Page 223 - Essentials of physical chemistry

Basic Spectroscopy 185

small integers; ponder that for a moment, integers. There are some excellent demonstrations of the
use of prisms to separate the wavelengths of light and examples of the H lines on the Internet (http:==
csep10.phys.utk.edu=astr162=lect=light=absorption.html) where you can see the colors of the lines as
well as the fact that the lines occur at specific wavelengths. At that time, the most accurate
wavelengths for the H spectrum were available from A. J. Angström (1814–1874), a Swedish
physicist for whom the wavelength unit is named (1.0 10 10 m ¼ 1.0 10 8 cm ¼ 1 Å). Balmer
was a mathematician who spent most of his career teaching at a school for girls, but at the age of
sixty in 1885, he succeeded in fitting the wavelengths of the H spectrum to the formula:

hm 2
,
m n
l ¼ 2 2
where he referred to his symbol ‘‘h’’ as the ‘‘hydrogen constant’’ but ‘‘m’’ and ‘‘n’’ as integers
(Balmer’s ‘‘h’’ is not the same as Planck’s ‘‘h’’). Using n ¼ 2 and m ¼ 3, 4, 5, 6, . . ., he fitted the
H spectral lines and even predicted the wavelength of a new line for m ¼ 7, which was later
observed at 397 nm by Angström. Later other series of lines, not in the visible part of the
electromagnetic spectrum, were discovered when n ¼ 3, 4, . . .. While we should appreciate the
intellectual accomplishments of these early scientists and the role of amateur astronomy in
the development of physics and chemistry, we leave further study of such history to interested
students with Ref. [1]. Instead, we have offered this discussion to emphasize the role of integers in
Balmer’s formula.
A significant extension of Balmer’s work occurred in 1888 when the Swedish physicist Johannes
Rydberg (1854–1919) developed a similar formula using the reciprocal of the wavelength:

1 2 1 1
¼ RZ ; n 1 < n 2 :
l n n
n ¼ 2 2
1 2
This formula applies only to atoms=ions with just one electron such as H, He ,Li ,Be , etc.,
3þ
2þ
1þ
where Z is the number of protons in the nucleus. Although the constant ‘‘c’’ has been standardized,
the value of R is the most accurately measured number in physical science with an relative
uncertainty of only 6.6 10 12 in the 90th Edn. of the CRC Handbook. The modern value of the
Rydberg constant ‘‘R’’ is 109737.31568527 cm 1 and early measurements could be made to at least
109737 cm 1 before Bohr derived his formula in 1913. Looking back at Rydberg’s work, it is clear
that the specific value of his constant is dependent on his choice of (1=l) units, and we will see that
this unit is still used in infrared spectroscopy. The use of the reciprocal square of integers is an
extension of Balmer’s formula.
In a later chapter, we will explore the events in physics between 1885 and 1913, which were
tumultuous in the wonder of the discoveries, but we want to focus here on the spectrum of the
H atom. Although the work by Planck in 1901 and the interpretation of the photoelectric effect by
Einstein in 1905 are very important to the overall development of modern spectroscopy, the next
breakthrough for understanding the spectrum of the H atom occurred in 1913 with the theoretical
model of Niels H. Bohr (1885–1962), a Danish physicist who received the Nobel Prize for this
work in 1922 (Figure 9.4). Bohr was a ‘‘pencil-and-paper’’ theorist who made a major discovery
by postulating the quantization of angular momentum. Note that earlier work by Planck in 1901
had postulated that energy exists as small chunks of size e ¼ hn, which was an earth-shaking
concept that few people really believed. However, Planck’s treatment led to a model of the
broad spectrum emitted from a hot body with a fit of his theoretical data points to the experimental
data that was essentially ‘‘exact’’ and thus hard to refute. We do not know how Bohr arrived at the
idea that angular momentum is quantized but we can note that momentum is embedded in
some energy formulas. We can see that if momentum is quantized that will also quantize the