Hydrogen’s ionization · Estonian PhO 2015 #6
Physics · mechanics
Problem
What is the smallest kinetic energy of a free proton that is able to ionize a hydrogen atom? Assume that the electron inside the hydrogen atom is still and that the electromagnetic interaction between the atomic nucleus and the free proton is negligible. Hydrogen’s binding energy is , proton’s mass is and electron’s mass .Intuitive explanation
This is a threshold collision problem where you need to conserve both energy and momentum. The key insight is that ionization requires enough kinetic energy not just to break the atomic bond (), but also to account for the recoil of the products—energy gets "wasted" accelerating the system as a whole. Work in the center-of-mass frame or use the invariant mass approach: the minimum kinetic energy occurs when all products move together after collision, leaving just enough energy to overcome the binding energy while satisfying momentum conservation.
Intuitive explanation
This is a threshold collision problem where you need to conserve both energy and momentum. The key insight is that ionization requires enough kinetic energy not just to break the atomic bond (), but also to account for the recoil of the products—energy gets "wasted" accelerating the system as a whole. Work in the center-of-mass frame or use the invariant mass approach: the minimum kinetic energy occurs when all products move together after collision, leaving just enough energy to overcome the binding energy while satisfying momentum conservation.