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)?