Determination of Carbon-13 by Infrared Spectrophotometry of Carbon Monoxide Robin S. McDowell Los Alamos ScientiJc Laboratory, University of California, Los Alamos, N. M . 87544 The R(13) lines in the vibrational fundamentals of l2C0 and laCO occur at 2193 and 2144 cm-’, respectively, and have been selected as being sufficiently free from interference to serve for the determination of the ratio. From the theory of line intensities and a consideration of the effect of spectrometer resolution on the line shape, one can establish that lac/%= 1.023 (s‘A‘/sA)z, where I is the spectral slit width, A is the peak absorbance, and the primed quantities refer to %O. A series of analyses on samples with lSC/12C in the range 0.01-0.20 gave results which agreed to within about 2% of mass spectrometric determinations and had relative standard deviations of about 6%.
THIS PAPER REPORTS a straightforward and reasonably accurate method of analysis of 13Cin low concentrations which does not involve the use of a mass spectrometer. It might be expected to be useful in those laboratories which lack a mass spectrometer but which have an infrared spectrophotometer capable of resolving 2 cm-l or better. The method uses the vibrational fundamental of carbon monoxide, which consists of a series of vibration-rotation lines separated by about 2B = 3.8 cm-1, where B is the rotational constant. An examination of this fundamental indicates that the line R(13) should be the most suitable for quantitative analysis: in 13C0this line occurs at 2144.0 cm-I ( I ) , nearly in the middle of the 7.7 cm-1 space between the first Pand R-branch lines of l 2 C 0 ; and in l 2 C 0 it occurs at 2193.4 cm-1, well above the region in which 13C0 absorbs at the pressures used here. None of the other vibration-rotation lines is so free from interference, and accordingly R(13) was used for the development of the analytical method which follows. THEORY
The intensity (line strength) of a P- or R-branch transition in a parallel vibration-rotation band of a linear molecule is (293)
where a is the absorptivity in cm-1, N the number of molecules per cm3, vm the wavenumber of the line in crn-’, gz the statistical weight due to nuclear spin, Qr the rotational partition function including nuclear spin, m = J 1 for the R-branch and m = J for the P-branch, J is the rotational quantum number of the ground state, F (J) = BJ ( J 1) the rotational energy in cm-l, (u’l MI u ” ) the matrix element of the electric moment, T the temperature, and c, h, and k the usual fundamental constants.
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(1) W. S. Benedict, R. Herman, G. E. Moore, and S. Silverman, Astrophys. J., 135, 277 (1962). (2) D. M. Dennison, Reu. Mud. Phys., 3, 280 (1931); and correction. S . L. Gerhard and D. M. Dennison. Phvs. Rev., . 43., 197 (19331, footnote 1. (3) H. H. Nielsen, in “Handbuch der Physik,” Springer-Verlag, Berlin, 1959, Vol. 37, pp 173-313; especially p 239. I
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The electric moment matrix element for the fundamental of a harmonic oscillator can be written (4)
where bM/bQ is the derivative of the electric moment with respect to the normal coordinate and v, is the vibrational frequency. The normal coordinate for a diatomic molecule, in terms of the change in the internuclear distance r, is Q = p112 Ar, where p = mlmp/(ml mz) is the reduced mass of the molecule. Since we can write for the electric moment M = qr whereq is the effective electric charge, we have
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We can drop the last factor in Equation 1 since it will be negligibly different from unity in the spectral region we are concerned with; and if we write (for a heteronuclear diatomic molecule) Qr = g1kT/hcB = 8n2kTgzpr2/h2, we have
Of the constants in this expression, r and q should be the same for 1zCO and ‘ T O , to a very good approximation. If, then, we consider a given line in the fundamental band of each molecule, we should find that
S a Np-2(vm/yo)e- F ( J )hc/k l’
(2)
The integrated intensity of the line, which appears in Equation l , is difficult to obtain directly because the linewidth, which is determined primarily by collision broadening at the pressures we will be concerned with, is of the order of 0.010.1 cm-1, and hence much narrower than the spectral slit width of the spectrometer. As a result one does not observe the true band contour, but rather the convolution of the true band contour with a spectrometer slit function whose exact nature may be uncertain. This problem has been considered who show that if the line in in detail by Nielsen et al. (3, question has a true line shape which can be described by a Lorentzian function, a(v)
=
ad2
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6S
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(v - vm)2 62 T[(V - v,)2 621 where a,is the maximum absorptivity (Le., the absorptivity at the peak) and 6 is the half-width at half intensity; then provided certain other conditions are met the maximum absorbance can be written A, = b h = 10glo(l/Tm)= C(GbS)’I2/s
(3) Here Tm is the minimum transmittance, b the path length, s the spectral slit width of the spectrometer, and C is a constant (4) E. B. Wilson, Jr., J. C . Decius, and P. C. Cross, “Molecular Vibrations,” McGraw-Hill Book Co., New York, 1955, p 165. ( 5 ) J. R. Nielsen, V. Thornton, and E. B. Dale, Rev. Mod. Phys., 16, 307 (1944).
ANALYTICAL CHEMISTRY, VOL. 42, NO. 11, SEPTEMBER 1970
whose value depends upon the exact form assumed for the slit function. Equation 3 is valid if the following three conditions are met: 6 *6 6bS - < < I ; 5 < < 1 ; -T S 0.11 d b c m - 1 (where b is in cm), and hence is satisfied for a 10-cm cell if the spectral slit width is greater than about 0.4 cm-l. We combine Equations 2 and 3 and note that the Lorentzian half-width 6 is the same for the two molecules when they are components of a mixture. Letting a prime denote quantities pertaining to W O , we have
"
N
= 1.023
(x) s'Atm
2200
2190
2150
I
I
I
2140 cm-' I
-
4
m
8
'Yo
$4 I
A = 0.174 s = 1.36cm"
l3co
0.198
A'= 0.0759 s'= 137cm-'
Figure 1. Typical analysis of a sample with 1aC/12C = 0.194; Perkin-ElmerModel 521 spectrophotometer
(4) Table I. Results of Trial Analyses
for the 13CO/'2CO ratio from theR(13) lines of CO at 298 OK, where the constant 1.023is ( p ' / p ) I (vo'vm/vovm') exp [ - ( B - B') J(J l)hc/kr] and we have used the molecular constants of Lagemann et al. (7). Note the exponent in Equation 4: Beer's law, which requires a linear relation between N a n d A m , is not applicable to the case of a single rotational line in the spectrum of a gas.
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EXPERIMENTAL
Standard 10-cm absorption cells were used, and the pressure was adjusted so that satisfactory determinations of the base lines could be made: pressures of the order of 100 torr or less were generally satisfactory. Samples were run on both Perkin-Elmer Model 521 and E-14 spectrophotometers with spectral slit widths of 0.9-2.1 cm-1 and 0.3-1.0 cm-l, respectively. The 2193 cm-' and 2144 cm-' lines were scanned at slow speed. (Note: It is necessary to pay careful attention to the dynamic response of the instrument and to ensure that the scan speed is slow enough for the pen to give a good approximation to the true transmittance.) The mechanical slit widths were noted at each of the peak positions and the corresponding spectral slit widths were calculated from information in the instrument manuals supplied by the manufacturer. The zero lines were determined with an opaque shutter in the beam, and the base lines were determined using the transmission maxima on either side of the line of interest. At spectral slit widths greater than about 1 cm-l, the low-frequency maximum for the 2144 cm-I line showed significantly greater transmittance than did the high-frequency maximum ; best results were obtained by using the position of greater transmittance as the position of the base line. The measured transmittances were converted to absorbances and the isotopic ratios calculated from Equation 4. Figure 1 shows a typical analysis. RESIXTS AND DISCUSSION
Table I summarizes the results of analyses on a sample with the natural isotopic abundance and two other samples whose isotopic composition had been determined by mass spectrom(6) R. H. Hunt, R. A. Toth, and E. K. Plyler, J. Chem. Phys., 49, 3909 (1968). (7) R. T. Lagemann, A. H. Nielsen, and F. P. Dickey, Phys. Reu., 72, 284 (1947).
Known
13c0/12co
laCO/l*CO No. of
ratio
analyses
found, and standard deviation
0.011 0.079 0.194
7 8 11
0.009 =!= 0.002 0.077 i 0.005 0.191 + 0.010
Range 0.007-0.013 0.069-0.084 0.174-0.203
etry. Spectral slit widths for all samples covered the range 0.3-2.1 cm-', and there was no dependence of the results on the instrument used or on the slit widths. Except for the first sample, the mean error is about 2 % and the relative standard deviation is about 6%. For the first sample, the pressure necessary to get a good measurement of the T O line was higher than optimum for establishing the base line of the l2COline, and both accuracy and precision are somewhat poorer: this will generally be true of samples with a very low 13C/12Cratio. The method can be seen to give results of useful accuracy. It has the advantages that it avoids the necessity of integrating peak areas and that it is absolute: i.e., there is no need to determine an analytical curve. The only requirement on the spectrometer is that the resolution be sufficient to separate the individual lines enough for a proper base line to be established; this requires spectral slit widths of about 2 cm-1 or better. Once the proper operating conditions are established for the spectrometer used, a single analysis can be carried out in 5 minutes or less. Since the volume of a 10-cm infrared cell is typically about 100 cm3, about 5-10 mg of carbon is sufficient for an analysis. ACKNOWLEDGMENT
This problem was suggested by Bert M. Tolbert of the University of Colorado, The carbon monoxide samples and the mass spectrometer analyses were provided by B. B. McInteer and J. G. Montoya of this laboratory. Figure 1 was drawn by Dale E. Armstrong. RECEIVED for review April 6, 1970. Accepted July 7, 1970. This work was supported by the U. S. Atomic Energy Commission.
ANALYTICAL C H E M I S T R Y , VOL. 42, NO. 11, S E P T E M B E R 1970
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