# 19.2: Pressure-Volume Work

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- 13700

Work in general is defined as a product of a force **F** and a path element **ds**. Both are vectors and work is computed by integrating over their inner product:

\[w = \int \textbf{F} \cdot \textbf{ds}\]

Moving an object against the force of friction as done in the above dissipation experiment is but one example of work:

\[w_{friction} = \int \textbf{F}_{friction} \cdot \textbf{ds}\]

We could also think of *electrical* work. In that case we would be moving a charge *e* (e.g. the negative charge of an electron) against an electrical (vector) field \(\textbf{E}\). The work would be:

\[w_{electical} = \int e \textbf{E} \cdot \textbf{ds}\]

Other examples are the stretching of a rubber band against the elastic force or moving a magnet in a magnetic field etc, etc.

## Pressure-volume (\(PV\)) work

In the case of a cylinder with a piston, the pressure of gas molecules on the inside of the cylinder, \(P\), and the gas molecules external to the piston, \(P_{ext}\) both exert a force against each other. Pressure, (\P\), is the force, \(F\), being exerted by the particles per area, \(A\):

\[P=\frac{F}{A}\]

We can assume that all the forces generated by the pressure of the particles operate parallel to the direction of motion of the piston. That is, the force moves the piston up or down as the movement of the piston is constrained to one direction. The piston moves as the molecules of the gas rapidly equilibrate to the applied pressure such that the internal and external pressures are the same. The result of this motion is work:

\[w_{volume} = \int \left( \dfrac{F}{A} \right) (A\,ds) = \int P\,dV\]

This particular form of work is called **pressure-****volume (\(PV\)) work** and will play an important role in the development of our theory. Notice however that volume work is only * one form* of work.

It is important to create a sign convention at this point: positive heat, positive work is always energy you put in into the system. If the system decides to remove energy by giving off heat or work, that gets a minus sign.

In other words: * you pay the bill*.

To comply with this convention we need to rewrite volume work (Equation \(\ref{Volume work}\)) as

\[w_{PV} = - \int \left( \dfrac{F}{A} \right) (A\,ds) = - \int P\,dV \]

Hence, to decrease the volume of the gas (\(\Delta V\) is negative), we must put in (positive) work.

Thermodynamics would not have come very far without cylinders to hold gases, in particular steam. The following figure shows when the external pressure, \(P_{ext}\), is greater than and less than the internal pressure, \(P\), of the piston.

If the pressure, \(P_{ext}\), being exerted on the system is constant, then the integral becomes:

\[w = -P_{ext}\int_{V_{initial}}^{V_{final}}dV = -P\Delta V \label{irreversible PV work} \]

For a system that undergoes work at constant external pressure, we can show the amount of work being done using a \(PV\) diagram.

Equation \(\ref{irreversible PV work}\) only holds under isobaric conditions. If the outside pressure changes during compression we must *integrate* over all contributions:

\[ w = - \int_{V_{initial}}^{V_{final}}P_{ext}dV\]