The Rydberg Formula

Johannes Rydberg

Johannes Rydberg

In Sampling the Hydrogen Atom I introduced the idea that the orbital velocity of the electron is close to light speed and is itself affected by relativity. Derivation of the Rydberg formula based on this model is very straightforward and I think rather elegant.

The story starts with Joseph Jakob Balmer (1825-1898) who was a Swiss mathematician and numerologist and who, after his studies in Germany, took up a post teaching mathematics at a girls’ school in Basel.  A colleague in Basel suggested that he take a look at the spectral lines of hydrogen to see if he could find a mathematical relationship between them.  Eventually Balmer did find a common factor h = 3.6456*10-7 [a] which led him to a formula for the wavelength of the various spectral lines.

Equation 1 Equation 1

Where m is an integer with values of 3 or higher

Balmer originally matched his formula for m = 3,4,5,6 and based on this he predicted an absorption line for m = 7.  Balmer’s seventh line was subsequently found to match a new line in the hydrogen spectrum that had been discovered by Ângström.

Balmer’s formula dealt with a particular set of spectral lines in the hydrogen atom and was later found to be a special case of a more general result which was formulated by the Swedish physicist Johannes Rydberg.

Equation 2

Where λ is the wavelength of the spectral line

RH is called the Rydberg constant for hydrogen

n1 and n2 are integers and n1<n2

By setting n1 to 1 and allowing n2 to take on values of 2, 3, 4 … ∞ the lines take on a series of values known as the Lyman series.  Balmer’s series is obtained by setting n1=2 and allowing n2 to take on values of 3, 4, 5… . Similarly for other values of n1 series of spectral lines have been named according to the person who first discovered them and so:

n1                 n2                    Series

1                    2 … ∞              Lyman series
2                    3 … ∞              Balmer series
3                    4 … ∞              Paschen series
4                    5 … ∞              Brackett series
5                    6 … ∞              Pfund series
6                    7 … ∞              Humphrey’s series

Other series beyond these do exist, but they are not named.

By substituting different values for R, it was found that Rydberg’s formula worked for all so called hydrogenic[b] atoms.

The value of RH can be found by considering the case where n1=1 and n2=∞, a condition which represents the maximum possible change in energy level within the hydrogen atom.  RH is then the wavelength of the absorption line associated with such an energy change  and was calculated to have a value of 1.097*107

This was subsequently found to be given by the formula:

Equation 3 Equation 3

Where m0 is the rest mass of the electron and α is the Fine Structure Constant.

In Sampling the Hydrogen Atom I suggest that the orbital velocity of the electron is very close to the speed of light.  In the lowest or base energy state it has an orbital velocity of 99.997331% c, while in the theoretical infinite energy state it has a velocity equal to c.  Throughout all of its various energy states the orbital radius remains the same, constrained by Planck’s constant.  This means that its potential energy remains constant and all the various energy levels then exist as a result of changes in the kinetic energy of the electron.

The kinetic energy of an object of mass m, orbiting at velocity v is given by the equation:

Equation 4 Equation 4
Hence the difference between the energy of the electron in the base energy state and in the maximum energy state, referred to as the energy potential of the hydrogen atom is:
Equation 5
From the equation for Gamma in Special Relativity
Equation 6 Equation 6
We can easily compute an expression for c2-v2
Equation 7 Equation 7
Where v is the velocity in the base energy state and γ corresponds to this value of v.
But in Sampling the Hydrogen Atom I showed that for the base energy state
Equation 8 Equation 8

Hence the maximum energy potential for the hydrogen atom is

Equation 9 Equation 9

For higher energy states the energy potential is the difference between the energy in that state to that of the energy state where n=∞ and v=c.  For such energy states the value of γ is given by

Equation 10 Equation 10

We can therefore calculate the energy potential for any energy level n:

Equation 11 Equation 11

And hence the difference between the energy of any two levels is given by

Equation 12 Equation 12

Where m > n.

It can be seen that this is structurally similar to the Rydberg formula.  The essential difference is that here we are dealing with energy, while Rydberg dealt with wavelength. To convert from energy to wavelength (the reciprocal of frequency) we must divide both sides by 2πcħ 

Equation 13 Equation 13

The Rydberg formula.


[a] h here is not to be confused with Planck’s constant
[b] A hydrogenic atom is one which is ionized such that it has only one orbiting electron.  In theory, at least, any atom can be ionized so as to become hydrogenic.

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