# What makes nuclear binding energy so much stronger than chemical energy

The strong force acting between quarks and responsible for holding protons together is 100 times stronger than the electromagnetic force. How come the nuclear binding energy derived from the strong force is millions time stronger than chemical energy. (rather than 100 times)

• Are you asking what makes a force stronger than another force? Or are you saying, why does the electromagnetic force not pull apart atoms? – jhobbie Jun 22 '14 at 19:36
• I mean why are the nuclear energy so large while the forces involved are "only" 100 times stronger than the electromagnetic force – user51921 Jun 22 '14 at 19:40
• this reminds me of the "conflict" over "vis viva" and momentum (one is quadratic in velocity, other is linear). D'Alembert (contributed in) clarified the issue making clear how each is applied (wikipedia) – Nikos M. Jun 23 '14 at 4:25

In both cases the potential energy for two interacting unit charges is $$U = -\alpha\frac{\hbar c}{r}$$ The strong force is between color charges and has $\alpha \approx 1$, while the electric interaction is between electric charges and has $\alpha \approx 1/137$.

However the gluons, which carry the color force, are themselves charged. This means that the interaction energy between distant color charges is dominated by self-interactions among the gluons going back and forth between them. This gives color confinement, which means that the color force only occurs between color charges that happen to be very close together — $r$ is small, so $U$ is large.

I'm not sure it's not quite right to directly compare the color force between quarks to the electrical interaction between a nucleus and its electrons. The most important degree of freedom inside the nucleus is the pion, which has a Yukawa potential and (I suppose probably) a different coupling constant. However if you note that a nuclear radius is typically a factor of $10^5$ smaller than an atomic radius, and that the $\alpha_\mathrm s$ and $\alpha_\mathrm{em}$ differ by a factor of 100, you get an energy ratio of somewhere around $10^7$. That takes you from sub-eV to few-MeV, which does have the right scale.

Chemical forces are the result of the higher order moments, mainly of the electric field, of the neutral atoms. Look at these orbitals:

The five d orbitals in ψ(x, y, z)2 form, with a combination diagram showing how they fit together to fill space around an atomic nucleus.

These anisotropic shapes in space allow for spill over attractive and repulsive fields that are collectively called Wan der Waals forces. , spill over forces from the combined electric fields of the electrons and nucleus .

Correspondingly the nuclear force between protons and neutrons in the nucleus are spill overs of the strong force as it appears in the "geometry" of each nucleus. As the strong force is color confined , as rob describes in his answer, these forces do not exceed the fermi distances which characterize the size of the nucleons. They still are due to gluon exchanges and these are highly confining.

How come the nuclear binding energy derived from the strong force is millions time stronger than chemical energy. (rather than 100 times)

In this article the energy of intermolecular forces due to the VdWs forces is estimated as 1/r^6 , r distance between molecules. I do not have a functional estimate for the spill over forces of the strong interactions, but it is evident that the inter nucleon distance, of the order fermi, will play a corresponding role. The difference in the dimensions of the atomic orbitals and the nucleus dimensions allows for the large difference in the energies manifested as chemical and nuclear, overcoming factors coming from coupling constants just by "geometry".