Microwave Spectra and Chemical Analysis B. P. DAILEY, Columbia University, New York, N . Y . of only a small permanent electric dipole moment,
Absorption spectroscopy in the microwave region offers a new and useful tool in chemical analysis and process control. Molecular absorption spectra in the microwave frequency range are usually due to molecular rotation. To aid in understanding these spectra the relevant principles of molecular dynamics are discussed. The microwave spectrograph possesses extreme resolution, permitting actual absorption line shapes to be observed. The use of a single resolved molecular rotation line for quantitative analysis of an organic molecule in a complex mixture is proposed. The limitations of this analytical technique due to low intensities, the presence
excessive chemical reactivity of the absorption cells, etc., have been explored. The advantages and disadvantages of microwave spectroscopy in various applications are contrasted with those of the mass spectrometer and the infrared instruments. Molecular rotational spectra are characteristic of the molecule as a whole and do not possess features, as do vibrational spectra, characteristic of individual groups of atoms within the molecule. The extreme resolution available suggests that microwave techniques might be suitable for isotopic analyses in some instances.
2 cm.-', although for the most part any given investigator has been able to work in only a portion of that band. I t is convenient in discussing spectra in this region to express frequencies in megacycles (1 Mc. = lo6 cycles; 29,978 Mc. = 1 cm.-I). I t is possible with not very elaborate equipment to measure the frequency of a rotational line to 0.01 Mc. (6), which for a line falling a t 30,000 Mc. is better than one part in 10'. Lines separated by no more than 0.1 Mc. clearly have been resolved by microwave spectrographs (11). The pattern of the rotational spectrum of a molecule depends on the molecular symmetry. Spectra for a representative group of molecules are shown in Figure 1. In the absence of perturbing effects the frequency spacing of the rotational lines of linear and symmetric rotor molecules is linear. This can be seen to be true for all the molecules in Figure 1 except sulfur dioxide. This asymmetric rotor has a complex rotational spectrum rich in lines with no obvious pattern of frequencies. The figure would be more representative if all but one of the molecules were asymmetric rotors, as this is the class of molecules met most frequently in analytical work. A line occurs in the rotational spectrum of a molecule when a transition is made between two stationary energy states in such a manner &s to interact with electromagnetic radiation. The frequency of the emitted or absorbed line is related to the difference in energy of the rotational energy states by the equation
ODERX analysts, who have made good use of molecular spectroscopy in the infrared and Raman regions, have shown considerable interest in the recent development of spectroscopy in the microwave frequency range. Just as the electronic spectra of molecules occur largely in the visible and ultraviolet and vibrational spectra in the infrared, transitions between the rotational energy states of molecules are to be observed most conveniently in the frequency range from 0.3 to 20 cm.-' Little purely analytical work has been done in this field as yet, but such applications of microwave techniques should find increasing importance in the near future. Workers in the field of microwave spectra preparing very small samples of organic molecules containing enriched isotopes find the microwave spectrograph the most convenient analytical tool available. A considerable number of laboratories are now equipped with microwave spectrographs with which research groups are studying problems of molecular structure and the determination of the spins and moments of various nuclei. As a result of this work understanding of the basic principles underlying molecular rotational spectra has been much improved. This article presents certain of these principles briefly, in order that analysts searching for new instrumental methods may be able to evaluate the possibilities and limitations of chemical analysis with the microwave spectrometer. This new technique offers attractive possibilities. -4lthough the occurrence of a sharply resonant line spectra in this frequency region is limited to dilute gases, any compound having a vapor pressure of as much as lo-' mm. may be worked with if as much as 10-8 mole is available. The resolution of the microwave spectrometer is so great that in the microwave region so far investigated there is room for 5,000,000 noninterfering rotational lines. In principle, a t least, 1000 or more different complex organic molecules could be quantitatively determined from a sample smaller than 1 microgram without harming the sample in any way. Such performance cannot be expected of equipment in use a t the present time, but is possible in principle and indicates why this field should be of interest to analysts. Furthermore, rotational spectra when observed a t such high resolution are extremely sensitive to the over-all structure of molecules in a way that is different in important respects from infrared and mass spectra. On the other hand, microwave spectra have certain fundamental limitations which prevent detection of whole classes of molecules.
Y
= (W,
- W,)/h
(1)
where h is Planck's constant. A solution of the quantum-mechanical problem of the energy levels of a rigid symmetric rotor gives the following expression for the rotational energy
_ Wr hc where Wr C
-- F ( J , K )
= BJ(J
+ 1) + ( A - B ) K 2
= rotational energy in ergs = velocity of light = rotational energy in cm.-1
F(J,K ) J h / 2 i ~ = total angular momentum K h / 2 a = the component of the angular momentum parallel to the axis of symmetry of the molecule
MOLECULAR DYIVAMICS IB
Rotational spectra have in the past several years been observed by microwave techniques over the frequency range from 0.2 to
= moment of inertia of the molecule about an
axis through its center of gravity and perpendicular to the molecular symmet,ry axis 540
V O L U M E 21, NO. 5, M A Y 1 9 4 9 IA
54 1
= moment of inertia of the molecule about its
axis of symmetry
+ C ) J ( J + 1) + l/z(A - C)E,
or
F(J7) = ~/z(A
(6b)
A > B > C Microwave spectra are obtained as absorption spectra, by noting the amount of microwave radiation transmitted through a gas sample as a function of frequency. The quantum-mechanical selection rules for transitions between the rotational energy levels of a symmetric rotor are
AJ = + 1
AK = 0
(3)
A molecule ahich is only accidentally a symmetric rotor can have in addition transitions for which AK = +1 if there is a component of the dipole moment perpendicular to the symmetry axis of the molecule. With the selection rule given in Equation 3 a very simple formula is obtained for the frequencies of the rotational transitions v
em.-' = 2B(J
+ 1)
(4)
where J is the angular momentum quantum number for the lower rotational state. For a linear molecule in the ground vibrational state K = 0, so that thc expression for the rotational energy reduces to
F(J, K ) = BJ(J
+ 1)
(5)
The expression for the frequencies of the rotational transitions has the same form as Equation 4. The calculation of the rotational energy levels of an asymmetric molecule is a somewhat more complicated problem although techniques have been worked out for the lower rotational energy levels which are normally of chief interest (4, 5, 9). Equations of the following form are encountered in these calculations
mc x 103 40
30 -C%NC 1+2
51
41
31
21
32
+2 22
po.,
34 35 36
5g
I
6o Figure 1.
PF, 2+3
37 38
3cs 3 + 4 49
39
gj
40
+
A4san example, as the molecule 016C12S32 has a spectral line occurring a t 24,325.92 Rlc. corresponding to the rotational transition J = 1 +J = 2, its moment of inertia is 137.9 X 10-40 g. sq. cm. The moment of inertia of a molecule is defined as Zmir,2 where the m, are the masses of the individual atoms and the ri are the perpendicular distances to the axis running through the center of gravity of the molecule. For a specific molecule this can be transformed into an expression involving the atomic weights of the various atoms making up the molecule and the internuclear distances. In the case of OCS this expression has the form
of the J = 1 +J = 2 transition where J is the rotational quantum number. The frequencies and derived moments of inertia have been obtained for several of the isotopic forms of the OCS molecule (Table I).
30
Molecule 0 lSC12S32 0 16C 1 2 5 9 3 016C12S34
0 16C 1 3 s 3 2 016C11S32
33
tso2
Sample Microw-ave Rotational Spectra
< IC
W , and E , are closely related quantities which depend in a complicated manner on A , B , C, and J , and for a given J assume 2J 1 different values. These formulas and the sample spectra shown in Figure 1 indicate the considerable increase in complexity of rotational spectra with increasing asymmetry of the molecule. For a linear molecule or for a symmetric rotor it is possible to obtain the moment of inertia of the molecule directly from the frequency of a single identified line. The relation used is
Table I.
50
5 7 C Br8'47 Cl$Br7g 58 48
IA< Is
Moments of Inertia for OCS Y
1
+
2, Rlc. 24,325.92 24,020.3 23,731.33 24,247 . 8 2 24,173 .O
I B X 10-40 G. Sq. Cm. 138.0 139.7 141.4 138.4 138.8
If it is assumed that the internuclear distances are the same for these molecules, a set of simultaneous linear equations may be set up, any two of which are sufficient to determine the moleculzr parameters. The internal consistency of su2h data indicates that internuclear distances accurate to 0.01 A. may be determined in this manner. Equation 7 , relating the frequency of rotational transitions to the moment of inertia, serves to point out one of the limitations of microwave spectroscopy as an analvtical method. Symmetric rotor and linear molecules with small moments of inertia will have widely spaced spectral lines and for molecules with moments less than about 30 X 10-40 g. sq. cm. will have no rotational transitions falling within an accessible portion of the mi6owave frequency range. This limitation does not hold, however, for asymmetric molecules obeying a different set of selection rules. I t may also be seen from the simple relation between the moment of inertia and the frequencies of the rotational transitions that the rotational spectrum is a function solely of the overall structure of the molecule and does not contain, as does the infrared vibrational spectrum, features characteristic of groups within the molecule, Thus in analysis the two techniques, micro-
ANALYTICAL CHEMISTRY
542
wave and infrared, are in many cases natural complements rather than competitors. From the infrared spectrum the presence of characteristic groups may be determined and from the microwave spectrum their arrangement n-ithin the molecule. The foregoing treatment of the rotational spectra of molecules has given an exaggerated impression of their simplicity. Actual rotational ipectra are frequently complicated by various types of fine structure, some of which have been encountered for the first time in the microwave region because of the extreme resolving power available. In some instances, when there is no specific intereyt in the fine structure, it may be neglected by operating the spectrometer a t less than maximum resolving power. In many cases this is not practical and the fine structure serves to decrease the intensity of the observed spectral line. For a molecule whose total intensity for a given rotational transition is small fine structure, splitting may reduce the intensity of the individual lines below the observable limit. In discussing OCS the occurrence of individual rotational transitions for each isotopic variety of the molecule was pointed out. This type of fine structure should have little effect in reducing the intensity of a line in the rotational spectrum for atoms which have only a single abundant stable isotope such as C12, HI,0l6, S32, etc. Holyever, for a molecule involving bromine the intensity of each individual line would be only half that of the total intensity of the rotational transition. An important type of fine structure, first observed in the microwave spectrum of ammonia (2, s),is due to nuclear quadrupole coupling. This perturbation of the rotational energy of a molecule occurs if atoms possessing nuclear quadrupole electric moments and nuclear spins greater than 1/2 are present. Commonly occurring nuclei having these properties are N14, C137, C1S5, Rr81, Br79, 1127, and 5 3 3 . The expression for the change in rotational energy diir to nuclear quadrupole coupling ( 1 ) has thc form A V = eQqF(J, K , I )
where e is the electroiiic charge, Q is the nuclear quadrupole moment, p is the gradient of electrostatic potential produced by all charges in the molecule except those inside a small spherr surrounding the nucleus of interest, and F ( J , K , I ) is a function of the angular momentum and nuclear spin quantum numbers. This type of fine structure splitting may have a very pronounced effect on the spectrum, because for a molecule containing more thau one chlorine, bromine, or iodine the intensity of the strongest component may be less than 0.1 of the total intensity for the rotational transition. This coupling of the molecular and nuclear properties of molecules is of great interest to physicists, for in many cases it provides the best method of determining the nuclear spin and quadrupole moment of some nucleus of interest. Although the foregoing discussion has been based on the assumption of a rigid nonvibrating molecule, a t best we can deal with molecules in the ground vibrational state in which they still possess 0.5 quantum of vibrational energy for each normal vibrational mode. Furthermore, a t ordinary temperatures excited vibrational states will be appreciably populated. Because for an actual molecule the rotational constants have a dependence on the vibrational quantum numbers of the following form ( 7 )
where the cyi are constants small in comparison t o Be, which is the rotational constant for the molecule in its equilibrium position, and di is the degree of degeneracy of the vibration, there will be a separate rotational spectrum for each vibrational state which is sufficiently populated. For a molecule with a number of lowlying vibrations observed a t higher temperatures, the vibrational
fine structure may appreciably reduce the intensity of the strongest line in the group corresponding to a single rotational transition. One of the lonest lyiiig vibrations of a number of cnnimon molecules is of a special type. These are the torsional oscillations possible in nioleculei such as methanol, which have internal rotating groups. The presence of such groups in a molecule not only gives rise to a rotational hpectrum having a complicated fine structure but if there is a component of the dipole moment along an appropriate axis can give rise to new series of lines corresponding to free rotation and to the interaction of the rotation of internal groups and of the moleculr as a nhole. The rotational tranbitionz of rigid symmetric rotors are degenerate bc.cxuhe of the existence of separate energy levels corresponding to the 2J+l diEerent values of K for each J value. The actual molecule is not rigid and undergoes an amount of centrifugal distortion x hich is proportional to the speed of niolecular rotation. As a result, the K degeneracy is removed to some extent. This effect is not large for the lower rotational states of svnimetric rotor molecules and can usually be neglected. Effects due to centrifugal distortion play a much more important role in the higher rotational states of asymmetric molecules. For a molecule that is not quite an accidentally symmetric rotor the K degeneracy may also be imperfect. In treating a moleculr M hich is even moderately asymmetric K is no longer a proper quantum number and there are 2 J + 1 sublevels of different energy for each value of J . Saturally the intensities of transitions involving these discrete sublevels will be less than for the corresponding transition between degenerate level,q in a svmmetric rotor. In the absence of external electric and magnetic fields, each rotational energy level J , K is made up of 2 J + 1 dkgenerate sublevels. The removal of these degeneracies in an external field gives rise to the Stark and Zeeman effects in molecular spectroscopy. For symmetric rotor molecules the Stark splitting of the rotational levels can he large. These effects have proved useful in the interpretation of microwave spectra and have served as the hasis of modulation techniques R hich have increased the available wnsitivitv of mirrowavr qpectrometers.
SPECTRAL IXTEh SITIES
The analytical applications of microwave spectroscopy depend strongly on the intensity of the available spectra and the sensitivity of the available spectrometer. These factors determine the class of molecules which may be detected, the size of sample required, and the maximum possible dilution ratio. A formula for the intensity of a line in the rotational spectrum of a molecule has been obtained as follows:
where y ‘Pv
= absorption coefficient in cm.-* = quantum-mechanical matrix element for the
absorption transition averaged over all values of the magnetic quantum number, m S(YO, V) = a shape factor for the spectral line with Y the working frequency and yo the resonant frequency for the transition n = number of molecules per cubic centimeter in the lower state of the transition i --+j c, k , and T = velocity of light, Boltzmann constant, and absolute temperature of the gas, respectivelv 1
V O L U M E 21, N O . 5, M A Y 1 9 4 9
543
r = mean time between molecular collisions
As the formula indicates, a molecule with zero dipole moment has rotational transitions with zero intensity. Thus symmetric molecules such as carbon dioxide and methane cannot be detected with the microffave spectrometer and in general hydrocarbons and other classes of molecules with dipole moments of less than 0.1 Debye unit will offer serious difficulties. Because the intensity varies inverselv with the molecular diameter, large molecules will have weak absorptions. Calculations for symmetric rotors are slightly more complex but still straightforward. The following exprrssions may be uied for the dipole moment matrix element and the partition function
r = - 1 27r Av
Iu
=
half-width of the spectral line n =
Sge-E'lh T ___2
.I
= total number of molecu1t.s per cc. g = statistical weight of the lower state
It'
= energy of the lowcr state Z = rotational partition function for the molecule
\Z hen the working frequency is exactly equal to the resonant fiequency and the second term in the expression for S ( V , ,v ) 1s neglected, as is permissible a t pressures brlow a few millimeters of m i w w v, the shape factor becomes simply S(u,, vo) = 47r2r .\laking the appropriate substitution-, the following expression for ymnaresults 87r2 pt, 2yO. S y e - J T - l k T Yrnax. = ____ 3ckT Av
z
fi oi a linear molecule this expression can be put in a particularlv simple form. Substituting as follows:
p = degree of symmetry of the p-fold axis of svmmetrp of the molecule -4and B = the rotational constants in X c .
For a molecule with a threefold axis of symmetry-e.g., a t 300"K.-the partition function reduces to
zrot.=
1.8x 103
CH8C1,
[&2]1'2
with A and B now expressed in cm.-'
S = 0.9658 X l o L qP,, T
__-
--
~-
Table 11. Transit ion J = 1+2 J = 2-3 J = 3+4 J = 4 + 5 J = 1-2 J = 4-n
Jntensitj of Rotational Transitions for Carbonj1 Sulfide u , RIc. y Calrd , Cm - 1 T, K 24326 3648Y 48632 60814
24320 60814
8 X 13 0 14 4 106 2 b
18 8 291 3
300 300 300 300 200 200
From this relation it mav be seen that the intensity of the rotational transitions of nonlincar molecules (as the expression for the partition function of an asymmetric molecule has a similar form) is inversely proportional to the moments of inertia. There are, then, a number of factors that serve to reduce the strength of the rotational absorptionspectra of large molecules: an effect due to the large molecular diameter, one due to the large moments of inertia, and frequently most important of all the effect of fine structure due to quadrupole coupling, excited vibrational states, hindered internal rotation, etc. At present the most complex molecule for which a fairly intense microwave absorption spectrum has been obtained is pyridine. It may be worth while to obtain an idea of the range of intensities of practical interest. The most intense molecular absorption so far observed in the microwave region is probably the J = 3, K = 3 line in the ammonia inversion spectrum, having a y of approximately 10-8 cm.-' Substituting this value in the fundamental absorption equation
I = I, r h e significance of this equation can perhaps best be understood
by means of a specific example. I n the case of the OCS molecule g = 0.75 Debye unit, G~ = 0.52 X in c.g.s. units, and M 1 / 2 = 7.75. The observed experimental value of A j s 12 Mc., which corresponds to a molecular diameter u of 10.7 A., so that d is 115.0. The Ads observed for molecules with appreciable
dipole moments are usually considerably larger than the values calculated from molecular diameters obtained from electron diffraction, viscosity measurements, etc. 'I'he calculated intensity of several rotational transitions of OCS are given in Table 11. I t may be readily seen that a weak spectral line can be made much more easily observable by cooling the gas or by working with a higher rotational transition at a higher frequency. Unfortunately, a t present experimental difficulties encountered at the higher frequencies make the advantages of working a t these frequencies somewhat less obvious.
c-12
it may be seen that this represents an absorption of only approximately 40T0 of the incident power in a 10-meter absorption path length. The weakest absorption so far observed corresponds to a y a t room temperature of approximately cm.-' This sets a value of lo6 as the maximum dilution ratio possible with equipment now available. Considering the short period of time in which intensive development of microwave spectroscopy has been under way, it seems unduly pessimistic to consider the cm.+ figure an ultimate limit. A calculation has been made, however, of the theoretical limiting sensitivity of a microwave spectrometer (IO)resulting in the following equation
ANALYTICAL CHEMISTRY minimum detectable absorption coefficient of the gas attenuation coefficient in cm.-l of the wave guide base of natural logarithms porn-er introduced into Rave guide before absorption takes place noise figure of the receiver band width of the receiver The assumption is made in deriving this equation that the optimumlength of guide (27
