Interpretation of the Van der Waals Equation

[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!

 

[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 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/ .)

 

[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.

 

[atyy]

Thanks!

 

[PJC01]

The Van der Waals equation is a modification of the ideal gas law that takes into account the finite size of molecules and the attractive forces between them. The term "a/V^2" represents the correction for the attractive forces, while the term "b" represents the correction for the finite size of molecules.

The interpretation of the equation is based on the idea that when you decrease the volume of a gas, the molecules will be closer together and therefore experience stronger attractive forces. This leads to a decrease in pressure, as the molecules are not able to exert as much force on the container walls.

However, the decrease in pressure is not proportional to the decrease in volume. This is because the attractive forces are not evenly distributed throughout the volume, but are strongest near the surface of the molecules. As a result, the correction term "a/V^2" is added to the pressure term to account for this non-linear relationship.

On the other hand, the correction for the finite size of molecules is represented by the term "b". This term is subtracted from the volume term, because as the volume is decreased, the actual volume occupied by the molecules becomes a larger percentage of the total volume. Therefore, the correction is subtracted from the volume to account for the space that is already occupied by the molecules.

In summary, the seemingly counterintuitive addition of "a/V^2" to the pressure term and subtraction of "b" from the volume term in the Van der Waals equation can be explained by considering the non-linear relationship between attractive forces and volume, as well as the finite size of molecules. These corrections allow for a more accurate prediction of gas behavior at high pressures and low temperatures, where the ideal gas law breaks down.