a column of finer-mesh resin and eluting with a low ammonia concentration. Quantitative Measurements. The peak areas on the recorder chart were proportional to the quantities of’ solutes added, within the errors of measurement, which were about k 3 Z . Peak areas must be calibrated for each solute individually. At first sight, one would expect a given mass of triethanolamine to give a bigger peak than the same mass of di- or monoethanolamine, for the refractive indices of the three amines are, respectively, 1.4852, 1.4776, and 1.4541, compared to 1.3325 for water. However, the refractive index of a mixture depends on mole fraction rather than weight fraction, and calculation shows that on a weight basis the three amines affect the refractive index almost equally. Calibration is essential, because peak areas may be influenced by displacement of ammonia from the resin. We did not try to see how small a quantity of amine could be measured, but one can form an impression of the sensitivity from the first small peak in Figure 7. This peak corresponds to 0.1 mg of triethanolamine in the injected sample. The refractometer attenuation was 4 X, and it should be possible to detect one-tenth of this amount, namely, 0.01 mg, under favorable conditions. CONCLUSIONS
These chromatographic separations of ethanolamines and aziridines illustrate generalizations we have made before ( 4 ) . The strongest binding to the metal-loaded resins occurs with primary amino groups; carbon atoms attached to the amino nitrogen or to the adjacent carbon atom obstruct the binding and cause the compounds to be eluted earlier. Thus diethanolamine is eluted before monoethanolamine, and N-ethyl and 1-methylaziridines are eluted before ethylenimine. We must consider the coordination of the metal ions with hydroxyl groups, however, as well as their coordination with nitrogen atoms. T’riethanolamine is more strongly bound
by the nickel resin than is dimethylethanolamine, and the sugar amines glucosamine and galactosamine, which we have tested, are held almost as strongly as monoethanolamine in spite of their “obstructed” amino groups. Copper ions seem to coordinate more strongly with hydroxyl groups than nickel ions, to judge by the elution orders of aziridine-ethanolamine mixtures (see Table I). Chromatographic partition ratios depend on various factors, of which the metal-ligand interaction is only one, but this is probably the main effect in the systems studied here. We have shown that metal-ethanolamine complexes are only slightly less stable in an ion-exchange resin than in aqueous solution (15). We could use metal-ligand association constants in solution as guides to ligand-exchange chromatography if we had the data. For Ni(I1) and monoethanolamine, Kl = 950 (15). We do not have the constants for di- and triethanolamine, and they may be too small to measure by the usual techniques. Whatever the reason, monoethanolamine is held much more strongly than di- and triethanolamine, and our method of analysis is very effective for measuring small amounts of monoethanolamine in the presence of much di- and triethanolamine. It is almost as effective for measuring small amounts of di- and triethanolamine in the presence of much monoethanolamine. Our method is the only one we have seen reported for chromatographic analysis of simple aziridines. RECEIVED for review July 7, 1972. Accepted October 2, 1972. Work was supported by the U S . Atomic Energy Commission, Contract AT-(1 1-1)-499, by the National Science Foundation, Grant No. GP-25727, and by a fellowship awarded to one of us (T-J.H.) by the National Science Council of the Republic of China. (15) L. Cockerell and H. F. Walton, J . Pliys. Cliern., 66, 75 (1962).
Application of a Statistical Method to Isotopic Analysis General Principles C. Genty Commissariat a I’Energie Atomique, Centre &Etudes de Bruy2res-le-ChBte1, B. P. 61,92-Montrouge, France The statistical theory has been applied to the analysis of a mixture of isotopic compounds of a same species. Simple molecules like XA, and more complex molecules like XA,-YA,+ZA,~~ and XA,-YA,+XA, in which the atom A can be randomly chosen among p isotopes are studied. In each case, the number of isotopiccompositions for a given molecular species, the probability to obtain a given molecule, and the required isotopic ratios are calculated. The isotopic ratios of the whole mixture can be calculated very simply by measuring (p - 1) independent molecule concentration ratios. The advantages of this method which does not involve any absolute measurement and does not depend upon the number of isotopes in the mixture are shown. The theoretical application of the reported method to a mixture of hydrogenated, deuterated, and tritiated propanes i s examined.
REPLACING A(*)atoms of a type XA,(’’ molecule more or less completely with A!1),A ’ * ) ,. . . A(i), . . , AfP’isotopes causes the number of isotopic species to increase rapidly.
Isotope analysis problems are greatly complicated by the very large number of such species, while determination of the isotopic ratios, i.e. of ratios of the following type: [A(”] =
Number of A’’) atoms-
(1)
number of A(‘’atoms 1=1
often raises very arduous problems. In most cases direct measurement of the A‘’) values is impossible, so that in order to obtain the required isotopic ratios the analyst must either measure the individual concentrations of each species-which soon becomes very difficult because of the number of speciesor resort to chemical transformations in the material under analysis. The isotopic ratio calculation method based on the statistical hypothesis makes it possible to obtain the required values without chemical transformations, with the help of a ANALYTICAL CHEMISTRY, VOL. 45, NO. 3 , MARCH 1973
505
small number of measurements. This hypothesis, which assumes that the various isotopes have equal chances of combining, has long been known and is very often used for studying fragmentation in mass spectrometry or distribution problems (1-13). However, it would appear that this hypothesis has not been used to solve the complex problems of isotope analysis, and to determine the values of the isotopic ratios such as A(') ( I ) . It involves the use, for the exchange reactions between molecules, of theoretical equilibrium constants which cannot supplant the experimental constants but which, in the absence of the latter, will as a rule provide the required isotopic ratios with a good degree of approximation. It is based on the premise that systems exhibiting chemical inertia are equilibrated a t the outset. SIiMPLE MOLECULES This section will consider the case of a molecule XA, in which all the atoms A are equivalent and can be chosen among the p isotopes A") . . A(') , . A',) that have equal chances of combining. Number of Distinct Molecule Species. This is given by the number of combinations repeating p objects taken n at a time,
or :
L1
=
+
p - I)! n ! ( p - l)!
(n
XA(l),, . . . A(i),t
...
satisfying the general formula XA,,
According to Equation 1, the probability of choosing a n atom A(') is equal to [A!')], so that the probability P of obtaining the molecule I is:
which according to Equation 4 gives:
(5)
Number of molecules I , thereTotal number of molecules XA, fore represents the molecular fraction of form I in the mixture of species and is proportional to the concentration of I. Isotopic ratios. Measuring the ratios between the probabilities of obtaining distinct molecules of type XA, will enable the required isotopic ratios to be achieved. Consider two molecules :
P1, which is equal to
XA''),,
There are 126 distinct species for n = 4 and p = 6 and a further 10 for n = 3 a n d p = 3. Probability of Obtaining a Given Molecule. Let (I) be a specific molecule. (1)
Let Fl be the ratio of the number of ways of obtaining the molecule I to the total number of combinations.
.
,
According to Equation 3, the ratio between the molecular fraction for these two molecules in the mixture is given by :
RI =
Pl!
Pi!
"
~
ai! , ,
'
'
'
P,! [A(1)lal-(3,
a , ! , . . a,!
,
[A('),]"'-@'
A!p),p
(5
a i = n).
The
i=1
number of ways Nl of obtaining this molecule can be calculated as follows: there are C",, ways of choosing a1 atoms A f l ) among n atoms, C"-"',? ways of choosing a2 atoms A") among n - a1remaining atoms, and so on. Let Nl = C",, C"-*' . . . Cn-"1 . . -aL- a% ' . ' C"-"'.. - O p - l L,p or more simply :
...
and XA(l)B,. . .
. A'i' a i . . .
,
, ,
[A(P)]op-& ( 6 )
It may be seen that determination of ( p - 1) independent ratios of this type makes it possible to find the individual values of [A(')] . . , [A(i)]. . . [A',)], the pth relation being given by the relation: P
[A(i)] = 1 i=l
1
Q?
Ni
n! 1
a1!a2!
..,
ai!
,
. . a,!
(3)
The total number of all possible combinations for the set of formulas XA, is given by the number of arrangements, or s 1=
p".
P,
and
The new ratio is given by:
(1) J. L. Garnett and W. A. Sollich-Baurngartner, J . Pliys. Cliem., 68,3177 (1964). (2) J. Turkevich. D. 0. Schissler. and P. Irsa. J . Phys. Colloid Cliem., 55,1078 (1951). (3) R. J. Hodges and J. L. Garnett, J . Pliys. Cliem., 72, 1673 (1968). (4) G. C. Bond, QucIrr. Rec., 8,279 (1954). (5) J. Turkevich. F. Bonner, D. Schissler, and P. Irsa, Discrrss. Furuduy Soc., 8,352 (1950). (6) J. Turkevich, L. Friedman, E. Solomon, and F. M. Wrightson, J. Amer. Cliem. Soc., 70,2638 (1948). (7) J. R. Anderson and C. Kernball. Proc. Roy. SOC.Lo/xio/I, Ser. A , 223,361 (1954). (8) C. Kernball, J . Cliem. Soc., 1956, 735. (9) C. Kemball and P. B. Wells, J . Clicm. Soc. ( A ) , 1968, 444. (10) M. Bolder, G. Dallinga, and H. Kloosterziel, J . Cutrrl., 3, 312 ( I 964). ( 1 1 ) F. Fiquet Fayard. Brill. Soc. Cliini. Belg.. 73, 373 (1964). ( I 2) P. L. Corio. J . Pliys. Client., 74, 3853 (1970). (13) G. J. Martin, M. T. Quemeneur, and M . L. Martin, Brill. Soc. Chim. Fr., 1970.4082. 506
In practice, it is preferable whenever possible to measure ratios of molecular fractions such as XA(l ) ,,A(2),?A(3) ,3 . . . A(i),i . . . A'?')
It furnishes the value of [A(1)]/[A(2)] directly.
ANALYTICAL CHEMISTRY, VOL. 45, NO. 3, MARCH 1973
COMPLEX MOLECULES
The case of more complex molecules can also be resolved by the same method. Let XA,,-YA,,-ZA,~~ be a molecule in which the A-atoms attached to a given atom are equivalent and can be chosen randomly among the p isotopes that have equal chances of combining. Two contingencies must be considered, depending on whether the analyzing apparatus enables ratios to be achieved with empirical formulas (like XYZA'l),, . , A!')", . , , A(Pla,) or structural formulas (like , . , A"' U P - YA(l),,i . . . . ., XA"),, . . , A',' a ' p ZA('),., , A'i),,,,, . . A(P)a,~,). ,
,
,
Empirical Formulas. I n this case consideration is given only to formulas of the type XYZA“),, . , , A(ijai . . . A(p),, in which it is not stipulated to which central atom X, Y, or Z, the A-atoms are attached. Since all the molecules satisfy the general formula XA,-YA,,-ZA,~~, the following relation is obtained: P
at = m
+ m’ + m”
a=1
The total number of different species is given by the product L X A , x LYA,’ X LzA,” o r :
La
=
(m
+p
- l)!(m’ + p
- l)!(m’’ m!m”!m”’! [ ( p - l)!I3
+ p - I)!
(12)
Probability of Obtaining a Gicen Molecule. The first step is to calculate the number of different ways of obtaining the combinations XA,, YA,l, ZA,,., ciz., according to Equation 3:
Thus it is easy t o revert to the case of simple molecules by replacing n in the previous formulas 2 to 6 by m m‘ m”,which gives: NUMBER OF DISTINCT MOLECULE SPECIES.
+
+ +
( m m’ ___ ( m m’
L1 =
+ m” + p - l ) ! + m”)!( p - l ) !
+
(7)
PROBABILITY OF OBTAINING A GIVEN MOLECULE. Pp
(m = Ql!
...
+ m’ + m”)!
a1!
.
..
The number of ways of obtaining molecule I1 is given by the product N X A , X N Y A , ~x NzA,“, or:
X
ap!”+n~’+m’‘j
[A(lj]al. . . [A(ij]ai . . . [A(pj]rrp (8)
Nz
=
ISOTOPIC RATIOS. The ratio between the concentrations of molecules
a1!
. . . a,! . . ,
is given by the previously established formula 6, or:
The total number of possible combinations for the set of formulas XA,-YA,-ZA,. is given by Sz = p!rL-’rL‘-’rL”). The ratio of the number of ways of obtaining molecule I1 to the total number of combinations is given by:
R3 =
,
Pi!
[A(l)l,l-pl
,
a1!
. , . a,! . . . a p !
,
’
’
PP.!
PI!
’
Jt will be seen too that a knowledge of ( p - 1) independent values like Equation 9 above will enable the isotopic ratios [A‘ljj . . . to be calculated. Structural Formulas. GENERAL CASE. Number of Distinct Molecule Species. In order to obtain the number of different formulas of type 11, the procedure is to write that the molecule satisfies the general formula XA,-YA,,-ZA,(,.
, ,
.
Atp)
with
The next step is to calculate the number of distinct isotopic species corresponding to each of the formulas XA,, YA,,, ZA,’,, or according to Equation 2 :
LXA, =
+ p - l)!
m ! ( p - l)!
LPA,,, = -
+p
- l)! m ” ! ( p- l)!
(m’
+ p - l)! LZA,,,= (m” ______ m”!(p - l ) !
.
, , Cyp”!
in! m ” !m”’!
[A(ij]ai-@i, , , [A!PJIap-fi~ (9)
(:m
ai/’!
F2 =
...
(11) XA‘”,, . . , A(’),; . . . AfP),p-YA‘l)at, . . , A’7, , . . AiP),,p-ZA(lj , , ,
m ! m’! m”! . , . ai’! , . . ap’!al’”! , , ,
ap!al’!
al!
. . . a , ! .. . a p ! a 1 ’ !
The probability of obtaining molecule I1 is therefore given by the relation : p3 = F2[A(1)]a?+al’+01’’ . . . [A(Z~]C~+CZL’+C~L’‘ ...
[A(Pj]ap+as’+olp’’ (14) Isotopic Ratios. As before, the ratio between the probabilities of obtaining two given molecules is a function solely and a knowledge of of the individual isotopic ratios ( p - 1) independent values makes it possible to determine all the isotopic ratios. SPECIAL CASE. An important special case is that of the identity of the terminal atoms, namely molecules of the form: XA,-YA, ,XA‘C ,,,, , , , A‘i j , t , , (IV)
.
, ,
A(P),p8,
XA(l)mlu, , , A(Qaiu, , , A(ij,Li. . . XA(l),, . . . A(i)ai. . . A(p)ai,
are identical and must now be counted a s a single species. To obtain the total number of distinct molecules, it is therefore necessary to subtract from general formula 12 oneANALYTICAL CHEMISTRY, VOL. 45,
NO. 3, MARCH 1973 * 507
half the number of molecules with non-identical terminal combinations. Let L4 be the unknown number of distinct molecules, La the number of molecules given by general formula 12, LO the number of molecules with non-identical terminal combinations, and LI the number of molecules with identical terminal combinations.
1
L4 = La - -LO 2
where L3 = LD
+ LI
CONSEQUENCES OF THE STATISTICAL HYPOTHESIS
Measurement of ( p - 1) independent molecule concentration ratios invariably enables the p isotopic ratios of the mixture to be ascertained. The method uses ratio measurements only and consequently avoids all absolute determinations. The ratio between the molecular fractions of two species in a mixture containing only a certain number of isotopes is not altered b,v adding other isotopes. To see this, consider a mixture containing only i isotopes and consider two molecules XYZA(1)a,A(2)a:, ,
whence L4
=
1 -
2
(L3
XYZA(’)p,A(2)p,, . , A(’)pi.
+ LI).
L I , the number of molecules with identical terminal combinations, is obtained by writing that for each terminal combination XA, can be made to correspond at the other end of the chain with an identical terminal combination XA, irrespective of the central combination YA,,, or:
According to Equation 9, the ratio between the molecular fractions of these two species is given by:
with i
LI
=
L X A , X LYA,,
x
+
=
nand
[Acz!] = 1 ,=I
1=1
If compounds containing only isotopes of the k > i order are added to the above mixture, the concentration of the i first isotopes will be divided by a number a and become
+
which gives for L i : L4 =
2
/3,
=
1=1
(p nz - l ) ! ( p m’ - l ) ! LI = X m!(p - l ) ! m’!(p - l ) !
1 ~2
2
a,
Also, according to Equations 10 and 1 1 ,
+
,
[Ail)] new mixture
+
(___p m - l ) ! ( p m‘ - l ) ! X m!(p - l)!m’!(p- l ) !
=
[A‘l’lformer mixture U
or
Probability of Obtaining a Giuen Molecule. When the molecule is of the type with identical terminal combinations (i.e. when a , = ai’‘, with 1 i p ) , the probability of obtaining a given molecule can be determined by means of general formulas 13 and 14 by substituting m for m” and a ifor a,’‘, or
<
