Real Gases: Using the Van der Waals Equation

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  • 0:07 Real Gases Behave Differently
  • 1:55 Using the van der…
  • 5:40 Lesson Summary
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Lesson Transcript
Instructor: Sarah Friedl

Sarah has two Master's, one in Zoology and one in GIS, a Bachelor's in Biology, and has taught college level Physical Science and Biology.

To understand real gas behavior we use the van der Waals equation. This allows us to account for the volume and attractive forces of gas molecules. In this video lesson you'll see this put into action, and understand how it is different from the ideal gas law.

Real Gases Behave Differently

In another lesson, we learned how real gas behavior deviates from ideal gas behavior under non-ideal conditions. They behave differently because these non-ideal conditions are pretty stressful - very high pressures and densities, and very low temperatures. In order to account for this change we need to modify the ideal gas law slightly. These changes include constants that quantify the volume of the gas molecules as well as the attractive forces between those molecules, both of which are considered negligible under ideal conditions.

Originally, the ideal gas law looks like this: PV = nRT. P is the pressure in atmospheres, V is the volume of the container in liters, n is the number of moles of gas, R is the ideal gas constant (0.0821 L-atm/mol-K), and T is the temperature in Kelvin.

For real gases, we make two changes by adding a constant to the pressure term (P) and subtracting a different constant from the volume term (V). The new equation looks like this: (P + an2)(V-nb) = nRT. Here, a is the constant for the attraction between the molecules of a given gas (think a for attraction), and b is the volume those molecules take up inside the container.

You may have noticed a few extra ns tagging along with those two constants, and that is because a and b are values for one mole of that gas. So we have to multiply the constants by the total number of moles to get the correct value. And remember, the constants are different for each gas since each gas has different properties.

Using the Van Der Waals Equation

This new equation is called the van der Waals equation, in honor of Dutch scientist Johannes van der Waals, who did a lot of hard work to figure out how real gases behave. Though this equation looks different than the ideal gas law, you can still solve it in the same way as long as you know the other variables. The difference here is that you also need to know the values for the constants a and b, which, luckily for us, are available and can easily be looked up.

To see how real gases behave let's start with a simple example using the ideal gas law. Say we have the gas CO2. We have 1.00 moles (n), at 273 K (T), and it's in a container with a volume of 22.4 L (V). To solve for the pressure (P), we simply rearrange the equation and do the math.

What we find is that:

Now let's find the pressure using the van der Waals equation. Remember, everything is the same except we now have to incorporate a and b. For CO2,

Here's how it looks now:

What we find now is that P = 0.995 atm. That's pretty close to our first answer, so what's going on here? Well, at normal temperatures and pressures, we will get essentially the same result using either the ideal gas law or the van der Waals equation. So what we need to do is put this gas to the test under some pretty extreme conditions.

Let's try it again, but this time, let's change the volume of the container so that it is much smaller, say 0.100 L. We expect the pressure to increase because there is less space for the gas to take up, meaning the molecules will bump into the container's walls more often.

Using the ideal gas law, we find that

That's a lot of pressure!

Now let's see how this deviates when we use the van der Waals equation for real gas behavior. If we plug in our variables we get:

Our pressure now is only 31.5 atm. Do you see how P is very different than if we use the ideal gas law?

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