Non-ideal behavior of gases

By now you've probably heard a lot about the ideal gas law, and you may have a sense of how to use the ideal gas equation to look at the relationships between pressure ($P$‍), volume ($V$‍), moles of gas ($n$‍), and temperature ($T$‍). But when do gases follow the ideal gas law and why? What if we want to study a gas that behaves in a “non-ideal” way? When we use the ideal gas law, we make a couple assumptions:

$1.$‍We can ignore the volume taken up by the imaginary ideal gas molecules
$2.$‍The gas molecules do not attract or repel each other

However, we know that in real life, gases are made up of atoms and molecules that actually take up some finite volume, and we also know that atoms and molecules interact with each other through intermolecular forces.

One way we can look at how accurately the ideal gas law describes our system is by comparing the molar volume of our real gas, $V_m$‍, to the molar volume of an ideal gas at the same temperature and pressure. To be more specific, at some temperature we can take $n$‍ moles of our gas and measure the volume it takes up at a given pressure (or measure the pressure for a known volume). We can also calculate the molar volume of the ideal gas at the same temperature and pressure, and then take the ratio of the two volumes.

This ratio is called the compressibility or compression factor, $Z$‍. For a gas with ideal behavior, $V_m$‍ of the gas is the same as $V_m$‍ of an ideal gas so $Z=1$‍. It turns out that this is reasonably accurate for real gases under specific circumstances that also depend on the identity of the gas. Let’s look at the compressibility $Z$‍ for a couple different gases.

This graph shows the compression factor $Z$‍ over a range of pressures at $273\text{ K}$‍ for nitrogen (${\text{N}}_2$‍), oxygen (${\text{O}}_2$‍), hydrogen (${\text{H}}_2$‍), and carbon dioxide (${\text{CO}}_2$‍). For all of the real gases in this graph, you might notice that the shapes of the curves look a little different for each gas, and most of the curves only approximately resemble the ideal gas line at $Z=1$‍ over a limited pressure range. Also, for all the real gases $Z$‍ is sometimes less than $1$‍ at very low pressures, which tells us that the molar volume is less than that of an ideal gas. As you increase the pressure past a certain point that depends on the gas, $Z$‍ gets increasingly larger than $1$‍. That is, at high pressures the $V_m$‍ of the gas is larger than $V_m$‍ of the ideal gas, and $V_m$‍ of the real gas increases with pressure. Why is that?

At high pressures, the gas molecules get more crowded and the amount of empty space between the molecules is reduced. How might this affect $V_m$‍ and $Z$‍? It helps to remember that the volume we use in the ideal gas equation is the empty volume that the gas molecules have to move around in. We usually assume that this is the same as the volume of the container when the gas molecules don’t take up much space. But what happens when this is not the case, such as at high pressures?

For a given pressure, the real gas will end up taking up a greater volume than predicted by the ideal gas law since we also have to take into account the additional volume of the gas molecules themselves. This increases our molar volume relative to an ideal gas, which results in a value of $Z$‍ that is greater than $1$‍. The error in molar volume gets worse the more compressed the gas becomes, which is why the difference between $Z$‍ for the real and ideal gas increases with pressure.

To examine the effect of intermolecular forces, let’s look at the compressibility of a single kind of gas at different temperatures.

For nitrogen, you can see that at $T=300\text{ K}$‍ and $400\text{ K}$‍ with pressures below $200\text{ bar}$‍, the curve looks relatively similar to what you would expect for an ideal gas. As you lower the temperature to $200\text{ K}$‍ and $100\text{ K}$‍, the curves look much less ideal. In particular, at low pressures we see that $Z$‍ for real gases is noticeably less than $1$‍ for $T=200\text{ K}$‍, and this effect is even more pronounced at $100\text{ K}$‍. What’s going on at lower temperatures?

Imagine our gas molecules bouncing around in the container. The pressure we measure comes from the force of the gas molecules hitting the walls of the container. Attractive forces between the molecules will pull them a little closer together, which effectively slows the molecule down a little before it hits the container wall.

This results in a decrease in volume if the pressure is constant compared to what you would expect based on the ideal gas equation. The decreased volume gives a corresponding decrease in $V_m$‍ compared to the ideal gas so $Z<1$‍. The effect of intermolecular forces is much more prominent at low temperatures because the molecules have less kinetic energy to overcome the intermolecular attractions.

We can use a number of different equations to model the behavior of real gases, but one of the simplest is the van der Waals (VdW) equation. The VdW equation basically incorporates the effect of gas molecule volume and intermolecular forces into the ideal gas equation.

where:

$P=$‍ measured pressure
$V=$‍ volume of container
$n=$‍ moles of gas
$R=$‍ gas constant
$T=$‍ temperature (in Kelvin)

Compared to the ideal gas law, the VdW equation includes a “correction” to the pressure term, ${\displaystyle \frac{a{n}^2}{V^2}}$‍, which accounts for the measured pressure being lower due to attraction between gas molecules. The “correction” to the volume, $nb$‍, subtracts out the volume of the gas molecules from the total volume of the container to get a more accurate measure of the empty space available for the gas molecules. $a$‍ and $b$‍ are measured constants for a specific gas (and they might have some slight temperature and pressure dependence).

At low temperatures and low pressure, the correction for volume is not as important as the one for pressure, so $Z$‍ is less than $1$‍. At high pressures, the correction for the volume of the molecules becomes more important so $Z$‍ is greater than $1$‍. At some range of intermediate pressure, the two corrections cancel out and the gas appears to follow the relationship given by the ideal gas equation.

In a nutshell, the ideal gas equation works well when intermolecular attractions between gas molecules are negligible and the gas molecules themselves do not occupy a significant part of the whole volume. This is usually true when the pressure is low (around $1\text{ bar}$‍) and the temperature is high. In other situations such as high pressures and/or low temperatures, the ideal gas law might give answers that are different from what we observe experimentally. In these cases, you can use the van der Waals (or a similar) equation to take into account the fact that gases do not always behave as ideal gases.