Page 287 - Academic Press Encyclopedia of Physical Science and Technology 3rd Analytical Chemistry

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Encyclopedia of Physical Science and Technology EN009N-447 July 19, 2001 23:3

804 Microwave Molecular Spectroscopy

state v. This fractional population is proportional to the III. EVALUATION OF THE
Boltzmann factor e −E/kT , where E is the sum of the ro- MOMENTS OF INERTIA
tational energy E J,τ and the vibrational energy E v .In
general, lowering the temperature increases the ground The first step in the study of a rotational spectrum is to
vibrational state population and hence α 0 , while increas- evaluate the moments of inertia, or rotational constants,
ing the temperature increases α 0 for transitions in ex- from which the rigid rotor spectrum (discussed in the fol-
cited vibrational states. Also, large molecules tend to have lowing section) can be predicted. The rotational problem
weaker spectra since F J,τ is decreased. Typical values of is treated mathematically in terms of a molecule-fixed axis
−5
α 0 in the centimenter-wave region are 10 –10 −7 cm −1 system with its origin at the center of mass of the molecule
or smaller. A weak line with a peak absorption coeffi- and its axes oriented along the principal inertial axes. With
cient as small as 10 −9 cm −1 can, however, be detected respect to these axes the moments of inertia are constant,
with a good Stark-modulated spectrometer. To be spe- and the intertia matrix is diagonal. The principal axes of
32
13
cific, the isotopic species 18 O C S in natural abun- inertia are designated by a, b, and c. The corresponding
dance has an α 0 = 3 × 10 −9 cm −1 and, hence, should moments of inertia are denoted by I a , I b , and I c , where, by
be observable with a signal-to-noise ratio of better than convention, the inertial axes are labeled so that I a I b I c .
3:1. In terms of the coordinates of the atoms in the principal
Space does not permit a discussion of the effects of nu- axis system, the principal moments of inertia are defined
clear spin on the intensity of rotational lines; however, a by
few comments are in order. For molecules with symme- 2 2
I a = m i b + c ,
i
i
try, identical nuclei are effectively interchanged as a re-
sult of the rotational motion. Since the total wavefunction I b = m i a + c , (5)
2

2

i
i
must possess certain symmetry with respect to exchange

2 2
of identical nuclei, this results in the rotational energy I c = m i a + b ,
i
i
levels having different nuclear statistical weights. This, in
where a i , b i , and c i are the coordinates of the ith atom of
turn, affects the relative intensities of the rotational tran-
massm i , andthe sumis overall atomsof themolecule.The
sitions. Such effects are often used to confirm a particular
principal axis system is also characterized by the auxiliary
molecular symmetry for large molecules. In the case of
relations: the center-of-mass or first-moment equation
SO 2 (e.g., where the nuclear spin I of oxygen is zero) half

of the allowed transitions are missing because of nuclear m i a i = m i b i = m i c i = 0, (6)
spin effects.
which insures that the origin is at the center of mass, and
The above expression for α 0 is the basis for applica-
the product of inertia relations
tions of microwave spectroscopy to chemical analysis. In

qualitative analysis, the frequencies of the rotational ab- I ab =− m i a i b i = 0,
sorption lines provide the basis for identification of the
I ac =− m i a i c i = 0, (7)
compound. In quantitative analysis, the intensities of the

absorption lines provide the basis for determining concen- I bc =− m i b i c i = 0,
tration of the compound. Briefly, the term in brackets in which insures that the inertia matrix is diagonal. In addi-
Eq. (1) is constant if one compares the intensity for the tion to the moment-of-inertia relations, the above equa-
same transition at the same temperature in both a stan- tions are useful in the evaluation of molecular structures.
dard and unknown sample. Therefore, one can write for The rotational energies, as we shall see, depend on the
the ratio of the mole fractions rotational constants designated by A, B, and C, with
A > B > C, and defined by
x 1 /x 2 = (α 0 ) 1 /(α 0 ) 2 · ( ν/p 1 )/( ν/p 2 ), (4)
h h h
A = , B = , C = . (8)
2 2 2
where ν/p is measured for each sample. The term 8π I a 8π I b 8π I c
(α 0 ) 1 /(α 0 ) 2 is given by the ratio of the peak line heights This definition gives the rotational constants in frequency
from a Stark spectrometer when the microwave power units, and the relation between A and I a is
level is held constant. The latter may be accomplished by
˚ 2
A (MHz) = 505,376/I a (amu A ), (9)
keeping the crystal current constant and avoiding power
saturation. From the known mole fraction of the refer- with similar expressions for B and C. This conversion
12
ence sample it is possible by making the above mea- factor is based on the C mass scale.
surements to evaluate the mole fraction of the unknown Thedifferenttypesofrotorsstudiedbymicrowavespec-
sample. troscopists may be classified according to the values of the