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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 ?

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So mathematically speaking, if we look at the equation for the energy levels of the hydrogen atom below,

$E=- \frac{ \mu e^4}{8 \epsilon^2_0 h^2 n^2}$

where n is the principal quantum number, $\mu$ is the reduced mass, and the rest are constants, we see that as the principle quantum number increases to infinity, the energy eigenvalues approach zero and the spacing decreases (1).

This arises from the electrostatic forces of attraction between the nucleus and electron. By consulting Coulomb’s law,

$F = \frac{k q_1 q_2}{r^2} $

we see that the magnitude of the electrostatic force is proportional to the square of the distance.

Therefore, as the energy level is closer to the nucleus, the spacing between each quantized level, gets greater because of the $r^2$ term.

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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.

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The effect is a consequence of the Coulomb law for the interaction of the proton and electron; potential energy $\displaystyle V(r)=-\frac{e^2}{4\pi\epsilon_o}\frac{1}{ r}$ producing (quantised) energy $\displaystyle E(n)=-hcR\frac{1}{n^2}$, where $n=1,2\cdots$. (That the energy does not depend on angular momentum quantum number $l$ is an accidental effect of the Coulomb potential). If another potential, say linear, quadratic etc. were to be used a different dependence on energy would be obtained.

The lowest energy is $-hcR$, but why is it not zero? The answer is that the uncertainty principle restricts this because as the energy gets lower the electron is confined into a smaller space, i.e. is closer to the proton, this means that the kinetic energy increases and so does the uncertainty in the momentum. So the energy is a compromise between the kinetic and potential energy and this applies to all levels. The existence of atoms would then seem to be a consequence of the uncertainty principle.

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