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Bohr's equation for the hydrogen atom

Bohr derived an equation to give the values of the energy levels in the hydrogen atom.
Bohr showed that the energy levels were proportional to 1/n2, where n is an integer.
When an electron falls from one energy level E2 to another level E1 radiation of frequency f will be emitted. Bohr showed that the energy difference (E2 - E1 = hf) was given by the equation:

E2 - E1 = hf = me4/[εoch3](1/n12 – 1/n22)

Where n1 and n2 are integers (1, 2, 3. . .) and the other constants have their usual meanings. The equation can be expressed in the form of wave number (1/λ) as:

1/λ = hf = me4/[εoch3](1/n12 – 1/n22)

The term me4/[8εoch3] is known as Rydberg's constant (R). R = 1.097 x 107 m-1.

Putting n1 = 1,2 or 3 will give us three series of energy changes and therefore three series of wavelengths for radiation emitted from a hydrogen atom:
n1 = 1 gives the Lyman series (ultraviolet)
n1 = 2 gives the Balmer series (visible)
n1 = 3 gives the Paschen series (infrared).

Quantum numbers

The full treatment of the electron energy in terms of quantum numbers is not predicted by the simple Bohr model.

Every electron within an atom is described by four numbers called quantum numbers:
(a) the principal quantum number (n), representing the energy level
(b) the orbital quantum number (L), which may have any integral value between 0 and n -1
(c) the magnetic quantum number (m), which may have any integral value between - L and + L
(NB it is normal to write the orbital quantum number as l but we are using L to make it clearly different from 1.)
(d) the spin quantum number (s), which may have values of + ½ or – ½ .

Related to these quantum numbers is the Pauli exclusion principle which states that no two electrons in an atom may exist in the same quantum state. That is no two electrons in an atom may have quantum numbers with the same value.

Pauli exclusion principle:   No two electrons in an atom may exist in the same quantum state

This important statement may be used to predict the numbers of electrons in the shells of an atom.

Let’s think about the K-shell and L-shell.
(a) In the K-shell, n = 1. The only possible values for L and m are 0, s can be + ½ or – ½, and so only two electrons can exist in this shell.

(b) In the L-shell, n = 2. In this shell eight electrons are possible, as shown by the following table:

n L m s   n L m s
2 0 0 + ½   2 0 0 – ½
2 1 -1 + ½   2 1 -1 – ½
2 1 0 + ½   2 1 0 – ½
2 1 +1 + ½   2 1 +1 – ½

One result of this principle is that you can never squeeze two particles together to such an extent that they occupy the same state - objects must have a finite volume! It also means that if the exclusion principle did not apply then all electrons in an atom would end up in the lowest possible energy state. Chemistry would be changed forever and the world as we know it would not exist.

© Keith Gibbs 2020