Describe the working of a Carnot engine and mention the expression for its efficiency.

A Carnot engine is a theoretical heat engine that operates on the principles of maximum thermodynamic efficiency, as described by French physicist Sadi Carnot in 1824. While idealized and not achievable in practice, the Carnot engine serves as a benchmark for the maximum efficiency any heat engine can achieve when operating between two temperature reservoirs.

The working of a Carnot engine can be described in the following steps:

Isothermal Expansion (Reversible): The Carnot engine begins with an isothermal (constant temperature) expansion process at a high temperature, T_H. During this phase, the working substance (often an ideal gas) absorbs heat, Q_H, from a hot reservoir at temperature T_H while doing work, W, on the surroundings. This expansion is reversible, meaning it can be reversed without producing any net change in the system or surroundings. The temperature remains constant throughout this phase.

Adiabatic Expansion (Reversible): After the isothermal expansion, the working substance undergoes an adiabatic (no heat transfer) expansion. During this phase, the gas continues to expand, causing its temperature to decrease without any heat exchange with the surroundings. This process is also reversible.

Isothermal Compression (Reversible): In this phase, the working substance is in contact with a cold reservoir at temperature T_C. It undergoes an isothermal compression process, releasing heat, Q_C, to the cold reservoir while work is done on it. This compression is also reversible and occurs at a constant temperature, T_C.

Adiabatic Compression (Reversible): Following the isothermal compression, the working substance undergoes an adiabatic compression process. During this phase, the gas is further compressed, causing its temperature to increase without heat transfer. Like the other processes, this compression is reversible.

The efficiency of a Carnot engine, denoted by η_Carnot, is given by the expression:

η_Carnot = 1 - (T_C / T_H)

Where:

η_Carnot is the efficiency of the Carnot engine (expressed as a decimal or percentage).
T_C is the absolute temperature (in Kelvin) of the cold reservoir.
T_H is the absolute temperature (in Kelvin) of the hot reservoir.
It's important to note that the Carnot engine represents an idealized theoretical limit for the maximum possible efficiency of any heat engine operating between two temperature reservoirs. In practice, real engines are never able to achieve Carnot efficiency due to various sources of energy loss and irreversibilities. However, the Carnot cycle provides a useful reference point for understanding the thermodynamic limits of heat engines.