J. L. Margrave, Chemistry Dept. and R. B. Polonsky,' Physics Dept.
University of Wisconsin Madison
Relative Abundance calculations for Isotopic Molecular Species
A
number of modern experimental techniques are now available to distinguish isotopic molecular species on the basis of chemical and physical properties: optical spectroscopy (infrared, visible ultraviolet), microwave spectroscopy, nuclear magnetic resonance, mass spectroscopy, various chemical reactions, and gas chromatography. In addition, compounds with synthetic isotope ratios are finding wide use in tracer and exchange experiments. It is therefore of interest to be able to calculate quickly the relative abundances of the various isotopic molecular species in any system from the available isotope abundance data. Such a calculation is described in the "Mass Spectrographic Computing Manual" of the Consolidated Engineering Co., for the C'Wla and HI-Hz isotopic species found in hydrocarbon work, but a general description of the calculation technique does not seem to be available in the literature. Although thc procedure is one which practicing mass spectrosPresent address: Massachusetts Institute of Technology, Cambridge, Mass.
copists have been using for years, it is probable that few outside of this discipline are familiar with it. It is presented here in the hope not only that some readers will find it of immediate practical value but also that it will provide for students a reference to an important chemical application of mathematical principles.2 Mathematically the problem is one of elementary probability theory and involves combinatory analysis.8 The basic principle is illustrated by the following: (1) the number of ways, ,C%, of taking any two atoms from a large number N ,
N o t e added in proof: In the new book, ''Mass Spectrometry and its Applications to Organic Chemistry," by J. H. BEYNON (Elsevier, 1960) a set of equations is presented on pages 294-302 for calculating peak heights at M, M 1, M 2, etc., from isotope abundance data. They are constructed by summing equations of the type discussed here for the various ways of getting molecules of mssses M,M 1, M +I?, eto. See any textbook of probability theory.
+
+
+
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335
and (2) the number of ways of taking two oxygen atoms from N/2 total is, thus,
As a. sample calculation consider BCll which can form the isotop& species BL0C1335, B11C133s,B10ClzasC137, 1311C1935C13T. B10C1~S7. and B" - > -B'OClWb3'. - - B11C1W1937. - " , ~ - - , Cl,a7. Let A, = B1O., A?- = B1l., B,- = Cla7.and Bt = C135 with abundances x, a, y, and dy, respectively, in the nomenclature of the preceding paragraph. 2
The probability of successfully getting two oxygen atoms from a mixture of (N/2) oxygen and (N/2) silicon atoms is given by:
-
~~~
number of B"CIa" molecules let R' = number of BWl#S molecules
R'
=
number of B"CIP number of BIoClam
number of -B R3 = number of B W I P number of BLLC!P
P,
R4 = number of B W I W l P
(N - 2) 1 lim N 4 - 1 - -4
=
number of BnCIP
Rs = number of B" C 1 9 W l ~ ~
Thus, for a mixture of a very large number of silicon and oxygen atoms in equal amounts, the probabilities of getting 02,SiO, and Si2are I/%, and respectively. The SiO value is given by:
number of B W I P Ro = number oi B W I W I P R1
=
number of B W I P number of B"CI2WI~
By calculation:
Y(Y - l)(Y - 2)
+
- 3ys 21?_ - 3dlyP 2dy
Y'
+
a
This calculation method applies directly for isotopic mixtures. For example, if B P and Br81are present in bromine gas in equal amounts, the relative abnndances and of BPBr7*,Br79Br81,and BrBLB+' are I/., respectively. The more general case is of interest since many elements have two, three, or more stable isotopes and the abundance ratios are usually irrational. Consider the general binary molecule A. B, formed from elements A and B which have three isotopes present in ratios A2/Al = a ; A3/A1 = b, B2/B1 = d and B3/B1 = e . Then, if the number of Al atoms = x and the number of B1 atoms = y, the probability of forming the isotopic species (A3)R(B3)m is given by: P
=
(number of ways of getting n& atorns)(nurnber of ways of getting mBi atoms)/nurnber of ways of getting n.4 atoms and mB atoms irom the total assemblage of N , A stoms and N . B atoms
=
a
(,,
[ t k
y' -
1
321' - 2y
- 3d2y2 - 2dy)]
=
and similarly,
Re
' =
(&)'Ri (&) =
The results of the calculations are given in Table 1. These relative abnndances are given by the binomial expansion of (1
+ a ) ( l + d)'
= 1
+ a + 3d + 3ad + 3d2 + 3nd2 + d8 + ad8.
Table 1.
Isoto~icSpecies of BCI,
Relativea abundance
Species
Relatives abundance
Species
-
[ n ! ( z A n)! ' m!(N? - m)!
wherex+ax+bx=N1andy+dy+ey=N1. A similar relation holds for (A,).(B,), and (A2).(B2), except for the different abundances of the isotopes. The ratio: number of (&)dB8), R = number of (A,),(B1),
I n!(z - n)! m!(?, - m)!
since the denominators are identical in the absolute probability expressions. 336
/
Journal of Chemicol Education
a = 4.00 and d = 3.06 me reasonable values for the BTL/BIo and Cl5S/CP7ratios, respectively.
In general, then, for an element with isot,opesA1 and Az of fractional abundances pl and pn, the relative probabilities of having [(A,),(A?),I where w
are given by the terms of (p,
+x
=
n
+ p2)nwhich are + 1 )] P,-P,.
(n)(n - l ) . . , ( n - z x!
[
Thus, for the molecule A,B, (two isotopes of A with
Table
fractional abnndances pl and pz and two isotopes of B with fractional abundances ql and q2) the general molccule is (A1)ED(A2).(B1)V(BZ)s where w x = n and y z = m and the probability for a given isotopic species is:
Relative
+
+
As an example, the various probabilities for BCls are calculated with pl = 0.8 for B", pz = 0.2 for BIO, q, = 0.246 for C18', and q2 = 0.754 for ClS5in Table 2, and compared with experimental data. I'inally, one may write
as the probability of the compound A,B, occurring with a atoms of isotope Al; b atoms of isotope As, etc., to f atoms of isotope As and u atoms of isotope B,, u atoms of isotope Bz, etc. to z atoms of isotope B6. The p's need only he proport,ional to the isotope fractions of the various isotopes of A, and the q's proportional to the isotope fractions of the various isotopes of B tmo provide relative abnndances.
At. No.
A
n
kt. No.
B
n
Diatrlbutlon of laotopof A Ylrn Nvhbevhbe~
2
a Courtesy R. W. Law, Callery Chemical Company, Callery, Pennsylvania.
Tables of percentage and relative abundances for most common gases have been calculated from these formulas through the cooperation of the Numerical Analysis Laboratory of the University of Wisconsin and the Wisconsin Alumni Research I.'oundation, and are available on microfilm^.^ A sample page from the tables (Fig. 1) present,^ the results of a calculation for H202based on the isotopic abundances H' = 99.98%; H2 = 0.02%; 016= 99.7.587%; 0'' = 0.0374%; and OLX= 0.2039%. "rite J. L. Margrrwe, Dept. of Chemistry, Univereity of Wisconsin, Madison, Wisconsin.
Mstribution of lootopel Of B Yass N w e r o
% Abundance
Relative Abundance
Figure 1. Sample page from table of imtopic obvndancer of molecular species. (To interpret the table, which war printed directly by the IBM 650 computer, for o general molecule, A d , note that columns 1 and 3 concern element A and its irotoper, columns 2 ond 4 concern element B and its irotoper, column 5 gives the per sent obvndonce for the vorious isotopic rpecier, ond column 6 giver the relotive obundanre with the most common rpecier orbitrorily assigned the obvndonce 1.0000000.1
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