Biomedical Engineering Reference
In-Depth Information
quantum number
n
i
to a final orbit of lower energy with quantum number
n
f
(i.e.,
n
i
>
n
f
)
, then from Eqs. (2.4) and (2.12) the energy of the emitted photon is
-
1
n
i
,
k
0
Z
2
e
4
m
2
hc
λ
=
1
n
f
h
ν =
+
(2.13)
2
where
λ
is the wavelength of the photon and
c
is the speed of light. Substituting
the numerical values
3)
of the physical constants, one finds from Eq. (2.13) that
1
λ
= 1.09737 × 10
7
Z
2
1
m
-1
.
1
n
i
-
(2.14)
n
f
When
Z
=
1
, the constant in front of the parentheses is equal to the Rydberg con-
stant
R
∞
in Balmer's empirical formula (2.1). The integer 2 in the Balmer formula
is interpretable from Bohr's theory as the quantum number of the orbit into which
the electron falls when it emits the photon. Derivation of the Balmer formula and
calculation of the Rydberg constant from the known values of
e
,
m
,
h
,and
c
pro-
vided undeniable evidence for the validity of Bohr's postulates for single-electron
atomic systems, although the postulates were totally foreign to classical physics.
Figure 2.3 shows a diagram of the energy levels of the hydrogen atom, calculated
from Eq. (2.12), together with vertical lines that indicate the electron transitions
that result in the emission of photons with the wavelengths shown. There are in-
finitely many orbits in which the electron has negative energy (bound states of the
H atom). The orbital energies get closer together near the ionization threshold,
13.6 eV above the ground state. When an H atom becomes ionized, the electron
is not bound and can have any positive energy. In addition to the Balmer series,
Bohr's theory predicts other series, each corresponding to a different final-orbit
quantum number
n
f
and having an infinite number of lines. The set that results
from transitions of electrons to the innermost orbit
(
n
f
= 1
,
n
i
= 2, 3, 4,
...
)
is called
the Lyman series. The least energetic photon in this series has an energy
-13.6
1
1
2
-13.6
-
3
4
1
E
=
2
2
-
=
=
10.2 eV,
(2.15)
as follows from Eqs. (2.4) and (2.12) with
Z
=
1
. Its wavelength is 1216 Å. As
n
i
increases, the Lyman lines get ever closer together, like those in the Balmer series,
converging to the energy limit of 13.6 eV, the ionization potential of H. The photon
wavelength at the Lyman series limit (
n
f
=
1,
n
i
→∞
) is obtained from Eq. (2.14):
10
7
1
1
λ
=
1
∞
10
7
m
-1
,
1.09737
×
1
2
-
=
1.09737
×
(2.16)
2
or
λ =
911
Å. The Lyman series lies entirely in the ultraviolet region of the electro-
magnetic spectrum. The series with
n
f
≥
3
lie in the infrared. The shortest wave-
10
7
)/9
m
-1
,or
length in the Paschen series
(
n
f
=
3)
is given by
1/
λ =
(1.09737
×
10
-7
m
λ =
8.20
×
=
8200
Å.
3
For high accuracy, the
reduced
mass of the
electron must be used. See last paragraph in
this section.
