Interaction energy and the van der Waals

The attractive forces discussed so far are inversely proportional to the sixth power of the distance of separation r of the molecules, so we write Eq. (26.33) in the form 

To solve this problem we fix our attention on a molecule at the center of a spherical container of radius R, having a volume v = 4nR /3. If there are N molecules in the container, then the number per unit volume is N. How many molecules are at a distance between r and r dr from the central molecule The volume of the spherical shell bounded by spheres of radii r and r -h irisdl heii = Tir (ir. The number, diV, of molecules in this shell is dN = N4nr dr. The energy of interaction of these molecules with the one at the center is Ui dN the average interaction energy of all the molecules with the one at the center is 

In any reasonable situation, the radius R of the container is very much larger than a, so that Eq. (26.35) reduces to 

If there are N molecules, the first one of a pair may be chosen in N way s, the second member may be chosen in iV — 1 ways, then the total number of pairs is the product of N(N — 1) since N is very large, this is effectively equal to N. But this enumeration of the number of pairs counts both the pair between molecules a and b and the pair between molecules b and a as being different so we divide by 2 and get The total interaction energy is 

Problem 10.1 required proof that dUldV) = a/F for a van der Waals gas. Comparing this result with Eq. (26.38) shows that the van der Waals constant a is given by 

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