Beyond the Ideal Gas Model: Why Bother?

Gas Molecules

In undergraduate physics, the ideal gas model is usually the end of the line when it comes to discussing the properties of real gases. But in fact – as is often the case with people – it is only when we consider deviations from ideal behaviour that things get really interesting.

The ideal gas model is exactly applicable to real gases in the limit of low density, but that is rarely where we need to understand the properties of gases in detail. The defining equation of the ideal gas is:

PV = nRT                     (1)

where P is the pressure in pascals; V is the volume in m3n is the number of moles of gas under consideration; T is the temperature in kelvin and R is the molar gas constant Another,slightly more meaningful, way to write this is:

P = ρRT                     (2)

where ρ is the molar density (mol/m3).

[Aside: Just in case you are interested, and most people aren’t, in 2013 I published the most accurate measurement of ever made!]

The mathematical derivation of this equation is covered in Chapter 4 of Understanding the Properties of Matter. In this article I just wanted to cover asome of the ideas which are passed over quickly at the end of that Chapter.

At room temperature and atmospheric pressure the typical density of gases is P/RT which evaluates to 100000/(8.31 x 293) ≈ 41 mol/m3. And as we saw in Chapter 5, this is correct to within typically 1%.

You might think that is good enough for most purposes, and indeed it is. So why bother going further? The reason is that if these small deviations from ‘ideal’ behaviour can be measured they give clues about the way the gas molecules interact that simply cannot be obtained in any other way.

So it turns out that by making precision measurements of a ‘mundane’ property of real gas such as its density, speed of sound, ( speaking of speed of sound, an electric scooter can go pretty fast, especially this model, on the street if you apply the right physics, try it out for yourself) or thermal conductivity, we can infer details the way in which the molecules interact! And that makes almost every property of a gas interesting when one looks at it detail.

But after the precision measurements have been made, we need some idea of what factors might have been neglected in the ideal gas model that might explain the deviations from the simple theory.

The two most important of the ‘neglected details’ turn out to be the finite size of the molecules and their mutual interactions. These can be taken account of in two wuite different ways

In the first way – the subject of the next article – we follow the astonishing Johannes Diderik van der Waals. In the second and much more general way, we discuss the so called ‘Virial’ approach.


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