### #76 Fundamentals--Carnot engine

Carnot cycle is an ideal cycle composing of 2 reversible isothermal and 2 reversible adiabatic processes, proposed by Sadi Carnot in 1824. It was the first quantitative analysis of how much work could be done by heat far before people realized the nature of heat. It also gave rise to the concept of entropy and the second law of thermodynamics. The main goal of this article is not to thoroughly explain the concept and derivation of Carnot cycle but to give a rapid remind of some key results.

The Carnot cycle starts from a bunch of gas sealed in a piston. The gas is isothermally expanded, adiabatically expanded, isothermally compressed, and finally adiabatically compressed to the initial state. During the isothermal process, the engine is in close contact with a hot bath or a cold bath to release or get heat from it. In terms of phase diagram, the Carnot cycle looks like this:

According to some basic thermodynamics, the following holds:
in which γ is the heat capacity ratio. From these, we could derive that
Take the logarithm of the above equation and multiply it with n*R(the ideal gas constant), we will get

There is no heat transfer during adiabatic processes, while the heat transfer during an isothermal reversible process could be easily derived:
Therefore if we define a quantity S:
We found that S is a state function for it is conservative in a cyclic process. In fact, this is a quantity called entropy.

The work that could be done by a Carnot engine is the area encircled by the Carnot cycle in the phase diagram. It could be easily calculated from the first law of thermodynamics:
while the heat that is uptaken from the hot reservoir could be calculated as:

Carnot engine is the most efficient engine that could be designed using pure gas expansion and compression. No heat or work is wasted. We could define its efficiency by:

However, these concepts are challenged in the Brownian Carnot engine. Stay tuned to explore more!