Potential energy in a molecule

Potential energy functions can take more complicated forms as well. For example the potential energy associated with the relative location of two atoms in a molecule with respect to one another could look something like Fig. 7-24 in the book. According to Eq. 18 we can actually derive the conservative force from knowledge of the potential energy function:

We see that the force between atoms vanishes at the minimum of the potential energy function so this is the stable equilibrium separation of the atoms. If the molecule is part of a gas then on average the potential energy is raised by over where T is the absolute temperature which is 300 Kelvin at room temperature and Joule/Kelvin. At room temperature the value is

From the graph we can read of the range of atomic separations covered in the thermal vibration of the molecule. We see that

1. Higher temperature gives a larger range of excursions from the equilibrium separation of the molecule.
2. Because the potential is flatter towards higher separation higher temperatures increase the average separation between atoms. This is the origin of the important phenomenon of thermal expansion which we observe best when atoms are part of a solid. Heat up the solid and its dimensions increase. This is because the average atomic separation increases with temperature, which we have traced to the fact that the potential energy increases more rapidly for compression than for expansion.

We introduce potential energy because it converts our work-kinetic energy theorem into a mechanical energy conservation theorem:
 
For one thing it is easier to remember this expression since it takes the form of a conservation law: It simply states that if only conservative forces act then the total mechanical energy which is the sum of the kinetic and potential energy is conserved. But perhaps more importantly, this expression is more readily generalized to handle considerably more complicated situations. In fact it survives in all regimes of physics whether we deal with the very many in thermodynamics (as we just saw in our discussion of thermal expansion), the very small in quantum mechanics, or the very fast in relativistic mechanics.

Whenever we use the consept of potential energy It is important to remember that it is only defined down to an arbitrary additive constant. Another way of stating this is to say that the choice of the configuration which is defined to have zero potential energy is arbitrary. Nonetheless we can only choose once in a given problem since we have to use the same definition on either side of Eq. 26. We now consider a few mechanical problems which are conveniently handled through the gravitational potential energy.