weight range 280 to 700. Very simple spectra, free from fragment ions, result, from which the qualitative composition and carbon number range of the main hydrocarbon series can be seen a t a glance and series such as di- and tricycloparaffins can be readily detected even at low concentrations. Relative sensitivities of the four main hydrocarbon seriesviz., normal paraffins, isoparaffins, cycloparaffins, and alkylbenzenes-have been obtained and found to be constant over the molecular weight region studied.
The computation involved in quantitative analysis of wax from its field ionization spectrum is much less than from the conventional electron bombardment spectrum and analyses of two reference waxes gave results in good agreement with those obtained by conventional mass spectrometric techniques. RECEIVED for review October 19,1967. Accepted January 18, 1968. Permission to publish this paper given by The British Petroleum Co., Ltd.
Mathematical Solution of the Mass Spectrometric Standard Mixture Problem John M. Ruth’ Research Department, Liggett and Meyers Tobacco Co., Durham, N . C . The quantitative analysis of a mixture by mass spectrometry may be carried out in two different ways, which use different data and different mathematical treatments. The familiar procedure requires the measurement of sample pressure during the recording of the spectra of the components. The alternative procedure permits the substitution of the spectrum and known composition of a calibration mixture for the pressure data, but the resulting mathematical problem is somewhat more complicated and had been solved only for the case of a binary mixture. The general solution for any number of components is derived here.
THEUSUAL METHOD of quantitative analysis of a mixture by mass spectrometry (1-3) requires that the sensitivity coefficients, St,, be obtained for each component, j , at each of the mass values, i, used in the analysis, the number, n, of mass peaks used being equal to the number of components in the mixture. The use of the least squares treatment in cases where the number of mass values exceeds the number of components need not be discussed here, since it is applicable t o any such set of linear equations. If the unknowns are taken t o be partial pressures, the sensitivity coefficients as defined above have the dimensions of peak height per unit pressure. A set of n simultaneous linear Equations 1 is then solved for the partial pressures, p j , of the components.
+ SIZPZ+ . . . + SzlPl + SZZPZ + . . . + S~npn Sllpl
SlnPn = =
HZ
(1)
. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . Snlpl
+ S ~ Z P+Z . . . + Snnpn
= H n
The product of the sensitivity coefficient, Sij, and the partial pressure, p j , is the contribution of component j to the height, Hi, of the mass peak, i , in the spectrum of the mixture. System 1 is more compactly represented in matrix notation as Present address: Entomology Research Division, U. S. Department of Agriculture, Beltsville, Md. (1) G . P. Barnard, “Modern Mass Spectrometry,” pp 214, 229, Institute of Physics, London, 1953. (2) V. H. Dibeler, “Mass Spectrometry,” C . A. McDowell, Ed., McGraw-Hill, New York, 1963, Chap. 9. (3) R. I. Reed, “Applications of Mass Spectrometry to Organic Chemistry,” Academic Press, London and New York, 1966, p 128.
Sp
=
H
(2)
and the solution as p = S-’H (3) As an alternative to the familiar method of sensitivity coefficients reviewed in Equations 1 to 3, there exists another, the standard mixture method. Table I presents in outline a comparison of the two. Each method requires that the identities of the components of the unknown mixture be established, that the spectra of the individual components be obtained, and that the spectrum of the unknown mixture be recorded. The difference in the experimental data required is shown by the fourth item under that heading. The method of sensitivity coefficients makes use of pressure measurements, while the standard mixture method avoids all pressure mea-
Table I. Sensitivity Coefficient and Standard Mixture Problems Sensitivity Coefficient Problem Standard Mixture Problem Required data Required data (1) Identities of components (1) Identities of components ( 2 ) Mass spectra of compo(2) Mass spectra of components, H i j at mass i nents, H , j at mass i (3) Mass spectrum of un(3) Mass spectrum of unknown mixture, H,i at known mixture, H,, at mass i mass i (4) Pressure of each pure (4) Mass spectrum of standcomponent during reard mixture, H,, at mass cording of its spectrum, i, and mole fractions, mi, to permit calculating Sij of its components Mathematical situation Mathematical situation Data listed above are sufficient Data listed above are sufficient to determine the cornposito determine the composition of the mixture tion of the mixture Theoretical problem Theoretical problem Derive a mathematical expresDerive a mathematical expression for the composition of sion for the composition of the mixture the mixture Solution (available in textbooks) Solution (derived here) Step 1. p = S-lH, Step 1. Q = a-IH, Step 2. y = b-lH, Step 2. x , = p~ ” I. Step 3. x >, Yj 7 2 Pi i-1
2 Yj
i-1
VOL 40, NO. 4, APRIL 1968
e
747
surements, which are replaced by the spectrum and the known composition of a standard mixture of the same components that are present in the unknown mixture. It has long been known that the data listed for the standard mixture method are mathematically sufficient to ensure the existence of a solution. However, the algebraic solution is in this case more complicated, and has not been reported for any but the simple two-component mixture. Johnsen ( 4 ) gives numerical approximations for some binary mixtures. Barnard ( I ) gives the algebraic solution for a binary mixture and states that the method can be extended to more complex systems, but does not do so. Robertson (5) also recognizes that the problem can be solved, but does not obtain the solution. Reed ( 3 ) has recently reviewed the subject, and his notation is used in the set of Equations 1. A general algebraic solution of the standard mixture problem is presented here, which can be as easily programmed for machine computation as that of the well known system 1. The first set of equations is obtained by applying Equations 1 to the standard mixture. First, p j is replaced by mjpc, where r n j is the mole fraction of component j in the standard mixture and p c is the unknown pressure of the mixture, to give Sfjrnjpc as the general term in the Equations 1. S m
SNQP~
=
(4)
The number of sensitivity coefficients Stj can then be reduced to n, since, for any pure component j , the numbers S i j are proportional to the corresponding peak heights Hi, in its spectrum (and in fact were obtained from them). Thus, for any component j and any mass peak i , if HE, is the height of the tallest peak in the spectrum, usually called the “base peak,” and r t j is the fractional form of the relative intensity,
Q = a-lHC
(12)
with the understanding that the reader may employ any method that he prefers to compute the solution. The second set of equations is then obtained by applying Set 1 to the unknown mixture and carrying out another transformation. Let p z be the unknown pressure of the unknown mixture, x j be the unknown mole fraction of component j in it, and p l j , be the unknown partial pressure of component j . Define the unknown ratio R of the pressures of the two mixtures by =
~z
Then the general term before.
in Array 1 may be rewritten as
Sijpj
StjP’j
(13)
RP~
=
SfjXjPZ
= rijSsjXjRPc
(14)
= rtjQjxjR
The numbers Q j are those found from Equation 12, computed from data obtained from the standard mixture and then used in the computation of the composition of the unknown mixture. They may be used without redetermination under the same conditions that apply in the repeated use of a matrix of sensitivity coefficients-that is, until they change because of some change in the behavior of the apparatus. Define (15)
btj = r i j Q j
and Y j
(16)
= XjR
Then SijP’j
and
(17)
= btjyj
and the second set of equations is Si1 =
r d B j
(6)
Only the sensitivity coefficients of the base peaks are now present. Then the general term S t j p j is further modified by substituting from Equation 6 into 4, to obtain a product containing the two unknowns SBjand p c : S t j P j = rtjssjmjpc
(7)
Define Qj
= SBjPc
(8)
(18)
b y = H,
in which the element Hzt is the measured height of mass peak i in the spectrum of the unknown mixture, and the solution is (19)
y = b-’H,
The third step is a normalization to remove the unknown factor R still present in the numbers y j . Since, for any mixture, n
and
c x j = 1
j=l
air = w r t j
(9)
we have
so that the general term in the left members of set 1 has, for
n
the standard mixture, been replaced by a i j Q f : =
UijQj
(1 1)
(4) S. E. J. Johnsen, ANAL.CHEM., 19, 305 (1947). (5) A. J. B. Robertson, “Mass Spectrometry,” Methuen and Co.,
748
ANALYTICAL CHEMISTRY
(20)
and, from the definition Equation 16,
in which the matrix element Hcr is the height of mass peak i in the spectrum of the standard mixture. The solution is conveniently indicated
London, 1954, p 84.
R
(10)
The first set of equations is then aQ = H,
=
,=I
j-1
SuPj
xjR
yj =
xj
=
Yj
7
c
j-1
Yj
Numerical results should be most economically obtained from Equations 9, 11, 15, 18, and 21. The writer has been accustomed to using the Gauss reduction to solve systems of linear equations such as the two sets represented by the matrix expressions (Equations 11 and 18), and Crout’s modification
of that procedure is frequently recommended (2, 6, 7). Although the solution in terms of determinants, according to Cramer’s rule, is not recommended for numerical work with systems of more than two unknowns, that solution in terms of the symbols defined above does yield a theoretical insight which is not immediately obtained from numerical operations. It shows clearly the nature of the functional dependence of the solution upon the experimental variables. If we solve system 11 by Cramer’s rule, put back m5rr5 in place of alj, and remove the mole fractions occurring as factors of the determinants, we have for each value of the subscript j Acj
Qj
=
r n m
to make clear the relationship between the two problems. The solution is mathematically exact, in the same sense as that obtained in the method of sensitivity coefficients. The assumptions of independence and additivity of the component contributions to the mixture spectrum are the same. The significant difference is that in the standard mixture method no pressure measurements are used. The experimental errors of pressure measurement are avoided, their place being taken by those involved in the preparation of the standard mixture. The criteria for choosing spectral peaks for the analysis appear to be the same in the two methods. SUGGESTED APPLICATIONS
taining only spectral data. As noted before, the use of determinants in the numerical computations is not recommended. The solution was derived from the set of Equations 1, partly because they provide a familiar starting point and partly
The standard mixture calibration is applicable whenever the necessary data are obtainable. In particular, it may be used as a substitute for the method of sensitivity coefficients when volatile materials are introduced into the ion source from a reservoir through a molecular leak. Although a comparison of the two methods under these conditions remains to be done, the elimination of reservoir pressure measurements may at times be desirable. Laboratories which do not specialize in quantitative analysis occasionally encounter the need for it, and some instruments are equipped with molecular leaks but not with micromanometers. Pressures calculated from sample weights, reservoir volumes, and temperatures, as found in some of the tables published by the American Petroleum Institute (8), are, for the purposes of this discussion, considered to be measured pressures. However, situations occur in which no pressure measurement is possible, as when a sample is introduced directly into the ion source by means of a probe. The use of probes in handling samples that are not sufficient in abundance, volatility, or thermal stability for the heated reservoir technique has become common, and many investigations of materials of biological origin would not have been done without them. When mixtures are found in such work, it frequently becomes desirable to calculate the composition. Even if the experimental difficulties of attaching a gauge to the proper region were solved, the partial pressure of such a probe sample might still not be measurable. In fact, the vapor of such a sample leaving a heated capillary exists as a sort of jet or spray which may be trapped, to an extent depending on the properties of the particular sample, on the first surface that it reaches. In such a case, the quantities p c and p z may be considered to be molecular population densities expressed in numbers of molecules per unit volume, averaged over the space through which the ionizing electron beam passes. The sensitivity coefficient Sij then has the dimensions of peak height per unit of volume concentration in that region of the ion source. In the use of a probe, where evaporation of a condensed phase is occurring, it is of course necessary to consider fractionation. If fractionation is suspected, where the calculated composition of the unknown mixture differs significantly from that of the standard mixture, a new standard mixture should be made up, approximating the composition found for the unknown. The spectrum of the new standard should then be used to recalculate the composition of the unknown mixture. The repetition of this step until the composition calculated for the unknown agrees with that of the last stand-
( 6 ) P. D. Crout, Trans. Am. Znst. Elec. Engrs., 60,1235 (1941). (7) F. B. Hildebrand, “Methods of Applied Mathematics,” Prentice-Hall, New York, 1952, pp 4, 503.
(8) American Petroleum Institute Research Project 44, “Selected Mass Spectral Data,” Texas A & M University, College Station, Tex.
r11rr2.. .r1,,-1Hc1r1,j+1.. .rln r21rzl. . . r ~ , , - ~ H ~ 2 r :. !.rzn ,~+1. Acj = , . . . . . . . . .. . . . . . . . . . . .
Azr y’ =
The determinant AZj is identical with Acj except in column j , where Hzt replaces H c i . Substitution of Qj from Equation 22 into 24 gives
or xjR
=mj
AZl
Acj
The only mole fraction present in the right member of Equation 25 is m j , that of the same component j in the standard mixture. Here y j is given directly in terms of m j and the spectra of the components, the known mixture, and the unknown mixture. The ratio R may be eliminated by the normalization Equation 21 as before, to give
By the use of Equation 13, and by replacing mjpC by p5 and x j p z by P I j , Equation 26 may be put into the form
which expresses the ratio of the partial pressures of component j in the two mixtures in terms of the two determinants con-
VOL. 40, NO. 4, APRIL 1968
749
ard mixture may prove to be worth while, since at that point, if enough conditions are held constant, deviations from ideal behavior may be expected to be the same for the two mixtures. For quantitative analysis, the probe is mainly of interest where a reservoir cannot be used, and only when the properties of the components are so nearly alike as to prevent a chromatographic separation.
ACKNOWLEDGMENT
The author thanks Vernon H. Dibeler for reading the manuscript* RECEIVED for review September 11, 1967. Accepted D e cember 18,1967.
Accurate Mass Spectrometric Determilnation of Low Concentrations of Carbon Dioxide in Nitrogen Ernest E. Hughes and William D. Dorko Institute for Materials Research, Division of Analytical Chemistry, National Bureau of Standards, Washington, D . C . 20234
A method for the rapid and accurate determination of low concentrations of carbon dioxide in nitrogen has been developed. The method is based on highpressure mass spectrometry in which the mass 44 to mass 28 ratio is compared to the same ratio in a carefully calibrated standard. Concentrations from 180 to 380 ppm were determined with an accuracy of better than 1%. The accuracy of the method depends on calibration of the mass spectrometer with a series of carefully analyzed standards. Any system which can be chemically converted to a mixture of carbon dioxide and nitrogen can be analyzed by this method with equivalent accuracy.
INCREASED INTEREST in possible long-term variations in atmospheric composition has led to a close scrutiny of the carbon dioxide content of air in various parts of the world. Accurate intercomparison of data from different observers and from different locations requires a carefully characterized reference gas whose carbon dioxide content approximates that of the atmosphere. In the past, most standards of this type were produced and distributed by the Scripps Institute of Oceanography under the direction of C. D. Keeling. The investigation described below was undertaken in anticipation of future demands which would seriously tax the facilities of the former supplier, The production of a standard gas mixture requires that there be available a method of analysis of the required accuracy which is suitable for the routine monitoring of the concentration of carbon dioxide in a large number of samples. The method described below is a result of a continuing investigation of the application of high-pressure mass spectrometry to gas analysis problems. In general, mass spectrometers for gas analysis have an inlet system in which the gas to be analyzed is confined at a known pressure and from which it is allowed to flow through a molecular leak into the ionization region. At inlet pressures below 0.5 mm, the measured current due to a particular ion is directly proportional to the absolute inlet pressure of the gas producing the ion. However, deviation from this direct proportionality begins to occur as inlet pressures are increased. The quantitative relationship is not the same for all gases so that empirical calibration data are required. This is well illustrated by the recent work of Suttle, Emerson, and Burfield (I) in which varia(1) E. T. Suttle, D. E. Emerson, and D. Burfield, ANAL.CHEM., 38, 51-3 (1966).
750
ANALYTICAL CHEMISTRY
tions of the sensitivity (ion yield per unit of inlet pressure) with increasing inlet pressure are given for a number of trace constituents in helium,. Some earlier work at the National Bureau of Standards in this laboratory ( 2 ) had indicated the same effects in mixtures containing low concentrations of benzene and butane in air. Further, it was observed that the sensitivity of the minor constituent tended to increase relative to that of the major. Inlet pressures in the region where sensitivity data for the individual components can be used reliably in calculating the concentration in mixtures are too low to yield ion currents which can be measured with sufficient accuracy to determine atmospheric carbon dioxide concentrations (0.03 mol %) with a reliability of better than about 3 0 z . Thus it was evident from the beginning that if mass spectrometry was to be the method of analysis, the inlet pressures would have to be much higher than for normal operation. The conditions necessary for the degree of accuracy required would have to be determined using gas mixtures with compositions known to better than 1 of the carbon dioxide present. EXPERIMENTAL
Instrumentation. The mass spectrometer used was a Consolidated Electrodynamic Corp. Model 21-103C. Several modifications were found necessary or desirable for analyses of low concentration mixtures. The ion pump normally used on the exhaust pumping system was supplemented with a two-stage mercury diffusion pump connected in parallel. The input resistor of the low sensitivity side of the amplifier was changed to allow a difference between low and high sensitivity of 1 to 50 rather than 1 to 10. A strip chart recorder was connected to the amplifier output to facilitate rapid data readout. For the experiments involving the three primary calibration mixtures, a McLeod gauge was attached to the inlet system to provide pressure measurements to 1 %. For other low Concentration mixtures prepared later, pressures were not measured with great accuracy but were duplicated from run to run by expansion from a small volume at a pressure duplicated to about 3 %. Samples of the low concentrations mixtures were admitted to the inlet system of the mass spectrometer at a pressure of about 1 mm. At this pressure and at an ionizing current of about 40 PA, both the ion current at mass 28 (actually t E / e = 28), read with the amplifier set at low sensitivity, and that at (2) J. K. Taylor, Ed., National Bureau of Standards Technical Note 403, U. S. Government Printing Office, Washington, D. C.,
Sept. 1966.
