The equation 1/wavelength = Rydberg's constant(1/(n final)^2-1/(n initial)^2
was first developed in another form by Balmer as an empirical equation to describe the observation of the visible spectrum of hydrogen produced and observable when hydrogen gas was excited by the addition of energy. The equation in the original form was modified by Rhydberg until it worked and could be applied to explain the spectrum of hydrogen by using integer values of n, only as suggested by Bohr in his postulates.
Bohr's postulates were:
Electrons can revolve around the nucleus in certain metastable orbits without radiating energy or falling toward the nucleus despite having opposite charges to the nucleus.
When an electron moves to a lower energy metastable state or orbital it emits energy in the form of electromagnetic radiation given by the relationship E = hf . If an electron moves to a higher metastable energy state it must gain a quantity of electromagnetic energy also given be the equation E = hf . That is the electrons movement from a specified higher energy state to a lower energy state always results in the emission of electromagnetic radiation of specific frequency being emitted. An electron moving to a higher specific energy level can only do so if the electron is able to absorb electromagnetic energy of a specific threshold frequency or a higher frequency.
An electron in a metastable orbit has an angular momentum that is an integer multiple of
******A basic derivation follows of rydberg's equation from the postulates follows******
In this classical model, the electron energy E is kinetic plus electric potential
E = (1/2)mv2 - ke2/r
where m is the electron mass, v its (tangential) velocity, k is Coulomb's constant, and r the 'orbit' radius. But the centripital force F is provided by the electrostatic attraction
mv2/r = F = ke2/r2 (1)
so substitution gives the classical result (exactly analogous to planetary mechanics):
E = - (1/2)ke2/r. (2)
Now we introduce de Broglie's contribution. In the 19th century, classical electromagnetism (Maxwell's equations) gave the momentum p of light as
p = E/c.
Using Einstein's quantisation of energy E = hf, we get the momentum of a photon
p = h/l
where l is the wavelength of the light. de Broglie speculated that this formula could hold for an electron also. Now the electrons in the H atom have sufficiently low energy that we may neglect relativistic effects, so de Broglie's speculation gives us for the electron:
mv = h/l.
Now if the electron wave is to give constructive interference in a circular orbit, one requires that an integral number n of wavelengths make up a circumference, so
2.p.r = nl = nh/mv, whence
v = nh/(2.p.r.m) (3)
Let's now solve (1) and (3) for r, and substitute in (2) to get the energy of the electron. Combining (1) and (3) gives
1/r = (4.p2.k2e2m)/(n2h2)
and substituting in (2) gives
E = - (2.p2.k2e4m/h2)(1/n2)
All the messy first factor is now seen to be the empirical constant. So the energy E of an electron in an orbit that has n waves around the circumference is
E(n) = -constant/n2
Now in this model, orbits with non-integral values of n are forbidden, because they correspond to destructive interference of electrons. So energy can only be absorbed or given to photons whose energy is E(n) - E(N) where n and N are both integers. So we can write the wavelengths l of the emitted or absorbed spectrum as
1/l = DE/hc = - R(1/n2 - 1/N2)
note: Above derivation does not need to be known
