An isothermal process is a process which is conducted in a manner such that the temperature remains constant during the entire operation.
The reversibly expansion of the gas takes place in a finite number of infinitesimally small intermediate steps.
To start with the external pressure, is arranged equal to the internal pressure of the gas P ext and the piston remains stationary.
If Pext is decreased by an infinitesimal amount dP the gas expands reversibly and the piston moves through a distance dl.
Since dP is so small, for all practical purposes
hence, P ext = P gas = P The work done by gas in one infinitesimal step (dw), can be expressed as:
dw = P x A x dl (A = cross-section area of piston)
P x dV