Piston and cylinder
A system of moving particles exerting a pressure P, quantified as a force per unit area, on the walls of the containing vessel.
In equations, the formula:

 dW = PdV \,

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.

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 P, and undergoes some transformation, the change will tend to be accompanied by a volume change dV, in accordance with Boerhaave's law (1720), thus causing the piston to expand (expansion work) or contract (contraction work), by an amount of vertical height dh.

In this scenario, the standard definition of pressure, first given by Dutch-born Swiss physicist Daniel Bernoulli in his 1738 Hydrodynamica, will be defined as the force F the particles of the container exert on the walls of the boundary per unit area A or:

 P = \frac{F}{A} \,

hence, the force will be given by:

 F = PA \,

where A is the surface area of the face of the moving piston head. Then, according to Gustave Coriolis' 1829 principle of transmission of work, the work done by (or on) the body will be given by the product of the force and the distance of the piston head moved:

 W = Fd \,

In this case the distance moved d will be dh:

 W = Fdh \,

and the force F will be PA, hence the work will be given by the expression:

 W = PAdh \,

where the work done by the body, being negative, with respect to the internal energy of the body, if the change of height dh of the piston is positive (piston moves up), or positive, with respect to the internal energy of the body, if the change of height dh of the piston is negative (piston moves down).

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:

 dV = Adh \,

Hence, with substitution into the previous to last expression:

 dW = PdV \,

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

 W = P \int_{n_1}^{n_2} d\sigma dn \,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 (d-sigma) and the boundary element dn. [2]
and the derivation is complete.

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 (d-sigma), be a surface element of the container or boundary, and let dn be a displacement element, i.e. the displacement of the surface element in a direction normal to the surface of the container. The work performed on the surface element by the pressure P during the displacement of the container from the situation A to situation B is:

 dW = P d\sigma dn \,

The total work performed during the infinitesimal transformation is obtained by integrating the above expression over all the surface σ of the boundary or container:

 dW = \int P d\sigma dn \,

where, if the pressure is found to be constant (which may not always be assume true when dealing with human pressures and human volume changes of human systems and human boundaries, e.g. as during the volume change associated with the rise and fall of Rome, the so-called beauty expansion or height expansion phenomenon, associated with personal space, etc.), can come outside of the integral:

 dW = P \int d\sigma dn \,

In this last expression, we note:

 dV = \int d\sigma dn \,

and with substitution into the previous to last equation:

 dW = PdV \,

and the derivation is complete.
PV diagram
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.

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:

 W = \int_{V_1}^{V_2} PdV \,

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

1. Fermi, Enrico. (1936). Thermodynamics (pgs. 5-6). Prentice-Hall.
2. (a) Thims, Libb. (2007). Human Chemistry (Volume One) (alpha molecules and Van der Waals force, pgs. 291-95). Morrisville, NC: LuLu.
(b) Thims, Libb. (2007). Human Chemistry (Volume Two) (mean girls, pg. 550). Morrisville, NC: LuLu.

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