# Ideal Gas Law

## Limits of the Ideal Gas Law

If you compress a gas (by reducing the volume) the ideal gas law gives a clear prediction of how pressure and temperature should change.

It doesn’t predict exactly how the change will distribute between pressure and temperature – however, if the compression is slow, and heat is allowed to exchange between the gas and its surroundings, temperature will remain constant and pressure will predictably increase under compression.

If we rearrange the ideal gas law, we should note that the ratio between PV and nRT is equal to 1.

[frac{pV}{nRT}=1]

If this remains true, the system is ideal.

## Compression under Real Conditions

However, this value may change, and no longer equal 1. The ratio here can equal Z, known as “compressibility”.

[frac{pV}{nRT}=Z]

When Z is approximately 1, the gas behaves ideally. The behaviour of “Z” is best described if we fix one of the three variables (most conveniently, temperature) and monitor the change of another (volume would be good) as we vary the other (pressure). We can then repeat that at different temperatures to show how Z changes.

Usually, as pressure increases, the real volume is lower than the ideal gas law predicts (Z<1). This is because molecules have attractive properties, and interact with each other. The attraction causes them to compress together more than predicted if they did not interact.

At higher pressures, the volume is greater than predicted by the ideal gas law (Z>1) as the gases are close enough to start repelling more than attracting.

At higher temperatures, the overall deviation is lower, and Z tends to be only greater than 1, and closer to 1. A real gas behaves more ideally at higher temperatures.

### Modifying the Ideal Gas Law

The ideal gas law is modified into the van der Waals equation by modifying the pressure to take into account molecular attraction (with parameter “a”), and by modifying the volume to take into account the size of molecules (with parameter “b”).

These two parameters are empirical, and just fitted from data.

[left(p+frac{n^2a}{V^2}right)left(V-nbright)=nRT]

Parameter “a” that modifies pressure is also divided by volume squared, because there is a volume dependence on attractive forces. All else being equal, larger molar volumes mean that the gases are less likely to interact with each other. This means that the strength of attraction and its effect is much lower at larger volumes.

You should also note that these two parameters have units, and will vary depending on whether they are molar quantities or per-molecule quantities.