# Distance Dependence of Van Der Waals Forces and his Equation of State

I've been trying to reconcile the form of the van der Waals equation of state and the equations from which they are derived. Stop me where I go wrong:

The VDW EOS includes two terms: one for the consideration of the excluded volume (which results in a higher pressure than predicted by the ideal gas equation) and another for the consideration of attractive forces of the gas (which results in a lower pressure than predicted by the ideal gas equation).

My confusion is mainly with the second term: a/v^2

If I recall correctly, this term is derived from the distance dependence of the potential energy from VDW forces. The explanation being, if v is proportional to r^3, v^2 is proportional to r^6, the potential energy due to attractive forces is proportional to v^2, and thus the reduction in pressure due to those attractive forces should be proportional to v^2.

So here it is: shouldn't the term actually represent the distance dependence of VDW forces? That is, shouldn't the final term of the VDW EOS be proportional to 1/r^7 (the derivative of the VDW force wrt. distance)?

• Your question is answered in the classic paper Hill, Terrell L. (1948). “Derivation of the complete van der Waals’ equation from statistical mechanics.” In: Journal of Chemical Education 25.6, p. 347. You need to understand some statistical physics to make sense of this though. Mainly the partition function and its uses. Do you? – frapadingue Oct 29 '18 at 8:34
• looks like I was just a few days ahead of lecture, thanks though! – Ugochukwu Nwosu Oct 31 '18 at 18:29