Physical Chemistry Third Edition

754 17 The Electronic States of Atoms. I. The Hydrogen Atom

to the motion of a classical particle orbiting the nucleus with positive values ofLz.

Since a constant can be added to the energy without effect, as discussed in Chapter 15,

we can add a constant toEand make the frequency of oscillation equal to any value

whatsoever, as mentioned in Example 15.6. Only the differences in the frequencies of

oscillation have meaning. It appears that a positive value ofEis required to make the

nodes of the de Broglie wave move around the nucleus in the “correct” direction.

EXAMPLE17.8

a.Calculate the frequency of oscillation of the 1sorbital of a hydrogen atom. Take the zero

of energy as the value of〈V〉for the 1sstate, equal to 2E 1 s(an arbitrary choice).

b.Calculate the frequency of oscillation of the 211 orbital of a hydrogen atom, using this

zero of energy.

c.Calculate the frequency of a photon given off if a hydrogen atom makes a transition from

then2tothen1 energy level of a hydrogen atom. Compare this frequency with the

frequency of the 211 orbital oscillation, with the frequency of oscillation of the 1sorbital,

and the difference between these two frequencies.

Solution

a.Relative to this zero of energy, the energy of the 1sstate is−E 1 , equal to 2. 179 × 10 −^18 J.

The frequency is given by Eq. (15.2-23)

v

E

h



2. 179 × 10 −^18 J

6. 6261 × 10 −^34 Js

 3. 289 × 10 −^15 s−^1

b.Relative to this zero of energy, the energy of this state is

E 2 −

2. 179 × 10 −^18 J

22

−(−2(2. 179 × 10 −^18 J)) 3. 183 × 10 −^18 J

v 2 

3. 183 × 10 −^18 J

6. 6261 × 10 −^34 Js

 5. 755 × 1015 s−^1

c. ν(photon)E^2 −E^1

h



3. 183 × 10 −^18 J− 2. 179 × 10 −^18 J

6. 6261 × 10 −^34 Js

 2. 466 × 1015 s−^1

This frequency does not match eitherν 1 orν 2 , but it does matchν 1 −ν 2.

ν 1 −ν 2  5. 755 × 1015 s−^1 − 3. 289 × 1015 s−^1  2. 466 × 1015 s−^1

The values of the frequencies of oscillation of the wave functions do not have any physical

meaning, but the differences in the frequencies do have meaning.

From this example we can deduce the same thing that we deduced in Example 15.6.

Only the difference in de Broglie frequencies is meaningful.

If we visualize an electron orbiting around thezaxis like a classical particle, it is

difficult to picture the motion of an electron in an orbital withm0, since there are

no nodal planes andLzhas the predictable value of zero. We can only visualize the

wave pulsating without nodes, and try to visualize the electron moving out from the