An isochoric process is a thermodynamic process that occurs in a constant volume. It is also called an isometric or isovolumetric process

In an isochoric process, the pressure of an ideal gas in a closed system is directly proportional to its temperature. In the case of real gasses, Charles' law is not obeyed.

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

## Isochoric Process Formulas

### Calculation of the work of an isometric process

An isochoric process does not produce work because it is produced at constant volume, that is, ΔV = 0. The process does not do pressure-volume work since such work is defined by

W = P·ΔV = P·0

### Calculation of the internal energy change of an isochoric process

Applying the first law of thermodynamics, we can know the variation of the internal energy of a thermodynamic system, considering that the amount of gas is constant.

The first law is a principle that reflects the conservation of energy and states that if work is done on a system or it exchanges heat with another, the system's internal energy will change. This principle allows heat to be defined as the necessary energy that the system must exchange to compensate for the differences between work and internal energy.

We can express the variation of internal energy with the following formula:

ΔU = Q - W

As we have mentioned before, the work done by the system is zero because the volume remains constant. Therefore, the increase in the system's internal energy is equal to the heat transferred (Q) to the system.

ΔU = Q

If an ideal gas is used and the amount of gas is held constant, then the energy added is proportional to an increase in temperature and pressure.

### Calculation of the heat released from an isovolumetric process

If we consider that the amount of gas does not vary, the energy variation will be directly proportional to the temperature variation:

Q=n·Cv·ΔT

where

Cv corresponds to the specific heat at constant volume.

In the formula, n is the amount of substance of gas (also known as the number of moles).

ΔT is the temperature variation of the process.

This formula comes from the ideal gas law

## Isochoric process examples

### Example in everyday life

An example of an isochoric process in everyday life is seen when we heat water in a pressure cooker. When we transfer heat to the container, we experience an isovolumetric process since the temperature increase is done at a constant volume.

If the same container were open, the water vapor would expand while maintaining its (atmospheric) pressure. Therefore it would be an isobaric process because it would be done at a constant pressure.

### Examples in thermodynamic cycles

An excellent example of an isochoric process is the Otto cycle used in heat engines that use gasoline. In this cycle, we can see two isochoric processes during the gasoline combustion phases and the expulsion of gases in which the gas does no work. In this example, the other two steps are adiabatic processes.

The other well-known cycle used in heat engines is the diesel cycle. The cycle diesel has only one isochoric process in the escape phase. Contrarily to the cycle Otto, instead of fuel combustion, here we have a compression of fuel in an isothermal process.

The Stirling cycle is also an example of an isometric process. The gas heating and cooling phases occur at a constant volume; hence, it is an isochoric process. The other processes of the cycle are isothermal processes in which the temperature remains constant.