Pressure volume work

Indicator diagram and P-V diagram
The graphical nature of pressure volume work, signified by the area of region abcd, was conceived when in the 1830s French physicist Emile Clapeyron began to use the indicator diagram, invented in 1796 by Scottish engineer James Watt and his employee John Southern, to quantify the mechanical work of the steam.
In thermodynamics, pressure volume work, or PV work, is the work done when a body expands outward, typically through the addition of heat to the body, against the surrounding atmosphere, quantified by the measure of the pressure P of the body multiplied by the change in volume dV :

δW = PdV

whereby upon integration of the definite integral:

W=\int_{V_i}^{V_f} P\,dV

equates to:

W = PΔV = P (Vf – Vi)

whereby the system pressure and initial and final volume give solution to the calculation of pressure volume work.

Pressure volume diagram
A 2011 derivation overview of pressure volume work, with a human example application from the 2004 film Mean Girls, by Libb Thims.
See main: dW = PdV
To derive the equation for pressure volume work, we start with the standard definition of pressure, loosely deriving from Dutch-born Swiss physicist Daniel Bernoulli’s 1738 book Hydrodynamica, as force per unit area:

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


F = PA

We then substitute this into French physicist Gustave Coriolis' 1829 principle of the transmission of work:

W = Fd

where mechanical work W is equal to the force F moving a point of mass through a unit of distance d, whence we have:

W = (PA)d

In the case of a geometric body of a piston and cylinder, as shown adjacent, the area A of the cylinder moved upwards through a distance d gives the change in the volume V of the working body of steam, whereby the pressure volume work is thus:

W = PV

or in Neumann notation:

đW = PdV

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


The general formulation of pressure volume work was introduced in 1834 by French physicist Emile Clapeyron. [1]

1. Clapeyron, Émile. (1834). “Memoir on the Motive Power of Heat”, Journal de l’Ecole Polytechnique. XIV, 153 (and Poggendorff's Annalender Physick, LIX, [1843] 446, 566).

External links
Pressure-volume work – Wikipedia.
Pressure volume work –

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