What is the work of isothermal reversible expansion of a perfect gas represented by?

The work done during an isothermal reversible expansion of an ideal gas is expressed by the equation ( w = -nRT \ln\left(\frac{V_f}{V_i}\right) ). This relationship arises from the first law of thermodynamics and the principles governing isothermal processes.

In an isothermal process, the temperature remains constant while the gas expands. For an ideal gas, the internal energy change (( \Delta U )) is zero during an isothermal expansion because internal energy depends only on temperature for an ideal gas. Therefore, any heat absorbed by the system (( q )) is equal to the work done by the system (( w )), leading to the equation:

[ q = w ]

To derive the expression for work, we consider the infinitesimal amount of work done ( dW ) by the gas during an infinitesimal volume change ( dV ) against an external pressure ( p ):

[ dW = -p dV ]

In a reversible process, the pressure of the gas can be expressed using the ideal gas law as ( p = \frac{nRT}{V} ). Thus substituting this into the work equation gives us:

[ dW = -