A system of moving particles exerting a pressure P, quantified as a force per unit area, on the walls of the containing vessel. |

is the differential expression for pressure volume work. If, to note, the equation for the pressure volume work becomes an inexact differential, which may occur as particle count approaches one, then it may become intuitive to use Neumann notation:

đW = PdV

where the d-crossbar đ derivative signifies that the work in this case is an inexact derivative.

Derivation

When a body (working fluid, working body, working substance, or system) is enclosed inside of the volumetric boundaries (or boundary surface) of a piston and cylinder, at a pressure

In this scenario, the standard definition of pressure, first given by Dutch-born Swiss physicist Daniel Bernoulli in his 1738

hence, the force will be given by:

where

In this case the distance moved

and the force

where the work done by the body, being negative, with respect to the internal energy of the body, if the change of height

In this last expression, we note that the product of the surface are A and the change in height dh of the piston equates to volume change dV, or:

Hence, with substitution into the previous to last expression:

An irregularly shaped system at an initial state volume A and final state volume B, shown indicated with a small surface element dσ (d-sigma) and displacement element dn. [1] | |

An example of human PV work (above, right): the alpha female, flanked by two beta females, and a forth gamma-alpha female, enter a system, at an initial state 1 (volume one), which trigger reaction, resulting in a transformation of the system to state 2 (volume two), described by a human molecular pressure P, directed radially outward from the alpha female (a Johannes van der Waals theory), an amount of human work, which can be quantified by the product of the pressure into the integration of the surface element dσ (d-sigma) and the boundary element dn. [2] |

Irregular shapes

If the body or system is not of the standard piston and cylinder geometry, e.g. the territory of Rome, the same derivation applies, albeit with slight adjustment to the volume calculation.

The adjacent diagram shows a body at a uniform pressure P enclosed in an irregularly shaped container, at an initial state, position, or volume A, which undergoes some transformation, e.g. territorial expansion or acquisition, heat addition, etc., and expands to a final state, position, or volume B.

In this calculation, we let

The total work performed during the infinitesimal transformation is obtained by integrating the above expression over all the surface

where, if the pressure is found to be constant (which may not always be assume true when dealing with

In this last expression, we note:

and with substitution into the previous to last equation:

and the derivation is complete.

A PV diagram showing a body transforming from state 1 to state 2 and the representative work done indicated by the shaded area under the curve. |

Integration

If the integration can be written in the form of the a definite integral, having a definite measurable initial state volume V1 and final state volume V2, then:

The integral, and hence the work done, can be represented by the shaded area under the curve.

References

1. Fermi, Enrico. (1936).

2. (a) Thims, Libb. (2007).

(b) Thims, Libb. (2007).