Frequency and Energy
These two formulas look quite unrelated. However, in the following proof, I'll show that Rutherford's equation becomes a good approximation to Bohr's at high energy levels. Based on Balmer's formula,
Bohr assumed that the angular momentum L of an electron in a hydrogen atom had to satisfy the condition
where n is a positive integer. If the electron is in a circular orbit
around the nucleus at the nth energy level, then its angular momentum
is also described by the equation
where m is the electron's mass and v and r are its speed
and orbital radius at this energy level. Therefore,
To find another relationship between r and v, we can apply
Newton's second law, F= ma, and the coulomb force, to the
electronthis has been done elsewhere.
Using Bohr's formula for the angular momentum, it turns out that
Then, plugging into equation (3),
Now the total energy of an electron at a particular n can be calculated:
With the values obtained for r and v in (5)
and (6), this can be simplified to
(B is related to the Rydberg constant, R.) Given this value for E, it is clear that
the difference between energy levels n+1 and n
is
For large values of n, this energy difference approaches
Now consider the orbital frequency of the electron. Its angular frequency at level n must be
or, plugging in the values for r and v from equations (5) and (6),
Using the value of A defined in equation (5),
This bears a striking resemblance to the energy difference derived in equation
(10). And remarkably,
for large values of n. What's remarkable about it? Well, recall that
for large n, which makes Rutherford's and
Bohr's formulas approximately equivalent. Therefore, the frequency of the
emitted radiation approaches the electron's orbital frequency as n
increases.

