Page 197 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

188 The hydrogen atom

Table 6.3. Rydberg constant for
hydrogen-like atoms
ÿ1
Atom R (cm )
1 H 109 677.58
2 H (D) 109 707.42
4 He ‡ 109 722.26
7 Li 2‡ 109 728.72
9 Be 3‡ 109 730.62
1 109 737.31

1 1
~ í ˆ RZ 2 ÿ , n 2 . n 1 (6:83)
n 2 1 n 2 2
where the Rydberg constant R is given by

ìe9 4
R ˆ (6:84)
3
4ð" c
The value of the Rydberg constant varies from one hydrogen-like atom to
another because the reduced mass ì is a factor. It is not appropriate here to
replace the reduced mass ì by the electronic mass m e because the errors
caused by this substitution are larger than the uncertainties in the experimental
1
4
‡
data. The measured values of the Rydberg constants for the atoms H, He ,
7 Li , and Be 3‡ are listed in Table 6.3. Following the custom of the ®eld of
9
2‡
spectroscopy, we express the wave numbers in the unit cm ÿ1 rather than the SI
ÿ1
unit m . Also listed in Table 6.3 is the extrapolated value of R for in®nite
nuclear mass. The calculated values from equation (6.84) are in agreement
with the experimental values within the known number of signi®cant ®gures
for the fundamental constants m e , e9, and " and the nuclear masses m N . The
measured values of R have more signi®cant ®gures than any of the quantities
in equation (6.84) except the speed of light c.
The spectrum of hydrogen (Z ˆ 1) is divided into a number of series of
spectral lines, each series having a particular value for n 1 . As many as six
different series have been observed:
n 1 ˆ 1, Lyman series ultraviolet
n 1 ˆ 2, Balmer series visible
n 1 ˆ 3, Paschen series infrared
n 1 ˆ 4, Brackett series infrared
n 1 ˆ 5, Pfund series far infrared
n 1 ˆ 6, Humphreys series very far infrared