Physical Chemistry Third Edition

14.4 The Old Quantum Theory 647

wheren 1 andn 2 are two positive integers andRHis a constant known asRydberg’s

constant, equal to 1.09677581× 107 m−^1 if the wavelengths are measured in a vacuum.

Classical physics was unable to explain this relationship.

Johannes Robert Rydberg, 1854–1919,
was professor of physics at the
University of Lund in Sweden, where he
received both his bachelor’s and
doctor’s degrees in mathematics and
where he spent his entire career. He
was originally an instructor in
mathematics but moved into
mathematical physics.

Rutherford’s discovery of the atomic nucleus showed that the negative electrons

must be orbiting around the nucleus. However, an orbiting electron would be accel-

erated and according to the electrodynamics of Maxwell should emit electromagnetic

radiation. It would lose energy and fall onto the nucleus and collapse the atom, emitting

various wavelengths of light as it fell. Classical physics was unable to explain either

the line spectrum of the hydrogen atom or the continuing existence of the atom.

In 1913 Bohr published a theory of the hydrogen atom, based on unproved assump-

tions. We regard his theory as the third part of the “old quantum theory.” A simplified

version of Bohr’s assumptions is:

1. The hydrogen atom consists of a positive nucleus of chargeeand an electron of
   charge−emoving around it in a circular orbit. The chargeehad been determined
   by Millikan to have the value 1.6022× 10 −^19 C.


2. The angular momentum (see Appendix E) of the electron is quantized: Its magnitude
   can take on one of the valuesh/ 2 π,2h/ 2 π,3h/ 2 π,4h/ 2 π,...,nh/ 2 π, wherehis
   Planck’s constant and wherenis an integer (a quantum number). Figure 14.12
   schematically shows the quantization of the angular momentum.


3. Maxwell’s equations do not apply. Radiation is emitted or absorbed only when a
   transition is made from one quantized value of the angular momentum to another.


4. The wavelength of emitted or absorbed light is given by the Planck–Einstein relation,
   Eq. (14.4-8), with the energy of the photon equal to the difference in energy of the
   initial and final states of the atom.


5. In all other regards, classical mechanics is valid.

6

4

2

0

Angular momentum (

h/

2 π

)

Figure 14.12 The Quantized Angu-

lar Momentum Values of Electronic

Motion in a Hydrogen Atom as Pos-

tulated by Bohr.

Niels Henrik David Bohr, 1885–1962,
was a Danish physicist who received
the Nobel Prize in physics in 1922 for
this work. He was also responsible for
much of the accepted physical
interpretation of quantum mechanics
and for the quantum mechanical
explanation of the form of the periodic
table of the elements.

We now derive the consequences of Bohr’s assumptions. The nucleus is much more

massive than the electron and the electron moves about it almost as if the nucleus were

stationary. (See Problem 14.29.) We now proceed as though the nucleus was infinitely

massive and therefore completely stationary. To maintain a circular orbit about the

nucleus, there must be a centripetal force on the electron:

Fr−meν^2 /r (14.4-11)

whereνis the speed of the electron,meis its mass, andris the radius of the orbit

(see Eq. (E-17) of Appendix E).

EXAMPLE14.5

Find the centripetal force on an object of mass 1.50 kg if you swing it on a rope so that the

radius of the orbit is 2.00 m and the time required for one orbit is 2.00 s.

Solution

ν

2 π(2.00 m)

2 .00 s

 6 .28ms−^1

|F|

mev^2

r



(1.50 kg)(6.28ms−^1 )^2

2 .00 m

 29 .6N

This force is about twice as large as the gravitational force on the mass, 14.7 N.