An isochoric process is a thermodynamic process in which the volume remains constant. Since the volume is constant, the system does no work and W = 0. ("W" is the abbreviation for work.) This is perhaps the easiest of the thermodynamic variables to control since it can be obtained by placing the system in a sealed container which neither expands nor contracts.

### First Law of Thermodynamics

To understand the isochoric process, you need to understand the first law of thermodynamics, which states:

"The change in a system's internal energy is equal to the difference between heat added to the system from its surroundings and work done by the system on its surroundings."

Applying the first law of thermodynamics to this situation, you find that:

delta-Since delta-*U* is the change in internal energy and *Q* is the heat transfer into or out of the system, you see that all of the heat either comes from internal energy or goes into increasing the internal energy.

### Constant Volume

It is possible to do work on a system without changing the volume, as in the case of stirring a liquid. Some sources use "isochoric" in these cases to mean "zero-work" regardless of whether there is a change in volume or not. In most straightforward applications, however, this nuance will not need to be considered—if the volume remains constant throughout the process, it is an isochoric process.

### Example Calculation

The website Nuclear Power, a free, nonprofit online site built and maintained by engineers, gives an example of a calculation involving the isochoric process.

Assume an isochoric heat addition in an ideal gas. In an ideal gas, molecules have no volume and do not interact. According to the ideal gas law, pressure varies linearly with temperature and quantity, and inversely with volume. The basic formula would be:

pV = nRT

where:

*p*is the absolute pressure of the gas*n*is the amount of substance*T*is the absolute temperature*V*is the volume*R*is the ideal, or universal, gas constant equal to the product of the Boltzmann constant and the Avogadro constant*K*is the scientific abbreviation for Kelvin

In this equation the symbol R is a constant called the universal gas constant that has the same value for all gases—namely, R = 8.31 Joule/mole K.

The isochoric process can be expressed with the ideal gas law as:

p/T = constant

Since the process is isochoric, dV = 0, the pressure-volume work is equal to zero. According to the ideal gas model, the internal energy can be calculated by:

*∆U = m c _{v }∆T*

where the property c_{v} (J/mole K) is referred to as specific heat (or heat capacity) at a constant volume because under certain special conditions (constant volume) it relates the temperature change of a system to the amount of energy added by heat transfer.

Since there is no work done by or on the system, the first law of thermodynamics dictates *∆U = ∆Q. *Therefore:

Q = *m c _{v }∆T*