Understanding Work Done in Thermodynamics

In thermodynamics, work done by a system is often represented on a pressure-volume (P-V) diagram. The amount of work done depends on the path taken by the process connecting the initial and final states.

The Work Formula \(\smallint {\rm{pdv}}\)

The formula given, \(\smallint {\rm{pdv}}\), represents the work done by the system during a volume change from an initial volume \(V_1\) to a final volume \(V_2\), where p is the pressure of the system at any given volume V. This integral calculates the area under the process curve on a P-V diagram.

Condition for Using \(\smallint {\rm{pdv}}\)

The fundamental requirement for the work done by a system to be accurately calculated using the formula \(\smallint {\rm{pdv}}\) is that the process must be a quasi-static process.

What is a Quasi-Static Process?

• A quasi-static process is an idealized thermodynamic process that proceeds infinitely slowly.
• At every infinitesimal step during a quasi-static process, the system is considered to be in thermodynamic equilibrium.
• This means that properties like pressure and temperature are uniform throughout the system at all times during the process.

When a process is quasi-static, the pressure 'p' in the formula \(\smallint {\rm{pdv}}\) is the well-defined thermodynamic pressure of the system, which is exerted uniformly on the boundary that is moving (causing the volume change dv).

Why Other Options Are Not Sufficient

Let's look at the other options:

• Isentropic Process: A process where entropy remains constant. While many isentropic processes (like a reversible adiabatic expansion) can be quasi-static, an irreversible adiabatic process is not quasi-static, and the work done cannot be simply calculated as \(\smallint {\rm{pdv}}\) based on the system's internal pressure.
• Isothermal Process: A process occurring at constant temperature. Similar to isentropic processes, an isothermal process can be quasi-static (reversible isothermal expansion) or non-quasi-static (rapid expansion against zero pressure). The \(\smallint {\rm{pdv}}\) formula is only valid if the isothermal process is quasi-static.
• Adiabatic Process: A process where no heat transfer occurs. Again, an adiabatic process can be reversible (and thus quasi-static) or irreversible (and not quasi-static). The \(\smallint {\rm{pdv}}\) formula applies only to the quasi-static (reversible) adiabatic case using the system's pressure.

The formula \(\smallint {\rm{pdv}}\) specifically calculates the boundary work for a system undergoing a quasi-static expansion or compression. In non-quasi-static processes (like a rapid expansion or compression), the pressure within the system may not be uniform, and the work done at the boundary may not be equal to \(\smallint {\rm{pdv}}\) based on the internal system pressure. The integral \(\smallint {\rm{pdv}}\) correctly represents the work done by the system on its surroundings only when the process is quasi-static.

Therefore, the work done during a process can be determined by \(\smallint {\rm{pdv}}\) when the process is quasi-static.