Biomedical Engineering Reference
In-Depth Information
Fig. 2.2
Schematic representation of electron (mass
m
,charge
-
e
) in uniform circular motion (speed
v
, orbital radius
r
)about
nucleus of charge +
Ze
.
electron of mass
m
moving uniformly with speed
v
in a circular orbit of radius
r
(Fig. 2.2), we thus write
(2.3)
mvr
=
n
,
where
n
is a positive integer, called a quantum number
(
n
=
1, 2, 3,
...
)
. [Angular
10
-34
J s (Appen-
dix A)]. If the electron changes from an initial orbit in which its energy is
E
i
to a
final orbit of lower energy
E
f
, then a photon of energy
momentum,
mvr
, is defined in Appendix C; and
=
1.05457
×
h
ν =
E
i
-
E
f
(2.4)
is emitted, where
ν
is the frequency of the photon.
(
E
f
>
E
i
if a photon is absorbed.)
Equations (2.3) and (2.4) are two succinct statements that embody Bohr's ideas
quantitatively. We now use them to derive the properties of single-electron atomic
systems.
When an object moves with constant speed
v
in a circle of radius
r
, it experiences
an acceleration
v
2
/
r
, directed toward the center of the circle. By Newton's second
law, the force on the object is
mv
2
/
r
, also directed toward the center (Problem 10).
The force on the electron in Fig. 2.2 is supplied by the Coulomb attraction between
the electronic and nuclear charges, -
e
and +
Ze
. Therefore, we write for the equation
of motion of the electron,
mv
2
r
k
0
Ze
2
r
2
(2.5)
=
,
where
k
0
=
10
9
Nm
2
C
-2
(Appendix C). Solving for the radius gives
8.98755
×
k
0
Ze
2
mv
2
r
=
.
(2.6)
Solving Eq. (2.3) for
v
and substituting into (2.6), we find for the radii
r
n
of the
allowed orbits
2
k
0
Ze
2
m
.
n
2
r
n
=
(2.7)
