# Isochoric Process

The isocoric process is a thermodynamic process that occurs in a constant volume. To carry out an isocoric process in a gas or liquid, it is sufficient to heat (cool) a substance in a container that does not change its volume.

In an isochoric process, the pressure of an ideal gas is directly proportional to its temperature. In real gases, Charles's law is not fulfilled.

The graphics are represented by lines called isochromes. For an ideal gas, they are straight lines in all the diagrams that relate parameters: T (temperature) V (volume) and P (pressure).

## Thermodynamics of the isochoric process

An isocoric thermodynamic quasi-static process is characterized by a constant volume, that is, ΔV = 0. The process does not perform pressure-volume work, since said work is defined by

where P is pressure. The sign convention is such that the system does a positive job in the environment.

If the process is not quasi-static, the work may be done in a thermodynamic process of constant volume.

For a reversible process, the first law of thermodynamics gives the change in the internal energy of the system:

Replace work with a change in volume gives

Since the process is isochoric, dV = 0, the previous equation now gives

Using the definition of specific heat capacity at constant volume,

,

Integration of both sides produces

Where cv is the specific heat capacity at constant volume, T 1 is the initial temperature and T 2 is the final temperature. We conclude with:

Isochoric process in the pressure volume diagram. In this diagram, the pressure increases, but the volume remains constant.

In a pressure volume diagram, an isocoric process appears as a straight vertical line. Its thermodynamic conjugate, an isobaric process would appear as a straight horizontal line.

### Ideal gas

If an ideal gas is used in an isochoric process, and the amount of gas remains constant, then the increase in energy is proportional to an increase in temperature and pressure. Take, for example, a gas heated in a rigid container: the pressure and temperature of the gas will increase, but the volume will remain the same.

## Practical application of isocoric process theory

With an ideal Otto cycle, which is reproduced approximately in a gasoline internal combustion engine, steps 2-3 and 4-1 are isocoric processes. The work done at the engine exit is equal to the difference in the work that the gas will produce on the piston during the third cycle (i.e. the work stroke) and the work that the piston dedicates to the gas compression during the second cycle.

In the Stirling cycle, there are also two isochoric measures. For its implementation, a regenerator has been added to the Stirling engine. The gas that passes through the filling in one direction emits heat from the working fluid to the regenerator, and when it moves in the other direction returns it to the work theme. The ideal Stirling cycle achieves reversibility and the same efficiency values as the Carnot cycle.

Last review: February 26, 2020