why does the gap between the energy levels of a hydrogen atom keep decreasing as we move further and further away from nucleus? why is the gap between the first two orbits greater than the gap between second and third orbit?
shouldn't the gap keep decreasing as the radius keeps increasing? Can someone explain this ?
Whatever the radius, the allowed energy levels are proportional to $1/n^2$.
The first gap is proportional to $1/1^2 - 1/2^2 = 0.75$.
The second gap is proportional to $1/2^2 - 1/3^2 = 1/4 - 1/9 = 5/36 = 0.139$. This is smaller than $0.75$.
The third gap is proportional to $1/9 - 1/36 = 1/12 = 0.083$. It is still smaller.