, and T_H is the absolute temperature of the hot reservoir.