Imagine a perfect gas contained by a rigid-walled cylinder equipped with a frictionless piston held in position by a removable external agency such as a magnet. There are finite differences in the pressure (P1>P2) and volume (V2>V1) of the gas in the two compartments, while the temperature can be regarded as constant.

If the constraint on the piston is removed, will the piston move? And if so, in which direction?

Common sense, otherwise known as dimensional analysis, tells us that differences in volume (dimensions L3) cannot give rise to a force. But differences in pressure (dimensions ML-1T-2) certainly can. There will be a net force of P1–P2 per unit area of piston, driving it to the right.

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The driving force

In thermodynamics, there exists a set of variables which act as “generalized forces” driving a system from one state to another. Pressure is one such variable, temperature is another and chemical potential is yet a third. What they all have in common is that they are intensive variables – those which are independent of the quantity of a phase.

Each intensive variable has a conjugate extensive variable – those which are dependent on the quantity of a phase – and together they form a generalized force-displacement pair which has the dimensions of work (= energy ML2T-2). Examples include pressure × volume, temperature × entropy, and voltage × charge.

But back to those intensive variables. Experience confirms that it is these generalized forces which are the agents of change. And spontaneous change results when there are finite differences in intensive variables between one system and another – the direction of change being determined by their relative values.

– – – –

The result is work

In the illustrated example above, the freed piston moves spontaneously to the right because of finite differences in pressure (P1>P2). As a result, the system consisting of the left hand compartment does PV work on the system consisting of the right hand compartment. Likewise, finite differences in other intensive variables such as temperature and chemical potential will act through their respective force-displacement pairs to perform work.

We can thus conclude that finite differences in intensive variables drive spontaneous change, and that due to the dimensionality of their respective conjugate extensive variables, spontaneous change results in the performance of work. It should however be noted that this work is not always useful work. The spontaneous diffusion of two gases into each other is a classic case, where it is difficult to imagine how the work could be usefully obtained.

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P Mander March 2015

  1. Joseph Nebus says:

    Reblogged this on nebusresearch and commented:
    I’d wanted just to point folks to the latest essay in the CarnotCycle blog. This thermodynamics piece is a bit about how work gets done, and how it relates to two kinds of variables describing systems. The two kinds are known as intensive and extensive variables, and considering them helps guide us to a different way to regard physical problems.


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