# Interpretation of the Van der Waals Equation

1. Jul 22, 2014

### nezahualcoyot

There is a silly detail about the interpretation of the Van der Waals (VDW) equation that I cannot fully understand. Say we have the Van der Waals equation for one mole:

(P + a / V^2 ) (V - b) = RT

The usual interpretation is that if you start from the ideal gas law PV=RT, you have to "decrease" the volume to take into account the finite size of molecules, so you replace "V" by "V-b". The attractive forces also reduce the pressure, so you... replace "p" by "p+a/V^2 " ? Why not "p-a/V^2 " ? Why if both pressure and volume are reduced, you subtract a quantity to volume but add a quantity to pressure? I know the equation is correct as it reproduces experimental results within its domain of applicability, but I would like an intuitive explanation for this. Thanks!

2. Jul 22, 2014

### atyy

There is a formal way to see that the "a" term does indeed represent an attractive force, by using the virial expansion. However, I don't know an intuitive explanation to supplement the formal way. Kardar's notes http://ocw.mit.edu/courses/physics/8-333-statistical-mechanics-i-statistical-mechanics-of-particles-fall-2007/lecture-notes/lec17.pdf [Broken] give the formal way and some explanations which are supposed to be intuitive, but I don't understand the latter.

(Kardar's full set of notes is at http://ocw.mit.edu/courses/physics/8-333-statistical-mechanics-i-statistical-mechanics-of-particles-fall-2007/lecture-notes/ [Broken].)

Last edited by a moderator: May 6, 2017
3. Jul 23, 2014

### nezahualcoyot

Thank you atyy. I got an answer for my question elsewhere. Its basically that, in the VDW equation, "V" stands for the ideal gas volume, as this is the quantity you measure experimentally (the volume of the container), but "P" stands for the real gas pressure, as this is what you measure.

To derive VDW from the ideal gas law, P = RT/V, the real gas pressure "P" will depend on the real gas volume, which is the ideal gas volume "V" minus a factor, so you have

P = RT/(V-b)

Finally you must subtract a factor from the real pressure to account for the attractive forces, so you get

P = RT/(V-b) - a/V^2

which is the VDW equation (P + a / V^2 ) (V - b) = RT.

4. Jul 23, 2014

Thanks!