###### By Don Lincoln, Ph.D., University of Notre Dame

## What are the main differences between the ideal gas law and the Van der Waals equation? And, why is the Van der Waals equation considered as an improvement over the ideal gas law?

### How Are Ideal Gas Law and Van der Waals Equation Different?

In 1873, Dutch physicist Johannes Diderik van der Waals came up with a modification of the ideal gas law.

The ideal gas law is written as PV=nRT, where P is pressure, V is volume, n is the number of molecules in units of moles, T is the temperature, and R is just a constant. Van der Waals equation is written in a slightly different way.

It is written as the quantity P plus n-squared times a divided by V-squared times the quantity V minus n times b equals nRT. So, that looks pretty similar to the one we’re used to. Let’s identify these extra terms and tie them back to the effects we’ve already discussed.

[P +a(n/V)^{2}] (V – nb) = nRT

The first term, the one with the ‘a’, is due to the attraction of molecules to one another. If molecules are attracted to one another, they don’t fly off as much and hit the walls and this reduces the pressure on the walls of the container. The symbol ‘a’ is a measure of the average attraction between molecules. If you combine it with the n-squared, it becomes the square number of molecules times the attraction between molecules. A differs for different substances, after all, not all types of molecules have the same strength of interactions. So, this modification is to take into account the stickiness of the individual molecules.

If you’re wondering where that square comes from, it’s because the net effect of the amount of stickiness is proportional to the density of molecules. This net effect is also proportional to the number of molecules that can interact by this stickiness, which is again proportional to density. And since the density is number divided by volume, the density times density results in a one over volume squared behavior.

The other term, the one with the ‘b’, is due to the fact that molecules don’t have access to the entire volume of the container. You have to subtract off the volume taken up by the molecules themselves. The symbol b is the volume excluded by a mole of molecules. When it’s multiplied by n, which is the number of moles, the result is the volume of all of the molecules in the volume. And since the size of molecules differs from substance to substance, this number also depends on the identity of the gas.

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### Comparisons under Low Pressure

So, van der Waals equation is a lot more complicated than the ideal gas law. How much does it matter? We can just do an example and figure it out. In chemistry class, a common set of parameters is called STP, short for standard temperature and pressure. The temperature that is chosen is the freezing point of water: 0° centigrade or 273.15° Kelvin.

The standard pressure is one atmosphere. If you want to know for an ideal gas what volume a single mole of gas will fill up using the combined gas law, you can calculate that very easily.

Put in a pressure of one atmosphere, a value of 0.082-liter atmospheres per mole for the ideal gas constant R, and an n of one mole with the standard temperature of 273.15 Kelvin, you find that the standard volume is 22.4 liters. We can also calculate the pressure by looking for P, which is nRT divided by V. Put in the values for nRT and V and you get the pressure of one atmosphere.

So let’s see what pressure we get with van der Waals equation. Let’s do carbon dioxide since the a and b parameters are large, which means a bigger deviation from the ideal gas law. We can solve for the pressure.

First, we divide by the V minus nb term and then we subtract from that a n-squared over V-squared term. We put in all the numbers and we find that the pressure using the van der Waals equation is 0.996 atmospheres, or just 0.4% smaller than using the ideal gas law.

So, you might be wondering if the difference is so small, why bother with the more complicated equation? The ideal gas law is actually pretty good that’s why we teach it. But remember that the ideal gas law is about the behavior of gases in a low pressure, high temperature, environment. That means that the van der Waals equation makes a bigger difference when the pressure goes up.

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### Comparisons under High Pressure

So, let’s imagine another situation. Let’s take the same mole of carbon dioxide, at the same temperature, which is the freezing point of water, or 273.15 Kelvin, and squash it.

At one atmosphere of pressure, the volume of the gas we’ve taken is 22.4 liters. Now, let’s squash the gas down to one-hundredth of its size. That means taking those 22.4 liters of carbon dioxide and squashing it to 0.224 liters. What pressure would that be? Well, in the ideal gas law, it’s easy to see. Since pressure is proportional to one over volume, then reducing the volume to a 100th the initial size would increase the pressure to a 100-times greater or 100 atmospheres.

However, you get a very different pressure by using the van der Waals equation. It’s only 52 atmospheres or about half of what you get using the ideal gas law. So, that’s a big difference. At high pressure, you need the van der Waals equation to get things right.

One might think that the pressure of a gas is always lower than predicted by the ideal gas law. That’s a sensible conclusion, but it’s hasty. It’s not true, well, not always.

Suppose we reduce the volume, even more, by another factor of four. Try to compress a mole of carbon dioxide into 0.05 liters. What pressure does van der Waals predict?

The ideal gas equation predicts a pressure of about 450 atmospheres, which is a lot. But the van der Waals equation predicts a pressure of a little more than 1600 atmospheres or more than the ideal gas law.

Why is that? Let’s remember what those new terms in the van der Waals equation mean. One is how the molecules bump into one another as they zoom around the container. But the other is when the molecules get pushed together enough that they start touching. It’s almost as if they are approaching a liquid phase. But, either way, the size of the individual molecule becomes important. And that’s what’s happening here.

That term in the van der Waals equation, the one that says V minus b times n? Well, for carbon dioxide, b is 0.04367 liters per mole. Since in our example we have one mole, we see that when the volume becomes 0.04367, the V minus bn term becomes zero. If you take this seriously, this suggests that when a mole of carbon dioxide is put in a volume of 0.04367 liters, there is no space left for the molecules to move.

And if you tried to squish the volume, even more, you’d be pushing molecules into other molecules, and pretty clearly what we’ve spoken about just doesn’t work. We didn’t take that eventuality into consideration when we did the calculation and it is inevitable that the equations would break down.

In fact, if you tried to go to a volume smaller than 0.04367 liters, that whole V minus bn term becomes negative, which means that the equation predicts a negative pressure. Negative pressure means that the material wouldn’t push out, it would push in. And it means we’ve pushed the equation too hard. It was never intended to work under those conditions.

In the same way that the ideal gas law is an approximation, van der Waals equation is also an approximation.

There is a better version called the Redlich-Kwong model, which is almost always better than the van der Waals equation, although in many cases they are pretty much the same. So, even though we’ve spent a lot of time talking about the van der Waals equation, it’s not the end of the line. There is more to learn. There is always more to learn.

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### Common Questions about the Ideal Gas Law or Van der Waals Equation

**Q: What are A and B in the van der Waals equation?**

The a and b are called van der Waals constants: The constant a provides a measure of the average attraction of the molecules, whereas constant b adjusts for the volume occupied by the gas particles.

**Q: Why does the van der Waals equation better approximate real gas behavior?**

The van der Waals equation improves upon the ideal gas law by accounting for the volume of the gas molecules and for the attractive forces present between the molecules.

**Q: What is the real gas equation?**

The ideal gas law or real gas equation is PV = nRT. P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature in Kelvin.

**Q: What is ideal gas behavior?**

A gas behaves more like an ideal gas at high temperatures and low pressure, conversely low temperature and high pressure lead to different behavior.