E. Hugentobler Institute of Physics and J. Loliger
Institute of Organic Chemistry University of Fribourg CH-1700 Fribourg, Switzerland
I
1
A General Approach to Ca~culatingkotope Abundance Ratios in Mass Spectroscopy
In the interpretation of mass spcotra, one important point is the analysis of the isotope peak pat,terns. This analysis leads, e.g., to sum formulas or label distributions and is commonly known to mass spectroscopists; rcfcrence can therefore be madc to text,books on mass spectroscopy.' The isotope abundance ratios for various combinations of carbon, hydrogen, nitrogen, and oxygen are extensively tabulated, e.g., by Beynon (4) and Beynon and Williams (5). Furthermore, many met,hods have been described for the calculation of the isotopc abundance ratios of isotopc clusters whose elemcnt,al composition is not tabulatcd.2 Beynon (10) gives cxplicit formulas for t,he ratios of PM+I/P,,Iand for Ppl+2/P~, ete., Margrave and Polansky ( 1 1 ) treat the problem of finding thr isotope abundance ratios more gcnerrtlly, Brauman (12) shows an ingenious least square analysis, Riedel (IS) and Seibl ( I / , ) present t ~ numerical o methods. Although many attempts have been made for the formulat.ion of a generally valid treatment3 the lack of an approach which is rcadily understood by chemi s t , ~prompted us to publish this relat,ivelysimple mathcmatical model. I t inevitably leads to the samn numcrical evaluations as the other t r e a h e n t s , but it presents a straightforward way to see what calculations must be done. Mathematicml M o d e l
Thc general valid treat,ment is developed by calculating the isotopc abundance ratios of thc hypothctical isotopc cluster AB, whose elemcnt,~consist,^ of the isotopes as listed in Table 1. The procedure is schcmatically shown in the figure. The contents of Table 1can bc exprcsscd in a mathematical way. Tronsformafion: Toble o f Isotope Abundance to Mathemoficol Expression
Table 1.
Abundance o f the Isotopes of Elements A and B
Msss number Abundance
M,, M, M, h..
h..
h,.
.. . Mq Mbn
...
hh,
hh.
Mbr
hr,
. .. ...
To simplify the notation we can write
ahcre only a finite number of coefficients h,. are different from zcro. If thn coefficients h,. are normalized to 1 , which means
-
C
ha.
=
P.(1) = 1
n=O
then h.., is cqual to the probability of the event: "The mass number ill,. of clement A equals n." I n general, this probability is given by pa. = h,./P, ( 1 ) . (If P. ( 1 ) = 1 , t,he polynomial is said to be normalized.) Calcubtion: Multiplication o f the Polynomiols
Considering now thc isotope cluster AB, one can ask for the probabilit,y pas, of finding a given value k for the mass number Ma,. The mass number MaOis simply given by M , = M , Ma. The formation of species with mass number Ma*is independent of the masses
+
See, e.g., Beynon (I), Budsikiewicz, ct al. (b), or Biemann (3). Several papers have been published with the aim of condensed form of isotope abundance ratios tables, see, ex., Lederberg (fi), Xendrick (7), Tunnicliff, W&worth, and Schissler (8) and Hemloberg and Cacper (8). 3 See, e.g, Beynon (lo), Margrave, Polansky (Il), and Brauman (12). If 3* sequence oi numbers (or functions) jg.1 n = 0, 1, 2 . . . is produced by expanding s. function P(z) in a power series in z, 1
m
The abundances h.. of t,he isotopcs of elcment A bccome t,he corfficients of a polynomial P,(x), the mass numbcrs, 4fam,the of the z. ~h~ polynomial is called the generating funct'ion4 of picmcnt A. The generating function of element B is Pdx).
610 / lournol of Chemicol Education
p(,)
C 8.2",
then P(Z) is c~lledthe "generating function" ol", of the sequence (8.1. The exponential e2, e.g., is theugen-ting function" of the reciprocal ffncturial numbers lln!. In the probalueinl in bility theory, generating funct,ions are problems involving discrete dist,ributions (a3 in this case). For details see Feller (16). =
-
-
Way II
Way I
Table 2.
Abundance of the Naturally Occurring Carbon and Oxvaen lsoto~es
Mass number Abundance % Table 3.
13 16 17 18 12 98.893 1.107 99.759 0.0374 0.2039 Abundance Ratios Calculated for Carbon Monoxide
28 Mass number 9865.4668 Abundance Abundance CI, 100.0
Calculation Division Ot Eguofionr
Mulfiplicafion 01 Egu.l,on.
30 31 29 114.1318 20.20568 0.225717 1.156882 0.20481 0.002288
Trmsformofion: Mathematical Expression to Table of lsotope Abundance of the Cluster
The generating function of the isotope cluster AB being known, a knowledge of the abundance of the isotope combinations automatically follows: The exponents of the variable x of the generating function Paa(x) are the mass numbers of the isotopc combinations in the cluster, the respcctivc coefficients are their abundances. Calculation of the lsotope Abundance Ratios of lsotope Clusters
Flow chort representation of the two opplicotionr of the mathemoticcl model. W o y I reprerent. the sequence of operations for the calculation of the isotope abundance ratios for on isotope clutter if oll the eiementol isotope abundmcet are known. Woy II ,hour the sequence of operations for the calculation of the isotope composition of o label, if the m a w spectrum and all but the label elemenb' itofope obundances ore known.
A[. and MBof the isotopes of the elements. Therefore, by elementary applicat,ion of the probability theory, one finds
The sum extends over all non-negative values of n and m = k. Let us now calculate m, consistent with n the generating function P.,(x) of the isotope cluster AB. By definit,ionwe have
+
Inserting t,hc expression (3) one finds
The significance of the result therefore is function of the isotope as it shows that the g~nerat~ing cluster AB is the product of the generating functions of t,he elements. The relation in eqn. (4) is easily generalized to any isot,opecluster, e.g., AmB,Co. . . .
The operations for the applications of the mathematical model for t.he calculation of isotope abundance ratios of isotopc clusters have bceu developed within the chapter "Mathematical Model." Way I of t,he diagram represents schematically the procedure for those calculations. Determination of the Isotope Distribution of a Label
The procedure for the calculation of t,hr isotope abundance of a labcl from a measured mass spcctrum is shown by way I1 of the diagram and is as follows: Equation (4) relates two known generating functions P,(x) and P,(x) of the elements A and B to the unknovn generating function Pah(x) of t,he cluster AB. The generating function P,(x) of an dement U with an unknown isotope composition (the label) could therefore be calculated by eqn. (6),5 (P"(z))* = P,"(x)/P,(x)
(6)
if P,(x) is the g~neratingfunction of the isotope cluster (measured by the mass spectrometer) and P,(x) is thc generating function for the cluster with the exception of t,he label (calculated by formula ( 5 ) ) and n is the number of t,imes the label U is containcd in the cluster. To exemplify the mathematical model, two examples are presented: The calculation of the isotope abundance ratios of carbon monoxide and t,he calculation of the deuterium distribution of part,ially denterated formaldehyde.' (1) Example: Calculation of the Isotope Abundance Ratios of Carbon Monoxide The first operation is the transformation of the isotope abundance tables of t,he two elements carbon and oxyFor restrictionssee, e.g., Biemann (lfi). 'Formaldehyde is chosen only for the sake of simplicity of the calculations. The massspectrum of the partially deuterated formaldehyde ss represented in Table 4 is entirely synthetic. A real mass spectrum of formaldehyde shows the base peak at M+-1 and the cdculation of t,he isotope distribution would therefore he different. Volume 49, Number 9, September 1972
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61 1
gen (Table 2) into their respective mathematical expressions
Toble 4.
Moss Spectrum of Portiolly Deuteroted Formaldehvde
+
Po(=) = 98.893~'~ 1.107~'~
+ 0.0374~"+ 0.2039~~8
Po(=) = 99.759~"
The generating function Pco(x) of the isotope cluster CO is then calculated by multiplying the generating functions PC(%)and Po(x) of the elements carbon and oxygen
Table 5.
Composition of Partially Deuterated Formaldehyde
Pco(x) = Po(x).Po(z) = 9 8 . 8 9 3 . 9 9 . 7 5 9 ~ ~ 98.893 ~ ~ ~ ~ .O.0374x'W7
+ + + + 1.1070.0374x1W7+ l.lO7.O.2 0 3 9 ~ ~ ~ 2 ' ~ 9865.4668sn8+ 114.1318~"+ 20.20568sz0+
98.893~0.2039~~~~" 1.107.99.759~~~~~~
=
0.225717~~1
The generating function Pco(x) of the cluster CO heing lmoum, it can he transformed into the table of mass numbers and abundances for carbon monoxide (Table 3). (8) Example: Calculation of the Deuterium Distribution of Partially Deuterated Formaldehyde The first operation is the transformation of the peaks of the mass spectrum of the partially deuterated formaldehyde (Table 4) into the generating function P,,(x) of formaldehyde, the peak intensities being the coefficients, the mass numbers the exponents of the polynomial P,a(x) P&) = 19.95d0 100.Osa'+ 8 1 . 0 1 + ~~~
+
1 . 1 3 0.168~~' ~ ~ ~
The isotope distribution of that part of forrnaldehydc not containing the label (hydrogen7) is just CO and has therefore already been calculated in thc preceding example, the polynomial is Pco(z) = 9865.4668~"4- 114.1318zao+
+
20.20568xa0 0.225717~~~
The generating function of the label (hydrogen') is easily calculated by the following division of the. two polynomials
This means, the partially deuterated forrnaldehydc is composed of 10% Do-, 50% Dl-,and 40% '0%-species as indicated in Table 5.
' The element with atomic number 1 ('H and 'H isotopes with an artificial distribution). SListing and further descriptions of the programs MASS and LABEL may he obtained upon request to the author (J. L.).
612
/
Journal of Chemical Education
Summary
The mathematical model shows how the isotope abundance ratios of an isotope cluster is calculated (way I) and how the isotope abundance ratios of a labcl is determined (way 11). A computer is very hclpful for the multiplications and divisions of polynomials, therefore two programs have been written. Both programs, MASS for the evaluation of the isotope abundance ratios of an isotopc clustcr, LABEL for thc evaluation of the labcl contents of a laheled compounds, are written in BASIC FORTRAN IV (IBM 1130)8. Acknowledgment
We thank Prof. Dr. R. Scheffold, University of Fribourg, for many helpful discussions and Prof. Dr. J. Seibel, Eidigenossische Technische Hochschulc, Zurich, for a valuable suggestion. One of us (J. L.) thanks Prof. Dr. E. Schumachcr, CIBA-GEIGY-Photochemie, Marly, Switzerland, for the permission of using the companys' computer facilities. Literature Cited (1) Reunon, J. H.. "Maas Spectrometry and its Appliestions to Organic Chemistry," Elsevier, Amsterdam. 1960. (2) R u o z r n ~ ~ w r cH., z . D ~ ~ n ~ sC.. s rAND , W z m . r ~ ~ aD. . H., "Mass S ~ e e trosooDy of Organic Compounds," Holden-Day. San Francisco. 1967. (3) BIEMANN. K.,"Mess Speetrometrv, Organic Chemical Ap~lieations, h4oGraw-Hill,New York. 1962. (4) Ref.(1)p. 486. (5) B ~ n r o a J. , H.. AN" Wrmmua, A. E.. "Mass snd Abundance Tables foruse in Mass Speotrometry." Elaevier. Amsterdam. 1963. (6) Lenennena. J., "Computation of Moleoular Formula for Masa Spectrometry." Holden-Day, San Franeisoo. 1964. (7) Kmnnrcu, E.,Anal. Chem., 35,2146 (1963). (8) T u n m c m ~ D. , D., W ~ o s w o n ~ P. n , A,. m o S c m s m n . D. 0.. ~ n o l . Chem.37. 543 (1965). (9) H m x ~ a e n a D., , m n C ~ s p ~ K., n , 2. A n d Chem., 227, 241 (1967). (10) Ref. (I) p. 293. (11) M A R G ~ A V E , J. L., AND POLANSXY, R. B., J. Camr. Eouc., 29, 335
p. 248. (16) Ref. (3)p. 223
