# Potential Energy Function for Two Atoms

The potential energy between two atoms, in a molecule, is given by $$U(x)=\frac{a}{x^{12}} -\frac{b}{x^6}$$ where $$a$$ and $$b$$ are positive constants and $$x$$ is the distance between the atoms. The atom is in stable equilibrium when: (NEET 1995)

Differentiating and equating to zero gives a single solution, which must then be the stable equilbrium: $$\frac{dU}{dx}=0\Rightarrow x=(\frac{2a}{b})^\frac{1}{6}$$

Question: Is the potential energy function here an actual one for quantities like bond length, or just made up for the sake of this question?

• Does this answer your question? Lennard-Jones Potential repulsion by nucleus nucleus or Pauli repulsion? Dec 18, 2023 at 15:28
• I wanted to identify the function (which the comments have done for me), not ask about the properties of that function. In general, I do not think the other questions/answers answer my question. Dec 19, 2023 at 12:12
• I would think the Lennard-Jones potential is an empirical potential but based on solid theoretical grounds. Basically, the $C_{12}/r^{12}$ term describes the short range Pauli repulsion and the $C_6/r^6$ term is a long-range attractive potential that you can associate to dipole-dipole interactions. I think originally was developed to describe the interatomic interactions in gases but its used extended to many fields.
– PAEP
Dec 23, 2023 at 20:22
• Although there are better interatomic potential functions, it gives you a good approximation of the isotropic part of the potential of Van der Walls complexes and in that constext an idea of their stability and bond length. You would find a more nuanced description in the interatomic potencial chpater of a Physical Chemistry textbook (e.g. Atkins, Engel, ...).
– PAEP
Dec 23, 2023 at 20:22

$$U(x)=\frac{a}{x^{12}} -\frac{b}{x^6}$$