# Lennard-Jones Potential

Proposed by Sir John Edward Lennard-Jones, the Lennard-Jones Potential is a mathematical approximation that illustrates the energy of interaction between two nonbonding atoms or molecules based off their distance of separation. The equation takes into account the difference between attractive forces (dipole-dipole, dipole-induced dipole, and London interactions) and repulsive forces.

### Introduction

Imagine two rubber balls separated by a large distance. Both objects are far enough apart to where they are not interacting. We can bring both balls closer together with minimal energy in order for them to begin interacting. The balls can continuously be brought closer together till they are touching. At this point, it becomes increasingly difficult to further decrease the distance between the two balls. Why? In order to bring the balls any closer together, we would need to add increasing amounts of energy because eventually, as the balls begin to invade each other’s space, they repel - the force of repulsion is far greater than the force of attraction.

This scenario is very much similar to what takes place in neutral atoms and molecules and can be best described by the Lennard-Jones Potential.

### The Lennard-Jones Potential

The Lennard-Jones Potential is given by the following equation:

$$V= 4 \epsilon \left [ {\left (\dfrac{\sigma}{r} \right )}^{12}-{\left (\dfrac{\sigma}{r} \right )}^{6} \right]$$   (Equation 2.1)

where V is the intermolecular potential between the two atoms or molecules. $$\epsilon$$ is the well depth and a measure of how strongly the two particles attract each other.  $$\sigma$$ is the distance at which the intermolecular potential between the two particles is zero (See Figure 1.2).  $$\sigma$$ gives a measurement of how close two nonbonding particles can get and is thus referred to as the van der Waals radius.  It is equal to one-half of the internuclear distance between nonbonding particles. And lastly, r is the distance of separation between both particles (measured from the center of one particle to the center of the other particle).

Equation 2.1 describes both the attractions and repulsions between nonionic particles. The first part of the equation, ($$\sigma$$/r)12 describes the repulsive forces between particles while the latter part of the equation, ($$\sigma$$/r)6 denotes attraction.

#### Bonding Potential

As mentioned earlier, the Lennard-Jones Potential is a function of the distance between the centers of two particles. When two nonbonding particles are an infinite distance apart, the possibility of them coming together and interacting is minimal.  For the sake of simplicity then, we can say that their bonding potential energy is zero. However, as the distance of separation decreases, the probability of interaction increases. The particles come closer together until they reach a region of separation where the two particles become bound and their bonding potential energy decreases from zero to become a negative quantity.  While the particles are bound, the distance between their centers will continue to decrease until the particles reach an equilibrium, which is specified by the separation distance at which the minimum potential energy is reached.

Now, if we keep pushing the two bound particles together passed their equilibrium distance, repulsion begins to occur, as particles are so close to each other that their electrons are forced to occupy each other’s orbitals. Therefore, repulsion occurs as each particle attempts to retain the space in their respective orbitals. Despite the repulsive force between both particles, their bonding potential energy rises rapidly as the distance of separation between them decreases below the equilibrium distance.

#### Stability and Force of Interactions

Very much like the bonding potential energy, the stability of an arrangement of atoms is also a function of the Lennard-Jones separation distance. As the separation distance decreases below equilibrium the potential energy becomes increasingly positive (force is repulsive). Such a large potential energy is energetically unfavorable as it indicates an overlapping of atomic orbitals.  However, at long separation distances, the potential energy is negative and approaches zero as the separation distance increases to infinity (force is attractive).  This indicates that at such long-range distances, the pair of atoms or molecules experiences a small stabilizing force. Lastly, as the separation between the two particles reaches a distance slightly greater than $$\sigma$$, the potential energy reaches a minimum value (force is zero).  At this point, the pair of particles are most stable and will remain in that orientation until an external force is exerted upon them.

### Practice Problems

1. The $$\epsilon$$ and $$\sigma$$ values for Xenon (Xe) were found to be 1.77 kJ/mol and 4.10 Angstroms respectively.  Determine the van der Waals radius for the Xenon atom.
2. Calculate the intermolecular potential between two Argon (Ar) atoms separated by a distance of 4.0 Angstroms (Use $$\epsilon$$=0.997 kJ/mol and $$\sigma$$=3.40 Angstroms).
3. Two molecules, separated by a distance of 3.0 Angstroms were found to have a $$\sigma$$ value of 4.10 Angstroms.  By decreasing the separation distance between both molecules to 2.0 Angstroms, the intermolecular potential between the molecules became more negative.  Do these molecules follow the Lennard-Jones Potential?  Why or why not?
4. The second part of the Lennard-Jones equation is ($$\sigma$$/r)6 and denotes attraction.  Name at least three types of intermolecular interactions that represent attraction.
5. At what separation distance in the Lennard-Jones Potential does a species have a repulsive force acting on it?  An attractive force?  No force?

### Solution

1. Recall from Section 2.1 that the van der Waals radius is equal to one-half of the internuclear distance between nonbonding particles.  Since $$\sigma$$ gives a measure of how close two nonbonding particles can get, the van der Waals raidus for Xenon (Xe) would be given by:

r = $$\sigma$$/2 = 4.10Angstroms/2 = 2.05 Angstroms

2. To solve for the intermolecular potential between the two Argon atoms, we use equation 2.1 where V is the intermolecular potential between two nonbonding particles.

$$V= 4 \epsilon \left [ {\left (\dfrac{\sigma}{r} \right )}^{12}-{\left (\dfrac{\sigma}{r} \right )}^{6} \right]$$

The data given are $$\epsilon$$=0.997 kJ/mol, $$\sigma$$=3.40 Angstroms, and the distance of separation, r=4.0 Angstroms.  We plug these values into equation 2.1 and solve as follows:

V = 4(0.997kJ/mol) [(3.40Angstroms/4.0Angstroms)12-(3.40Angstroms/4.0Angstroms)6]

V = 3.988(0.14-0.38)

V = 3.988(-0.24)

V = -0.96 kJ/mol

3. Recall that $$\sigma$$ is the distance at which the bonding potential between two particles is zero.  On a graph of the Lennard-Jones potential then, we can see that this value gives the x-intersection of the graph.  According to the Lennard-Jones potential, any value of r greater than $$\sigma$$ should yield a negative bonding potential and any value of r smaller than $$\sigma$$ should yield a positive bonding potential.  In this scenario, as the separation between the two molecules decreases from 3.0 Angstroms to 2.0 Angstroms, the bonding potential is becomes more negative.  In essence however, because the starting separation (3.0 Angstroms) is already less than $$\sigma$$ (4.0 Angstroms), decreasing the separation even further (2.0 Angstroms) should result in a more positive bonding potential.  Therefore, these molecules do not follow the Lennard-Jones Potential.

4. From section 2.1, dipole-dipole, dipole-induced dipole, and London interactions are all attractive forces.

5. See Figure C.  A species will have a repulsive force acting on it when r is less than the equilibrium distance between the particles.  A species will have an attractive force acting on it when r is greater than the equilibrium distance between the particles.  Lastly, when r is equal to the equilibrium distance between both particles, the species will have no force acting upon it.

### References

1. Atkins, Peter and de Paula, Julio. Physical Chemistry for the Life Sciences. New York, N.Y. W. H. Freeman Company, 2006. (469-472).
2. Royer, Donald J. Bonding Theory. New York, NY.:McGraw-Hill, Inc., 1968.
3. Chang, Raymond. Physical Chemistry for the  Biosciences. Sausalito, CA. University Science Books, 2005. (498-500)

### Contributor

• Rabia Naeem

15:35, 8 Jan 2014

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