Environmental Engineering Reference
In-Depth Information
k
c
¼
(4
pe
o
)
1
¼
9
10
9
Nm
2
/C
2
, must match
m
e
v
2
/
r
, which is the mass of the
electron,
m
e
¼
9.1
10
31
kg, times the required acceleration to the center,
v
2
/
r
.
The total kinetic energy of the motion,
E¼mv
2
/2
k
c
Ze
2
/
r
, adds up to
k
c
Ze
2
/2
r
.
This is true because the kinetic energy is always
0.5 times the (negative) potential
energy in a circular orbit, as can be deduced from the mentioned force balance.
There is thus a crucial relation between the total energy of the electron in the orbit,
E
, and the radius of the orbit,
r
:
E ¼k
C
Ze
2
=
2
r
:
ð
3
:
1
Þ
This classical relation predicts collapse (of atoms, of all matter): for small
r
the energy is increasingly favorable (negative). So the classical electronwould spiral
in toward
r ¼
0, giving off energy in the form of electromagnetic radiation.
Fortunately, the positive value of Plancks constant,
h
, keeps this collapse from
happening.
Bohr imposed an arbitrary quantum condition to stabilize his model of the atom.
Bohrs postulate of 1913 was of the quantization of the angular momentum
L
of the
electron of mass
m
circling the nucleus, in an orbit of radius
r
and speed
v
,asa
multiple of Plancks constant
h¼
6.6
10
34
J s, divided by 2
p
:
L ¼ mvr ¼ n
h
¼ nh=
2
p:
ð
3
:
2
Þ
Here,
n
is the arbitrary integer quantum number
n¼
1, 2
...
. Note that Plancks
constant, already described in Chapter 1, has the correct units, J s, for angular
momentum. This additional constraint leads exactly to basic properties of electrons
in hydrogen and similar one-electron atoms:
E
n
¼k
c
Ze
2
r
n
¼ n
2
a
o
=
2
mk
c
e
2
=
2
r
n
;
Z
;
where
a
o
¼
h
=
¼
0
:
053 nm
:
ð
3
:
3
Þ
The energy of the electron in the
n
th orbit can thus be given as
E
o
Z
2
/
n
2
,
n¼
1,
2
...
, where
E
o
¼ m
r
k
c
e
4
2
=
2
¼
13
:
6eV
:
ð
3
:
4
Þ
h
All of the previously puzzling spectroscopic observations of sharply de
ned light
emissions and absorptions of one-electron atoms were nicely predicted by the simple
quantum condition
n
1
1
n
2
Þ:
hn ¼ hc
=l ¼ E
o
ð
1
=
=
ð
3
:
5
Þ
The energy of the light is exactly the difference of the energy of two electron states,
n
1
and
n
2
in the atom. This was a breakthrough in the understanding of atoms, and
explained sharp absorption lines as are seen in Figure 1.2. For example, the hydrogen
n¼
3
-
n¼
2 transition emits red light at 656 nm; this is called the Balmer line, present
in light from the sun. In evaluating the wavelength in Equation 3.5 it is convenient to
note that
hc ¼
1240 eVnm, since
hc ¼
6.6
10
34
Js
3.0
10
8
m/s (1/1.6
10
19
J/eV)
¼
1.2375
10
6
eV m
1240 eVnm (Figure 3.1).
