# An Elaborate Explanation of the Isothermal Process

Omkar Phatak May 6, 2019
What is an isothermal process? What is the work done in such a process? Read to get all the answers.
When taking your first basic physics course, you will be introduced to various thermodynamic processes. An important one is the isothermal process.
Thermodynamics is a branch of physics that involves the study of energy changes, and primarily focuses on the connection between heat and mechanical work. It deals with the variation of macroscopic variables of a system, which include temperature, volume, pressure, and entropy.

### Definition

Any thermodynamic process is characterized by a change in the macroscopic thermodynamic variables associated with it. An isothermal process is one in which all developments occur at a constant temperature.
In the language of thermodynamics, it is a process which is characterized by ΔT = 0. It may occur because of contact with a thermal reservoir. The system exchanges heat and its state changes slowly (ΔQ ≠ 0), while maintaining a temperature, that is equal to the temperature of the reservoir.
The ideal gas law is stated as: PV = nRT, where P is pressure, V is the volume, and T is the temperature of the system, while 'n' is the number of moles of a gas. During an isothermal process, 'nRT' becomes a constant and the equation of state for the gas can be written as:

PV = Constant
This is the Boyle's Law in thermodynamics, which describes the variation of volume and pressure, under isothermal conditions. If a graph of change in pressure and volume is generated, for constant temperatures, a range of pressure and volume variations will satisfy the equation.
A graph can be generated, that displays the evolution of a system, under isothermal conditions. All of these curves are known as 'Isotherms'. In case of an ideal gas, during an isothermal process, the internal energy remains unchanged, as temperature remains constant (ΔU = 0).
However, since the first law of thermodynamics, in this case states that ΔU = ΔQ + ΔW, we reach the conclusion that ΔQ = - ΔW. This means that all the heat input of the system is used up to do work.

### What is the Work Done?

Let us see what is the work done in an isothermal process. For a reversible process, the expression for work done, is given by the following equation:

Wiso = - nRT ln (Vfinal / Vinitial )

Here, Wiso is the work done in an isothermal process, Vfinal is the final volume, and Vinitial is the initial volume of the system, that undergoes an isothermal change.
If the gas gets compressed, as the system undergoes an isothermal process (Vfinal - Vinitial < 0), then according to the given equation, the work done would be positive. That means positive work has been done by the surroundings, on the system.
Whereas, if the gas expands as the system goes through an isothermal process (Vfinal - Vinitial > 0), the work done is negative. In this case, work is done by the system, to expand.
It is an important thermodynamic process to study, that has wide applications in science and engineering. It is of particular importance when studying the Carnot cycle, which is one of the most important processes in engineering thermodynamics.