Proton-proton collision and virtual photons

TheTourist

a) Two protons, approaching each other with 3keV of energy each, collide head on. Calculate how close they get to each other.

b) Estimate the momentum transferred by one virtual photon at the closest approach.

c) Hence, estimate roughly how many virtual particles are exchanged during the collision.

For parts a) and b) I used Coulombs law for energy I calculated that they get to about 4x10-13m apart, and hence to estimate the momentum for one virtual photon I used The Uncertainty Principle (momentum) to be about 2x10-22kgms-2.
It is with part c) that I am struggling. I used Newtons second law combined with Coulombs law to calculate the change in momentum of the protons, and found (by taking the ratio) that there are 1/137 virtual photons exchanged. Ordinarily I would have shrugged this off immediately as wrong, however this is the fine structure constant (alpha), and it seemed too great a coincidence. The probability of each virtual photon transition is 1/SQRT(137), so for an emission of a photon and absorbance of photon, this would be 1/137, and hence one virtual photon is exchanged! Is this right? Does it even make sense?! The main thing that is making me doubt is the fact that change in momentum for one photon is much greater than change in momentum of the protons.

Thanks
1. The problem statement, all variables and given/known data



2. Relevant equations



3. The attempt at a solution