Biomedical Engineering Reference
In-Depth Information
the circumference has to be an integral number
n
of de Broglie wavelengths, that is
to say
n
h
p
=
2π
r
(13.5)
=
where
p
is the momentum (
μ
v
). Simultaneous solution of Equations (13.3) and (13.4)
gives allowed values for the radius and the speed
ε
0
h
2
n
2
πμ
Ze
2
r
n
=
(13.6)
Ze
2
2ε
0
hn
v
n
=
and also quantized energies given by
Ze
2
4πε
0
r
n
+
1
2
μ
v
n
ε
n
=−
(13.7)
μ
Z
2
e
4
8
h
2
ε
0
1
n
2
=−
Bohr referred to these orbits as
stationary states
because according to his theory the electron
did not radiate when it was in any one such state. When the electron passed from one state
(the initial state) to another (the final state), radiation was emitted or absorbed according
to the Planck-Einstein formula
hc
0
λ
|
ε
f
−
ε
i
| =
(13.8)
Bohr's theory predicted the existence of many series of spectral lines for the hydrogen atom
(for which
Z
=
1) whose wavelengths fitted the generic formula
R
H
1
1
λ
if
=
1
n
i
n
f
−
=
1, 2, 3, ...
n
i
>
n
f
The Balmer series corresponds to
n
f
n
f
3 had already
been observed in the infrared region of the electromagnetic spectrum by Paschen. Soon
after Bohr published his theory, Lyman identified the series with
n
f
=
2, and the series corresponding to
n
f
=
=
1 in the ultraviolet.
The
n
f
5 (Pfund) series were also identified. Bohr's theory also
predicted a first ionization energy (given by
R
H
hc
0
, approximately 2.16806
=
4 (Brackett) and
n
f
=
10
−
18
Jor
13.5 eV) that agreed well with experiment, and a physically reasonable value (52.9 pm) for
the radius of the lowest energy orbit.
Many series with high values of
n
have since been observed. For example, radio astro-
nomers are very interested in the
n
i
=
×
167 to
n
f
=
166 emissionwhich occurs at awavelength
of 21.04 cm.
Bohr's theory gives the following expression for
R
H
, in agreement with experiment:
μ
e
4
8ε
0
h
3
c
0
R
H
=
