The Lattice Model Contact Energy

The Lattice Model Contact Energy iv Approximates Intermolecular Interactions 

The lattice model quantity iv (for example, in Equations (19.6) and (16.4)) is u r ), the pair potential evaluated at the equilibrium bond septu ation distance r = r. In the lattice model the sum over all pairs in Equation (24.17) is treated as a sum over only nearest neighbors, 

Let s see what fraction of the total energy comes from the second shell and beyond. On the lattice, you can sum over shells numbered by integers r = 1,2, 5...You can look up these sums in math tables [81  

The total energy due to all neighbors beyond the first neighbor shell is only about 7.6% of the total. This is the size of the error that we make in neglecting all except nearest-neighbor interactions. 

If you had substituted any attractive short-ranged potential (one that falls off more rapidly than r ) into Equation (24.17), it would have led to the pressure p oc y- , just as in Equation (24.22). For this reason, many different types of interaction are collectively called van der Waals forces. They all lead to the same contribution to p(V). For longer-ranged forces, such as coulombic interactions, the integral in Equation (24.17) becomes inhnite, and Equation (24.20) w ould not apply. For long-range interactions, you need to resort to the methods of Chapters 20 to 23, such as the Poisson equation. 

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