Page 225 - Essentials of physical chemistry

Basic Spectroscopy 187

very clever but it is one of those treatments where you can do correct algebra without getting the
right answer unless you take a certain path.
Here, we come to another problem in that we will be dealing with electrical units, a field that has
developed over time in various laboratories and over several hundred years resulting in as many as
five different systems [3]. The modern SI units have tried to unify this situation but at the cost of
introducing a new annoyance in the form of a factor of 4pe 0 that pops up all over the equations. For
the simplicity of explaining this derivation, we will use a system of units that goes forward with the
energy in units of ‘‘electron volts’’ that is used by most physicists and nuclear physicists. Let us set
out the Bohr hypotheses and develop the formula before we analyze the meaning of the results. Bohr
assumed that the electron in the H atom moved in a (flat) circular orbital around a positive ion (Ze )
þ
in one of the ‘‘allowed orbitals’’ determined by the momentum quantization. For the H atom, Z ¼ 1,
but the derivation can be applied to He ion as Z ¼ 2, Li 2þ ion as Z ¼ 3, Ne 9þ as Z ¼ 10, or even
þ
U 91þ as Z ¼ 92; for any system with just one electron in orbit around a positive nucleus. The model
does not apply to more than one electron. The electrical interactions are a result of the e charge on

the electron and a positive charge of Ze on the positive nucleus. Note that at the time Bohr worked
þ
on his theory, the Rydberg formula cited above was well known to scientists as providing a very
accurate fit to the measured spectral lines of atoms or ions with only one electron. There is a lesson
here in how theorists function in using some fragmentary experimental evidence to check a
pencil-and-paper theory but we must admit that no one else had an explanation of why=how the
Rydberg formula worked or the physical principles behind it.

h
1. mvr ¼ n ¼ n , for n ¼ 1, 2, 3, . . ., that is, quantize the angular momentum of the
h
2p
h
electron (mvr ¼ n ).
mv 2 Ze 2
2. ¼ , this balances the centripetal force of the electron with the electrostatic attrac-
r r 2
tion. In fact, the electrostatic attraction of the electron by the positive ion continually pulls
(accelerates) the path of the electron into a curve just as a rock on a string is pulled into a
circular path.
The key step occurs right here, in that Bohr solved for the velocity in terms of the velocity
instead of taking the square root to find ‘‘v.’’ Thus, he used an unusual algebra step so that
he could insert the quantization of the angular momentum:

Ze 2 Ze 2
:
h
mvr n
v ¼ ¼
mv 2 Ze 2
, the total energy is the sum of kinetic (T) and potential (V)
2 r
3. E tot ¼ T þ V ¼
parts.
4. From 2, we see
2 2 4 2

2
2
Ze Ze mZ e Ze
2
r n n r
h
mv ¼ ¼ m ¼ 2 2 ¼
h
so
2 2
2
n
n h 2
h
¼ r(n, Z):
mZe Z me
r ¼ 2 ¼ 2
2
2 2 2 2 4 4
m Ze Ze mZ e 1 Z me
¼ E(n, Z)

h
5. E tot ¼ ¼ 2 2 1 ¼ 2
h
h
2 n n 2 h 2 n 2 n 2
Z me 2