Force and potential energy for two adjacent molecules

In this section we will look at just two adjacent molecules in a solid and consider the force between them and their potential energies compared with each other. We will call the separation of the two molecules at the equilibrium position where the net force between the two molecules is zero r_o. This point is also the one of minimum potential energy for the system. Therefore r_o would be the separation of the molecules when no external forces are acting in the solid – i.e. it is not being squashed or stretched.

In a solid all the molecules exert a force on each other and these forces can give an explanation of some of the elastic and thermal properties of a material.

Figure 1 shows how the potential energy of two molecules and the force between them changes with their separation.

The force at any point is found from taking the gradient of he potential energy curve, in other words F = -dV/dr, where V is the potential energy.

Two forces act between the molecules:
(a) the repulsive force which predominates at short distances
(b) the attractive force which predominates at long distances
You can see from the graph that when the molecules are close to each other the repulsive force predominates, while at greater distances the attractive force is larger. The resultant force Is:
(a) repulsive from 0 to M
(b) attractive from M to B but increasing with distance, and
(c) attractive from B to infinity but decreasing with distance.
There is a position where the two forces balance, shown by M on the graph. This is the equilibrium position for molecules in the solid.

The potential energy is a minimum at this point (as would be expected). Any disturbance from this position would produce a force tending to return the molecules to M. The force of attraction between the molecules increases as the molecules arc separated from M to B. The breaking point is at B, since beyond this point the force of attraction decreases with increasing separation.

For a molecule to be completely separated from its neighbour it must gain an amount of energy F, represented by CM on the diagram. The latent heat of vaporisation for the two molecules is CM when there is no residual attractive force. This length also represents the latent heat of vaporisation for the whole material.

In a solid the distance OM is some 2-3 x 10^-10 m and you can see that around this point the force between the molecules varies approximately linearly with distance.

The curves also explain the expansion of a solid with increasing temperature. If an amount of energy F is added to a molecule at C its P.E will rise to the level C', the energy appears as kinetic and potential energy and the molecule oscillates about G. However, since the potential energy - distance curve is not symmetrical this centre of oscillation is further from 0 than from M. This results in a mean separation of the molecules -that is, an expansion.