If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: Chemistry library>Unit 8

Lesson 3: Non-ideal gas behavior

# The van der Waals equation

By adding corrections for interparticle attractions and particle volumes to the ideal gas law, we can derive a new equation that more accurately describes real gas behavior. This equation, known as the van der Waals equation, can be used to calculate the properties of a gas under non-ideal conditions. Created by Sal Khan.

## Want to join the conversation?

• Wonderful lecture but sir,I can't understand how you put n/v^2
• Interesting question!
Let's examine proportional factors that we have.
1.) The attractive force is proportional to the density of molecules (the more molecules we have, the stronger the force), hence proportional to n/V
2.) On the same time not all molecules experience the same attractive forces. As it was said in the video, the attractive forces between the molecules in the center of the container cancel out, as they are getting "pulled" in many directions simultaneously.
Hence, the attractive forces are experienced only by those molecules which are closer to the wall of the container. The number of molecules which are "on the edge" of the container is also proportional to the density (the more molecules there are, the more of them will be located near the "edge"), hence we again have n/V
Therefore we have two equal proportionalities. After we multiply them (n/V)*(n/V) we get (n/V)2

I hope it helped!

P.S. If you wanna dig a little bit deeper, here's the link: https://chemistry.stackexchange.com/questions/70616/why-is-the-pressure-correction-in-the-van-der-waals-equation-proportional-to-n
• why did they need n^2 and v^2? couldn't they just use a single n and v or perhaps n^3 and v^3? Why did they choose the squared versions?
• It means that for a single molecule, its inward force is proportional to the concentration of other molecules. And the total inward force of all the molecules is also proportional to the concentration. So there is (n/v)^2.
• I don't understand how the real volume will be greater than the ideal volume. Suppose n ideal molecules occupy say "x" space, since they have no volume, the volume they occupy will be "x". But if n real molecules occupy say "x" space, the actual space available to the molecules to move will be less than x and so the volume they occupy will be less than x, won't it?
• You are on the right track. If n real gas molecules occupy a container of the same volume as one occupied by n ideal gas molecules, then the real gas molecules will occupy more space in that container; this means the real gas molecules will have less room to move around. Now what if we want to keep the pressure in both containers constant? This means we would have to give the real gas molecules the same amount of space to move around, thus we would need to make their container larger. So in essence, this whole question boils down to deciding on which value you want to keep constant; if you want the pressure of a real gas to be equal to that of an ideal gas, then you have to increase the volume of the real gas's container, but if you want to keep the volume constant, then the real gas will exert a higher pressure because it takes up more space in the container. :)
• Despite the previous questions and answers, I still do not understand why one is supposed to square n/v, I understand that there is a difference in density between the edge and the center of the container, but why would one not simply find the average density rather than square the value for density?
• what are the values and the units of 'a' and 'b'? i mean, since they are constants, they are bound to have some unit and value , right? help me if i'm wrong :)
• What is the difference between the two (n/v)s?
• As there is differece between (pressure real and pressure ideal) as Sal stated as Pressure real< Pressure ideal so following this this creates a variation between areas of their particular container in which they have been placed, this follows to density (n/v) and experimantally it is found to be some constant times (n/v) i.e. a(n/v)
• Why you take n/v whole square
• In the final "Van Der Waals Real Gas Equation," wouldn't the P and V values be the "ideal" values that you are adding adjustments to?

Example: [Vr - bn] ... wouldn't that be [Vi - bn] because you are adjusting the ideal?
• I think they are explaining wrong here. We for certainty know the equation is correct. But why one term(a) is added and the other term(b) is subtracted is not explained precisely. I too came here looking for this, but I guess I have to search more.
(1 vote)
• Density is mass/volume not (moles/volume) and what is the sense of considering it density and applying to derive the formula
• I think that the density should be considered as number density and not as mass density because in case of gases what matters more is how many 'number' of molecules we have. So, that's the first point.

Now the reason why we consider number density here:
First we are looking at just one gas molecule and it is experiencing a net inward (in this case) from not just one but many molecules. so we multiply 'a' by the number of molecules that we have per unit volume (you could also imagine a 1 cubic unit container). But we don't have just one molecule in the container. So, we multiply this by the number of molecules we have per unit volume to take into account all the molecules.
Hope it helps.