The ideal gas law is used to describe the behavior of ideal gases, but sometimes the conditions are such that gases behave differently. When this is the case we can use the van der Waals equation to describe the behavior of real gases under these non-ideal conditions.
The Ideal Gas Law
Cities, states, and countries all have different laws. This is because what works for some places doesn’t work for others. You wouldn’t want the speed limit in the middle of a busy city to be the same as it is on the highway because that doesn’t make sense.Scientific laws are different, though. We call them laws because they have been studied and experimented through and through, helping us understand, predict, and explain natural phenomena, like gravity.
Every once in a while, though, just like speed limits, these laws only make sense under certain circumstances, which we call ‘ideal’ conditions. When this happens, we can adjust the law ever so slightly so that it helps us understand the phenomena under not-so-ideal conditions.We find this to be true with gases. As we learned in another lesson, the ideal gas law, which is the scientific law that describes gas behavior, is written as PV = nRT. In this equation, P is the pressure of the gas; V is the volume of the container holding the gas; n is the number of moles of gas; R is the ideal gas constant (the same for all gases); and T is the temperature of the gas.Most of the time, this law is just what we need to understand the behavior of a gas.
We call gases that can be described by the ideal gas law ideal gases. Under ideal conditions, we can use this equation to find the pressure, volume, number of moles, or temperature of a gas as long as we know the other missing variables.However, a gas may act differently under not-so-ideal conditions, such as really high pressures and densities or at very low temperatures.
These gases that exhibit different properties under extreme conditions are known as real gases. And in order to describe this behavior, we need to deviate slightly from the ideal gas law and use a modified version of the equation.
Gas Behavior Assumptions
Luckily for us, the work here has already been done! Dutch scientist Johannes van der Waals realized that there was a problem with two of the assumptions that are inherent to the ideal gas equation.
The first assumption is that the molecules of a gas make up a negligible amount of the total volume of that gas. The second assumption is that the attraction between the molecules of a gas is also negligible.Let’s start with the first assumption. Under normal circumstances, we can roll with this assumption and use the ideal gas equation to describe gas behavior. But as a gas is compressed, the total volume inside the container becomes less. The molecules of the gas also become more significant in terms of the amount of space they’re taking up relative to the overall space of that container.
Think of it this way: all of the cars in the U.S. are currently spread out over the entire country, so each individual car only takes up a negligible amount of space relative to the total space available. But if you were to put all of the cars in the U.S.
into the state of Rhode Island, the space they take up suddenly becomes quite significant compared to the overall available space! The volume of each car can no longer be neglected, and the same is true for gas molecules.Compressing a gas also forces us to re-examine the second assumption. This is because gas molecules are, in fact, attracted to each other, especially when they are close together.
It’s like having a large room full of people who don’t know each other. At first, they are nervous, so maybe only one or two people talk to each other in the center of the room. But if you were to make the room smaller so that all of the people were forced closer to each other, more conversations start and people are less likely to spread to opposite sides of the room to avoid each other.The same is true for gas molecules. As a gas is compressed, the molecules are forced closer to each other, and they become more attracted to each other.
This also means that there is less pressure on the container because the molecules are huddling together in the middle instead of flying around hitting the walls.
The Van Der Waals Equation
So what can we do to adjust the ideal gas law for real gases? We again look to our two assumptions about gas molecules. First, since molecules do have volume, we need to deduct this from the total volume of the container. So instead of just V, we adjust the variable so that we have V-nb. Here, n is the number of molecules in moles (since this is proportional to the volume they take up), and b is a constant that is different for each gas equal to the volume one mole of particles from that gas occupies.
So with this first adjustment, the equation now looks like this: P(V-nb) = nRT. But we’re not done yet! We still have to adjust for the attraction between the molecules. To do this, we add another term to the pressure in the equation, which looks kind of messy but really isn’t that bad. The new term is an2/V2. V is still the volume, and n is still the number of moles of gas; a is just another experimentally derived constant (and is different for each gas, just like b) relating to the degree of attraction between the molecules.Once we put all of this together, our finished equation looks like this: (P + an2/V2)(V-nb) = nRT.
Not so bad, right? We call this the van der Waals equation in honor of all the hard work he did for us.One question you may be asking yourself is why the two constants, a and b, are different for each gas. Well, different gases have different size molecules. So the constant b relates to the different volumes these molecules will have. The degree of attraction, a, is also different because some molecules are more attracted to each other than others, so we need a specific number for each gas.
Using the ideal gas law is often a perfectly fine way to solve for the pressure, temperature, volume, or number of moles for a given gas. Under ideal conditions, this equation works because we can assume that the attractive forces of the gas molecules as well as the volume of those molecules are negligible. However, under extreme conditions, like very high pressures or very low temperatures, we need to let go of these assumptions and account for the volume and attraction of the gas molecules.This is because these two factors become more important under extreme conditions, just like taking all of the cars in the U.S.
and putting them into one city or smushing a group of party-goers into a smaller room. All of a sudden, there is less space overall, so the relative volume and attractive forces of the molecules have a greater effect.Dutch scientist Johannes van der Waals realized that there was a problem with the two assumptions that go along with the ideal gas equation. Using what we now call the van der Waals equation and known gas constants, we can easily solve for any of the same variables we would using the ideal gas law.
The difference is that, thanks to him, we can also understand gas behavior in less-than-ideal conditions.