Isothermal process

An isothermal process is a process which takes place at constant temperature (T = cte). This type of process occurs when the thermodynamic system (in this case an ideal gas) is in contact with a thermal reservoir (or heat bath).

A thermal reservoir is a thermodynamic system that can absorb or release heat without changing its temperature.

During an isothermal process, the temperature of the system is well defined for every state of the process that is, the gas goes through successive equilibrium states. An isothermal process is therefore a reversible process.

In these pages 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 (for example, a piston). The gas is in thermal equilibrium with a thermal reservoir, as shown in the figure below.

The process undergone by the gas is illustrated in the following PV diagram:

The work done by the gas when it goes from state A to state B is given by:

The equation of state of an ideal gas provides the relation between the pressure and the volume needed to calculate the integral:

By substituting it in the expression of the work and by integrating, we get:

Which is positive since the gas expands when going from state A to state B. This work is the shaded area in blue in the PV diagram.

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The change in the internal energy of the ideal gas when it goes from state A to state B is:

Because the initial and final temperatures are the same.

When an ideal gas undergoes an isothermal process, its internal energy does not change.

We use the First Law of Thermodynamics to find the heat exchanged by the ideal gas during the process AB:

Therefore, the ideal gas absorbs heat as it goes from state A to state B.

In view of the results obtained, it may seem unintuitive that a system can absorb heat without increasing its temperature. To understand this result, we must consider the heat capacity of an ideal gas as being infinite when it undergoes an isothermal process. Which means that its temperature will not change no matter how much heat it exchanges.

If the process were reversed (that is, if the gas were compressed isothermally) the work and the heat exchanged would be negative.

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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