Page 229 - Essentials of physical chemistry

Basic Spectroscopy 191

This unit may have come about due to the way some early experiments were done to measure
wavelengths and was important in the derivation of Rydberg’s formula but it is now established in
infrared spectroscopy. In particular, the combined Balmer–Rydberg formula in wave numbers is

1 1 1
n H ¼ 109,737:31568525 ; n ¼ 3, 4, 5, ...:
cm 2 2 n 2
We show the modern value that was initially known only to about six significant figures. However,
through the work of Johannes Rydberg (1854–1919), a Swedish physicist, this number is probably
the most refined constant in spectroscopy. Since H is a common element in space and much of early
spectroscopic data came from astronomy measurements, the Rydberg became a unit of energy. In
1913, Bohr compared his theoretical equation to experiment using the value of the Rydberg:

me 4
E(n, Z) ¼ 2 2
2n
h
for Z ¼ 1, so,

me 4 1 1 hc
2n 2 n l
E 2 E n ¼ 2 2 2 2 ¼ hn ¼
h
and then

(E 2 E n ) me 4 1 1 1
¼ n:
2 2
¼

¼
hc 2n (hc) 2 2 n 2 l
h
We can see that the Bohr’s factor of constants should agree with the experimental value of the
Rydberg. We have already noted there needs to be a small correction to the value of ‘‘m’’ since a tiny
center-of-mass correction should be applied (the electron and proton actually rotate about their
mutual center of mass which is very close to the position of the more massive proton).
2
2
4
me 4 me (4p ) (4p )(9:1093826 10 28 g)(4:803204 10 10 g 1=2 cm 3=2 =s) 4
¼ R H
3
2 2
3
10
¼
¼
2n (hc) 2h c 2(6:6260693 10 27 erg s) (2:99792458 10 cm=s)
h

1
Thus, we calculate R H ¼ 109,737:2794 , and we see that using modern values of the
cm
constants without the center-of-mass correction to ‘‘m’’ we get six figure agreement with the
experimental value. This was a major triumph for Bohr and the amazing agreement with the rydberg
constant made believers in the idea that angular momentum is quantized.
Following the triumph of the Bohr theory, let us consider some limitations of the Bohr model of
the atom. First, there are only flat circular orbitals and you have probably seen orbitals in organic
chemistry textbooks that have different 3D shapes due to further research since 1913. However,
because the DE (eV) ¼ E 2 E 1 values are correct, at least for (n ! n þ 1) transitions, the Bohr
model was a breakthrough in understanding the energy levels of atoms. Note also that these flat
orbitals do not give insight as to how atoms combine into molecules.
A second philosophical dilemma with the model is that the electrons in orbitals have angular
momentum, so they must be moving in some sense of the word and in fact we calculated the formula
for their velocity. However, in the theory of radio transmitters, the radiated signal is caused by
sending electrons moving back and forth in some sort of antenna. Thus, it is known that moving