# Is it possible to obtain higher order corrections to the ideal gas law when one allows realistic phenomena to make their way into the equations?

I had an interesting thought today that caused me to ask whether it'd be possible to make corrections to the ideal gas law via introducing terms derived from more realistic phenomena to make their ways into the equation.

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Van der Waals' Gas Equation not satisfactory? –  ABC May 13 '13 at 12:50
Rofl na it's perfect! Thanks! I didn't know it existed –  Brenton Horne May 13 '13 at 13:00

See van der Waals' Gas Equation :

$$\Bigg(P+\dfrac{an^2}{V^2}\Bigg)\Big(V-nb\Big)=nRT$$

$a,b$ are constant dependent on gas properties.

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You might want to add something relating a and b to microscopic properties: b is the "volume" occupied by single molecules, and an^2 V^-2 a pressure coming from intermolecular attraction. –  Chay Paterson May 14 '13 at 20:12

Readers might be interested to know that you can actually see such terms in macroscopic data. I allude to this at the very end of this answer, which discusses an analysis of (century-old) mercury vapor pressure data. When you plot the residuals (not shown there), they display a sharp upside-down "V" shape. As I recall, the rapid change of slope can be traced to non-ideal behavior at the atomic scale and even provides quantitative hints about the energy potential at very short distances. (It is not a van der Waals-type equation.)

That answer provides the data and working code to reproduce the analysis, so interested readers could pursue this idea in more detail. It would be very interesting to see how far one can go in teasing out such information.

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