Adiabatic process

An adiabatic process is a process which takes place without transfer of heat (Q = 0). This type of process occurs when the thermodynamic system (in this case an ideal gas) is enclosed in an adiabatic container with an adiabatic wall. Processes that occur very quickly and for which the system does not have time to exchange heat with its surroundings can also be considered to be adiabatic.

On this page we will discuss a reversible adiabatic process, also called isentropic process. The case of an irreversible adiabatic transformation will be treated in the Joule expansion page.

We will use the so-called Clausius convention to state the First Law of Thermodynamics.

Where W is the work done by the system on its surroundings.

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Consider n moles of an ideal gas enclosed in a container with a moving wall (a piston for instance) as shown in the figure below. The container is covered with an adiabatic wall.

The volume, pressure and temperature of the gas varies as it expands. First of all we are going to determine the equation that relates the pressure to the volume starting from the differential form of the First Law of Thermodynamics and using the equation of state of an ideal gas.

By definition the heat exchanged in an adiabatic process is zero. In addition, the work done by a gas enclosed in a container and the change in the internal energy of an ideal gas are given respectively by:

By substituting in the differential form of the first principle and by isolating, we obtain:

By differentiating the equation of state of an ideal gas we get:

Next we can equal both expressions for dT:

To simplify the second member of the equation, we use the Mayer’s relation:

The quotient CP/CV is called the heat capacity ratio (or adiabatic index) γ.

By integrating the previous equation between any two states A and B, we obtain:

Finally, the equation of a reversible adiabatic process of an ideal gas is:

That is, the product of the pressure by the volume raised to the heat capacity ratio has the same value for any state of the adiabatic process.

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As the heat capacity ratio is greater than 1, the curve which represents a reversible adiabatic process for a ideal gas has a greater slope (in absolute value) than that of the isotherm of an ideal gas. The following PV diagram shows the adiabatic process as well as the work done by the gas when it goes from state A to state B:

The work done by the gas (that corresponds to the shaded area in blue in the PV diagram) is calculated by integrating the expression of the work done by a gas:

By substituting the value of the constant, we get:

Finally, the work done by the gas is:

which is positive since the gas expands when it goes from state A to state B.

From the figure above, we can also observe that the temperature of the ideal gas is lower in state B than in the initial state A. Since point B is located on an isotherm that is below the one which passes through point A, it means that an ideal gas cools down during an adiabatic expansion.

The change in the internal energy of the ideal gas is given by:

Note that the expression that gives the change in the internal energy of an ideal gas is the same regardless of the process that it undergoes, since the internal energy is a state function. As the final temperature is lower than the initial temperature, the gas loses internal energy.

On the other hand, using the First law of Thermodynamics we can calculate the work in the adiabatic process knowing the initial and final temperatures, since:

With the help of the equation of state of an ideal gas we can express the adiabatic equation as a function of temperature and volume or of pressure and temperature:


Follow the links below to see how to calculate the work, heat and the change in the internal energy for the following four reversible processes undergone by an ideal gas:

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