Calculation of the entropy change in reversible processes

In this page, we will see how to calculate the entropy change of an ideal gas between any two states for the most common reversible processes.

The entropy change between any two states A and B is given by:

Adiabatic process

An adiabatic process is a process which takes place without transfer of heat (Q = 0).

Since the gas does not exchange heat, we have:

A reversible adiabatic process is also known as isentropic process, since the entropy of the system does not change.

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Isothermal process

An isothermal process is a process which takes place at constant temperature (T = constant).

If we apply the definition of the entropy change, we have:

This expression is valid for any thermodynamic system that undergoes an isothermal process. As a consequence, we can use it to calculate the entropy change of a heat reservoir.

For an ideal gas, the heat exchanged during an isothermal process is given by:

And, by substituting in the entropy change expression, we get:

During the isothermal expansion represented in the previous figure, the entropy of the ideal gas increases between states A and B. The entropy would decrease If the process were an isothermal compression.

Isochoric process

An isochoric process is a process which takes place at constant volume (V = constant).

The entropy change between states A and B is given by:

Where CV is the molar heat capacity at constant volume. During the the isochoric process shown in the figure above the entropy of the gas increases because the final temperature is higher than the initial one.

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Isobaric process

An isobaric process is a process which takes place at constant pressure (p = constant).

The entropy change between states A and B is given by:

Where Cp is the molar heat capacity at constant pressure. In this case the entropy increases because the final temperature is higher than the initial one.

Other processes

To calculate the entropy change undergone by an ideal gas when it goes from an initial state A to a final state connected by a process different than those described above (whether reversible or not), we can make use of the fact that the entropy is a state function.

Let’s assume that an ideal gas undergoes any process, such as the one represented in green in the figure below

Since we don’t know the equation describing this process, we cannot directly calculate the entropy change between the states A and B along it. But, because the gas’s initial and final temperatures are the same and since the entropy is a state function, the entropy change between these states will be the same if they are connected by an isothermal process. This is true regardless of the process that connects the initial and final states, as long as the temperature is the same for both. The entropy change between the initial and final states represented in the previous figure will always be:

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