Lennard-Jones Potential and Polar Extension 

The Lennard-Jones equation relates the potential energy of two non-polar molecules. 

It is a mathmatical approximation to observed effects, and as such has different forms: 

The General formula for Lennard-Jones Potentials: 

> restart;
 

> Phi_[lj]:= n*epsilon_[lj]/(n-m)*(n/m)^(m/(n-m))*((sigma_[lj]/r)^n - (sigma_[lj]/r)^m);
 

(Typesetting:-mprintslash)([Phi_[lj] := n*epsilon_[lj]*(n/m)^(m/(n-m))*((sigma_[lj]/r)^n-(sigma_[lj]/r)^m)/(n-m)], [n*epsilon_[lj]*(n/m)^(m/(n-m))*((sigma_[lj]/r)^n-(sigma_[lj]/r)^m)/(n-m)]) 

Shown to work best with (12,6), (9,6), and (28,7) 

> n:= 12; m:= 6;
 

(Typesetting:-mprintslash)([n := 12], [12]) 

(Typesetting:-mprintslash)([m := 6], [6]) 

> Phi_[lj];
 

4*epsilon_[lj]*(sigma_[lj]^12/r^12-sigma_[lj]^6/r^6) 

An extension to the potential energy between two polar molecules can be made by adding a term with r-3 dependence that is based on the  

direction and force of the polar dipole.  This extension is called the "Stockmayer Potential". 

> Phi_[p]:= 4*epsilon*((sigma/r)^12 - (sigma/r)^6 + delta*(sigma/r)^3);
 

(Typesetting:-mprintslash)([Phi_[p] := 4*epsilon*(sigma^12/r^12-sigma^6/r^6+delta*sigma^3/r^3)], [4*epsilon*(sigma^12/r^12-sigma^6/r^6+delta*sigma^3/r^3)]) 

Boltzman's Constant (in SI) 

> k:= 1.38066e-16;
 

(Typesetting:-mprintslash)([k := 0.138066e-15], [0.138066e-15]) 

For HCl: 

sigma = distance between the two molecules (in Angstroms) 

epsilon = lowest parameter energy, which is the maximum attraction between the two molecules when neglecting polarity 

delta = polarity correction term based on dipoles 

> sigma:= 3.36; epsilon:= 328*k; delta:= .34;
 

(Typesetting:-mprintslash)([sigma := 3.36], [3.36]) 

(Typesetting:-mprintslash)([epsilon := 0.45285648e-13], [0.45285648e-13]) 

(Typesetting:-mprintslash)([delta := .34], [.34]) 

> sigma_[lj]:= 3.305; epsilon_[lj]:= 360*k;
 

(Typesetting:-mprintslash)([sigma_[lj] := 3.305], [3.305]) 

(Typesetting:-mprintslash)([epsilon_[lj] := 0.49703760e-13], [0.49703760e-13]) 

> Phi_[p];
 

0.3750523633e-6/r^12-0.2606491076e-9/r^6+0.2336239309e-11/r^3 

Plot of potential energy between two polar molecules with and without the polar correction term.  As can easily be seen, the Stockmayer term 

takes into account the increased repulsion between dipoles of similar orientation.  This is shown at maximum repulsion. 

> plot([Phi_[lj], Phi_[p]], r= .95*sigma_[lj]..3*sigma_[lj], color= [blue,red], legend= ["Lenard-Jones Model","Stockmayer Potential"]);
 

Plot 

When the orientation of the dipole is switched to opposite orientation, the plot shows the added attraction between the molecules. 

When this change in total potential between two molecules, it has the ability to affect the total viscosity of a substance. 

> delta:= -delta;
 

(Typesetting:-mprintslash)([delta := -.34], [-.34]) 

> plot([Phi_[lj], Phi_[p]], r= .95*sigma_[lj]..3*sigma_[lj], color= [blue,red], legend= ["Lenard-Jones Model","Stockmayer Potential"]);
 

Plot 

>