Atomic Structure 139
and so the frequency of revolution isf (4.19)
Example 4.5
(a) Find the frequencies of revolution of electrons in n 1 and n2 Bohr orbits. (b) What is
the frequency of the photon emitted when an electron in an n2 orbit drops to an n1 or-
bit? (c) An electron typically spends about 10^8 s in an excited state before it drops to a lower
state by emitting a photon. How many revolutions does an electron in an n2 Bohr orbit make
in 1.00 10 ^8 s?
Solution
(a) From Eq. (4.19),f 1 (2)6.58 1015 rev/s
f 2 0.823 1015 rev/s
(b) From Eq. (4.17),2.88 1015 Hz
This frequency is intermediate between f 1 and f 2.
(c) The number of revolutions the electron makes is
Nf 2 t(8.23 1014 rev/s)(1.00 10 ^8 s)8.23 106 rev
The earth takes 8.23 million y to make this many revolutions around the sun.Under what circumstances should the Bohr atom behave classically? If the electron
orbit is so large that we might be able to measure it directly, quantum effects ought
not to dominate. An orbit 0.01 mm across, for instance, meets this specification. As
we found in Example 4.3, its quantum number is n435.
What does the Bohr theory predict such an atom will radiate? According to Eq.
(4.17), a hydrogen atom dropping from the nith energy level to the nfth energy level
emits a photon whose frequency is
Let us write nfor the initial quantum number niand np(where p1, 2, 3,.. .)
for the final quantum number nf. With this substitution,(^)
When niand nfare both very large, nis much greater than p, and
2 npp^2 2 np
(np)^2 n^2
2 npp^2
n^2 (np)^2
E 1
h
1
n^2
1
(np)^2
E 1
h
1
n^2 i
1
n^2 f
E 1
h
1
23
1
13
2.18 10 ^18 J
6.63 10 ^34 Js
1
n^2 i
1
n^2 f
E 1
h
f 1
8
2
23
E 1
h
2.18 10 ^18 J
6.63 10 ^34 Js
2
13
E 1
h
2
n^3
E 1
h
2
n^3
me^4
8 ^20 h^3
Frequency of
revolution
bei48482_ch04.qxd 1/14/02 12:20 AM Page 139