I think I can see how Bohr's idea of different energy levels sort
of goes with Balmer's formula, but I don't understand how angular
momentum fits in.
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Bohr knew that a photon's energy was equal to Planck's constant
times its frequency (this formula was discovered by Einstein
during his work on the photoelectric
effect). If
the Bohr model was correct, he also knew that an
emitted photon's energy was the same as the
difference between the upper and lower energy
levels involved. So he had a relationship between
the energy levels and the frequencies of the photons...
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But Balmer's formula specified the wavelength, not the frequency.
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Ah, but don't forget that the two are related. The speed of a
wave is equal to the product of its wavelength and its frequency,
as I was telling Kyla earlier. A
photon, or burst of electromagnetic radiation, travels at the
speed of light, c.
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Quite so--and since we know that
from Balmer's formula. Now, we can write the energy levels in terms of the kinetic and potential energy of the electrons:
where m is the electron's mass, and v and r are its speed and orbital radius at the upper and lower levels. |
I'm beginning to see where angular momentum could go into this
equation. If the electron is in a circular orbit,
then
right? |
Absolutely. Thus you can now write everything in
terms of
r and L:
To find out what r is, we can apply Newton's second law, F=ma, to the electron. The force on the electron can be found using Coulomb's Law:
If the electron is in uniform circular motion, its acceleration is
Substituting the value for v you obtained in equation (6) and solving for r, we find that
With everything in terms of L, we get the rather nice equation
which means, from equation (3), that
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The two sides of that equation look really similar. Inside the
parentheses, both sides have 1 over something squared minus 1 over
something else squared, and all that stuff outside
the parentheses on the left is just a constant.
So we should be able to pick some value of
L that would make the two sides be exactly
the same...
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That's just what Bohr did. It seemed logical to assume that
the squared terms on
the right were related to his idea of energy levels. He
associated each energy level with an
integer--called, originally enough, n--with
n=1 corresponding to the ground state (the
lowest possible energy level). Then the 2 and the
n in the Balmer series could represent
electrons falling from the nth level into
the second...
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I get it--so the 656 nm line would be produced by an electron
falling from the third energy level into the second, and so on.
And then the Lyman series would
come
from
electrons falling into the first energy level, and in
the Paschen series they'd be
falling into the third--this makes so
much sense! |  |
Doesn't it? Bohr realized that everything would work out
beautifully if he just assumed that the electron's angular
momentum in the nth level was equal to n times some
constant. To find the constant, all he had to do was find the
value that makes equation (13) true. It
turns
out that the one
that works is
If you plug in the values of all those fundamental constants--the speed of light, the electron's charge and mass, and so on--you will end up with just the value of the Rydberg constant that had been found experimentally. |
