Isothermal Processes: Definition, Formula & Examples

Understanding what different thermodynamic processes are and how you use the first law of thermodynamics with each one is crucial when you start to consider heat engines and Carnot cycles.

Many of the processes are idealized, so while they don't accurately reflect how things occur in the real world, they're useful approximations that simplify calculations and make it easier to draw conclusions. These idealized processes describe how the states of an ideal gas can undergo change.

The isothermal process is just one example, and the fact that it occurs at a single temperature by definition drastically simplifies working with the first law of thermodynamics when you're calculating things like heat-engine processes.

An isothermal process is a thermodynamic process that occurs at a constant temperature. The benefit of working at a constant temperature and with an ideal gas is that you can use Boyle's law and the ideal gas law to relate pressure and volume. Both of these expressions (as Boyle's law is one of the several laws that were incorporated into the ideal gas law) show an inverse relationship between pressure and volume. Boyle's law implies that:

\(P_1V_1 = P_2V_2\)

Where the subscripts denote the pressure (​P​) and volume (​V​) at time 1 and the pressure and volume at time 2. The equation shows that if the volume doubles, for instance, the pressure has to reduce by half in order to keep the equation balanced, and vice versa. The full ideal gas law is

\(PV=nRT\)

where ​n​ is the number of moles of the gas, ​R​ is the universal gas constant and ​T​ is the temperature. With a fixed amount of gas and a fixed temperature, ​PV​ must take a constant value, which leads to the previous result.

On a pressure-volume (PV) diagram, which is a plot of pressure vs. volume often used for thermodynamic processes, an isothermal process looks like the graph of ​y​ = 1/​x​, curving downwards towards its minimum value.

One point that often confuses people is the distinction between ​isothermal​ vs. ​adiabatic​, but breaking down the word into its two parts can help you remember this. "Iso" means equal and "thermal" refers to something's heat (i.e., its temperature), so "isothermal" literally means "at an equal temperature." Adiabatic processes don't involve heat ​transfer​, but the temperature of the system often changes during them.

The first law of thermodynamics states that the change in internal energy (​∆U​) for a system is equal to the heat added to the system (​Q​) minus the work done by the system (​W​), or in symbols:

\(∆U= Q – W\)

When you're dealing with an isothermal process, you can use the fact that internal energy is directly proportional to temperature alongside this law to draw a useful conclusion. The internal energy of an ideal gas is:

\(U = \frac{3}{2} nRT\)

This means that for a constant temperature, you have a constant internal energy. So with ​∆U​= 0, the first law of thermodynamics can easily be re-arranged to:

\(Q=W\)

Or, in words, the heat added to the system is equal to the work done by the system, meaning that the heat added is used to do the work. For example, in isothermal expansion, heat is added to the system, which causes it to expand, doing work on the environment without losing internal energy. In an isothermal compression, the environment does work on the system, and causes the system to lose this energy as heat.

Heat engines use a complete cycle of thermodynamic processes to convert heat energy into mechanical energy, usually by moving a piston as the gas in the heat engine expands. Isothermal processes are a key part of this cycle, with the added heat energy being completely converted into work without any loss.

However, this is a highly idealized process, because in practice there will always be some energy lost when the heat energy is converted into work. For it to work in reality, it would need to take an infinite amount of time so that the system could remain in thermal equilibrium with its surroundings at all times.

Isothermal processes are considered reversible processes, because if you've completed a process (for example, an isothermal expansion) you could run the same process in reverse (an isothermal compression) and return the system to its original state. In essence, you can run the same process forwards or backwards in time without breaking any laws of physics.

However, if you attempted this in real life, the second law of thermodynamics would mean there was an increase in entropy during the "forwards" process, so the "backwards" one wouldn't completely return the system to its original state.

If you plot an isothermal process on a PV diagram, the work done during the process is equal to the area under the curve. While you can calculate the work done isothermally in this way, it's often easier to just use the first law of thermodynamics and the fact that the work done is equal to the heat added to the system.

If you're doing calculations for an isothermal process, there are a couple of other equations you can use to find the work done. The first of these is:

\(W = nRT \ln \bigg(\frac{V_f}{V_i}\bigg)\)

Where ​V​_f is the final volume and ​V​_i is the initial volume. Using the ideal gas law, you can substitute the initial pressure and volume (​P​_i and ​V​_i) for the ​nRT​ in this equation to get:

\(W = P_iV_i \ln \bigg(\frac{V_f}{V_i}\bigg)\)

It may be easier in most cases to the work through the heat added, but if you only have information about the pressure, volume or temperature, one of these equations could simplify the problem. Since work is a form of energy, its unit is the joule (J).

There are many other thermodynamic processes, and many of these can be classified in a similar way to isothermal processes, except that quantities other than temperature are constant throughout. An isobaric process is one that occurs at a constant pressure, and because of this, the force exerted on the walls of the container is constant, and the work done is given by ​W​ = ​P∆V​.

For gas undergoing isobaric expansion, there needs to be heat transfer in order to keep the pressure constant, and this heat changes the internal energy of the system as well as doing work.

An isochoric process takes place at a constant volume. This allows you to make a simplification in the first law of thermodynamics, because if the volume is constant, the system can't do work on the environment. As a result, the change in internal energy of the system is entirely due to the heat transferred.

An adiabatic process is one that occurs without heat exchange between the system and the environment. This doesn't mean that there is no change in temperature in the system, though, because the process could lead to an increase or a decrease in temperature without direct heat transfer. However, with no heat transfer, the first law shows that any change in internal energy must be due to work done on the system or by the system, since it sets ​Q​ = 0 in the equation.