Modern Physics

Carnot Machine: Formulas, How It Works and Applications

The Carnot machine is an ideal cyclic model in which heat is used to do a job. The system can be understood as a piston that travels inside a cylinder compressing a gas. The cycle carried out is that of Carnot, enunciated by the father of thermodynamics, the French physicist and engineer Nicolas Léonard Sadi Carnot.

Carnot enunciated this cycle in the early 19th century. The machine is subject to four state variations, alternating conditions such as constant temperature and pressure, where a change in volume is evident when compressing and expanding the gas.


According to Carnot, by subjecting the ideal machine to variations in temperature and pressure, it is possible to maximize the yield obtained.

The Carnot cycle must be analyzed separately in each of its four phases: isothermal expansion, adiabatic expansion, isothermal compression and adiabatic compression.

Next, the formulas associated with each of the phases of the cycle carried out on the Carnot machine will be detailed.

Isothermal expansion (A → B)

The premises of this phase are as follows:

– Gas volume: changes from minimum volume to medium volume.

– Machine temperature: constant temperature T1, high value (T1> T2).

– Machine pressure: drops from P1 to P2.

The isothermal process implies that the T1 temperature does not change during this phase. Heat transfer induces gas expansion, which induces movement in the piston and produces mechanical work.

When expanding, the gas has a tendency to cool. However, it absorbs the heat emitted by the temperature source and during its expansion it maintains a constant temperature.

As the temperature remains constant during this process, the internal energy of the gas does not change and all the heat absorbed by the gas is effectively turned into work. Thus:

Carnot Machine: Formulas, How It Works and Applications 3

On the other hand, at the end of this phase of the cycle it is also possible to obtain the pressure value using the ideal gas equation. So, you have the following:

Carnot Machine: Formulas, How It Works and Applications 4

In this expression:

2 : Pressure at the end of the phase.

b : Volume at point b.

n: number of moles of gas.

A: Universal constant of ideal gases. R = 0.082 (atm * liter) / (mole * K).

T1: absolute initial temperature, degrees Kelvin.

Adiabatic expansion (B → C)

During this phase of the process, the expansion of the gas takes place without the need for heat exchange. Thus, the premises are detailed below:

– Gas volume: changes from the average volume to the maximum volume.

– Machine temperature: drops from T1 to T2.

– Machine pressure: constant pressure P2.

The adiabatic process implies that the P2 pressure does not change during this phase. The temperature decreases and the gas continues to expand until reaching its maximum volume; that is, the piston reaches the top.

In this case, the work performed comes from the internal energy of the gas and its value is negative because the energy decreases during this process.

Carnot Machine: Formulas, How It Works and Applications 5

Assuming that it is an ideal gas, the theory that gas molecules have only kinetic energy is maintained. According to the principles of thermodynamics, this can be deduced from the following formula:

Carnot Machine: Formulas, How It Works and Applications 6

In this formula:

BU b → c : Internal energy variation of the ideal gas between points b and c.

n: number of moles of gas.

Cv: molar heat capacity of the gas.

T1: absolute initial temperature, degrees Kelvin.

T2: Absolute final temperature, degrees Kelvin.

Isothermal compression (C → D)

At this stage, gas compression begins; that is, the piston moves into the cylinder, whereby the gas contracts its volume.

The conditions inherent in this phase of the process are detailed below:

– Gas volume: ranges from the maximum volume to an intermediate volume.

– Machine temperature: constant temperature T2, reduced value (T2 <T1).

– Machine pressure: increases from P2 to P1.

Here the pressure on the gas increases, then it starts to compress. However, the temperature remains constant and therefore the internal energy change of the gas is zero.

Similar to isothermal expansion, the work done is equal to the heat of the system. Thus:

Carnot Machine: Formulas, How It Works and Applications 7

It is also possible to find the pressure at this time using the ideal gas equation.

Adiabatic compression (D → A)

This is the last phase of the process, in which the system returns to its initial conditions. For this, the following conditions are considered:

– Gas volume: ranges from an intermediate volume to a minimum volume.

– Machine temperature: increases from T2 to T1.

– Machine pressure: constant pressure P1.

The heat source incorporated in the system in the previous phase is removed, so that the ideal gas increases its temperature while the pressure remains constant.

The gas returns to the initial conditions of temperature (T1) and its volume (minimum). Again, the work done comes from the internal energy of the gas, so you must:

Carnot Machine: Formulas, How It Works and Applications 8

Similar to the case of adiabatic expansion, it is possible to obtain the variation of gas energy through the following mathematical expression:

Carnot Machine: Formulas, How It Works and Applications 9

How does the Carnot machine work?

The Carnot machine works as a mechanism in which performance is maximized by varying isothermal and adiabatic processes, alternating the phases of expansion and understanding of an ideal gas.

The mechanism can be understood as an ideal device that performs work subject to variations in heat, given the existence of two sources of temperature.

In the first focus, the system is exposed to a T1 temperature. It is a high temperature that puts the system under stress and produces gas expansion.

In turn, this translates into the execution of mechanical work that allows the piston to move outside the cylinder and whose stopping is possible only through adiabatic expansion.

Then comes the second focus, in which the system is exposed to a temperature T2, lower than T1; that is, the mechanism is subject to refrigeration.

This induces heat extraction and gas crushing, which reach their initial volume after adiabatic compression.


The Carnot machine has been widely used thanks to its contribution to the understanding of the most important aspects of thermodynamics.

This model allows us to clearly understand the ranges of ideal gases subject to changes in temperature and pressure, which is a reference method when designing real engines.

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