KIKETICS OF ISOTOPIC EXCHdNGE REACTIOKS
777
The authors nish to thank Professor John E. Willard of the Department of Chemistry of the University of Wisconsin for making available the facilities of his radioactive tracer laboratory. REFERESCES (1) BANCROFT, W. D . , A N D CALKIS,J. B . : J. Phys. Chem. 39, 1 (1935). Z ,F . : J . Textile Inst. 28, T2i (l93i). (2) D A V I D S OG. (3) GLhDSTONE, J . : Jahresher. Chem. 6, 823 (1852). (4) HESS,K . : Z. angew. Chem. 38, 230 (1925). (5) JOYSER,R . A , : J . Chem. SOC. 121, 1511 (1922). (6) II.: J. SOC.Chem. I n d . 60, 177 (1931). (11) THIELE, E . : Chem. Ztg. 26, 610 (1901). (12) VIETIG, W.:Ber. 40, 38i6 (19071.
T H E KISETICS OF ISOTOPIC EXCHXSGE REACTIONS' T. H. NORRIS Department of C'hentistry, Oregon State College, Cowallis, Oregon Received July 6 , 1949
I t has been shown (4, 8) that, in a radioactive (or separate isotope) tracer study of the rate of an exchange reaction carried out at chemical equilibrium (as is normally the case), no matter Tvhat the actual kinetics, the course of any given run mill be governed by a firstorder rate law. Duffield and Calvin (4) have given an equation which may be used in computing the rate of an exchange reaction, but since, in practice, ne have found that the application of this equation and others like it tends to become rather confusing, it has seemed worthwhile to discuss this topic in some detail. Following the usage of Duffield and Calvin, \ye represent our exchange reaction by the equation (the asterisks having their usual significance) :
BX*
+ AX
=
.4X*
+ BX
in which the concentrations are represented by x = (AX*) y = (BX*)
a = (AX*) b = (BX*)
xo = x and yo = y at t xs = x and ys = y at t
+ (AX)
+ (BX)
=
0
=
P
Published with the approval of the Oregon State College Monographs Committee. Research Paper S o . 154, Department of Chemistry, School of Science, Oregon State College.
778
T. H . KORRIS
and for which R = rate of reaction, F = fraction of exchange equilibrium, and x = x / a ; y = y / b (see table 1). The equation given by Duffield and Calvin is equivalent to
R
=
 “(“>log t a+b
(1  F )
where F , the fraction of exchange equilibrium which has been attained at time t , is given by
or x 
F=
2,
Xh
50
when xo # 0
(2‘)
Equation 2 is a special case of equation 2‘. In actual practice one would most frequently employ equation 2‘ for separated isotope tracers and equation 2 for radioactive isotopes. It often happens in practice that one of the reactants is present in large excess. Obviously, in such a case, equation 1 may be simplified to:
R
= 
2.3a
 log
t
(1  F )
b >> a
(1’)
R , the rate of the reactionbetween AX (+ AX*) and BX (+ BX*), represents the total number of “exchanges,” both radioactive and nonradioactive, per unit time, in terms of the concentration units being used, e.g., moles per liter. It is, therefore, quite independent of the concentration of the tracer and evidently remains constant throughout a run. It is not, however, in any sense a “rate constant,” but is some function of any of the factors which may govern the rate of a chemical reaction, such, for example, as the concentrations a and b. Kotice, though, that a possible functional dependence on a and b is quite apart from the existence of these quantities in equation 1, l’, or 1”. Any kinetic study has as its object the ascertainment of the nature of this function; this is done by conventional kinetic methods, R being directly comparable from one run to another. For example, were two runs idehtical in all significant respects except concentrations, R should be the same in both if the reaction is zero order, but not othermise. Were our reaction bimolecular we would get R = (ab) (constant). In equations 2 and 2‘ the x’s, of course, are defined as concentrations, but, since they occur as a dimensionless ratio, they may be expressed in any units which, for a given run (but not necessarily from one run to another), are proportional to the concentration of the radioactive molecules being formed in the reaction. I n practice this means either the total activity or the specific activity
779
KIKETICS O F ISOTOPIC EXCHASGE REACTIONS
of the initially inactire fraction, \There the units might be respectively, for example, counts per minute, and counts per minute per mole (or, if all activities are measured in a common chemical form such as BaC03, the latter might be counts per minute per milligram). .Uternatively, if the tracer is a separated isotope, such as C’3 or O”,the x’s refer t o that compound Tvhich initially had the normal isotopic composition and, if equation 2 is used, will be expressed as mole T.IBLE 1 Expressions for F in various w i t s TOTAL ACTIVITIES
SPECIFIC ACIIYIIIES OR YOLE FRACTIOK EXCES5
(1‘)
1
RE5TRICTlON
x t

XW
x
(3’) 
=
Yo a
(4’) b (t 5 ’,)
X *

b
>> a
Yw
. x
Y Yo
a>b
YO
 Y,
(6’) ( 1
+ $)(I  in)* b>a a
>> b
*Expressions most often useful. t I n terms of mole fracfions,these expressions become
F =   x   0
*.,
XJ
yoy
Yo  Y.,
fraction excess, Le., mole fraction of the separated isotope minus the normal mole fraction as it occurs in nature. This amounts t o the same thing, of course, as substituting mole fractions for the x’s in equation 2‘. The expressions for F contain the quantity z,, which is often not determined experimentally,* so it is perhaps worthwhile to set down in table 1 a number of alternative expressions for F which may be used in order to obtain a value for 2 The determination of z, is required in some work and is always useful of the experimental method.
&s
a further test
780
T. €3. NORRIS
R from equation 1. These are all readily derived from the selfevident relationships3
1
YG
+ z0
= y
+ x = + zcc
(3)
and the defining equations for F , equation 2 or 2‘. Since, under normal counting conditions, the units of z and y, i.e., concentrations, are proportional to total activities (in any given experiment), z and y may equally well be expressed in terms of the latter, as indicated by the heading of the first column. The second column is given in terms of the quantities x = z / a andy = y/b, which, then, represent specific activities. Alternatively, the latter expressions apply if the units are mole fraction excess of stable tracer isotope (excess over the “natural” mole fraction). The subscripts have the same meaning as hitherto. I n table 1 expressions 1 t o 4’ correspond to cases nhere the rate of exchange is followed by measuring the activity (or isotopic composition) of the substance initially inactive (or of normal isotopic composition), whereas expressions 5 to 8‘ correspond to the alternative procedure of following the decrease in activity of the substance initially active. Expressions 9 and 9’ correspond to the procedure of following the activity of both of these fractions. This last may be used as an aid in testing the experimental procedure. Expressions 3 to 4’ and 7 to 8’ correspond, as indicated, to a large excess concentration of one of the reactants. The equalities in expressions 3 and 3’ correspond to the fact that for b >> a, y or y remains constant and equal to yo or yo.For this same reason expressions 7 and 7‘ are of no practical value; they are included for the sake of completeness only. Likenise expressions 3 and 4’ will frequently be of no value, since the ratio b / a or aice versa, may be difficult to establish accurately, if the concentration of the excess component is unknown (cf. the last paragraph of this paper). I t is to be noted that in these cases, this (excess) concentration is not needed, since it drops out of equations 1’ and 1” for R. Since, in addition, x, and y, are not normally measured directly, expressions containing them are of secondary importance. (One may determine z, or y indirectly, but this is tantamount to using one of the other expressions for F.) Thus we find, as the expressions which most fre3 Throughout this paper, i t is assumed t h a t radioactive decay is corrected for in conventional fashion. These relations and, in general, the quantitative validity of tracer methods depend on the assumption t h a t the tracer and “traced” isotopes are chemically identical. I t has long been recognized t h a t such an assumption in the case of the hydrogen isotopes is far from valid. Recent v o r k has shown t h a t other isotopes, with higher atomic weights, such as C13 and 0 4 , may also, in certain cases, exhibit a surprisingly large isotope effect, although the magnitude of the effect in various reactions is not yet clearly established (1, 2 , 7, 9 , 12, 13, 14). For instance, Yankxich and Calvin (14) report an effect in the decarboxylation of (214labelled bromomalonic acid which, if correct, could never be ignored. I t is evident t h a t in quantitative tracer studies, where such an effect is appreciable, it may not be neglected but should be corrected for.
KIKETICS O F ISOTOPIC EXCHANGE REACTIONS
i81
quently ~ 1 1 be 1 used, those marked with an asterisk. Substitution of expressions 6 and 6' in equation 1 leads with slight rearranging to the equation given by Davidson and Sullivan5 (3). Discussing now the concentrations a and b : inspection of the equations in which they occur indicates that these quantities must always be expressed in the same units. Furthermore, these must be gramatom units (see the discussion below in regard t o molecules such as AX*,etc.), e.g., pressure at some standard temperature, gramatoms per liter, etc. Since we are here dealing v i t h a firstorder rate law, it is possible t o obtain a value of R from the slope of a semilogarithmic plot giving the rate of approach t o equilibrium, but it is to be noticed that R is not equal t o this slope, as may be seen from equation 4 below. This procedure has the advantage that no knowledge of a zero time is necessary, but it entails a measurement of the extent of exchange for fixed conditions a t af least two different values of t . From equation 1 we obtain
For Q we may substitute, besides (1  F ) , (zm 2) or (y  y), where the x's and y's may be expressed as concentrations, mole fractions, specific activities, or total activities. Values for (1  F ) may be obtained from table 1,but it is perhaps worth noting that in many such cases the expression can be rearranged to a form with a constant denominator. In such a case, of course, only the numerator need be substituted for Q in equation 4.Thus, for instance, application of expression 4 (table 1) leads to a value for Q of (yo  z). The foregoing discussion has general ralidity for a homogeneous exchange reaction between a pair of molecules each containing one exchangeable atom. Friedlander and Kennedy ( 5 ) discuss the case \There one or both of these molecules have more than oneexchangeable atom (e.g., AX2or BX,). As these authors indicate, the applicability of the treatment here under discussion remains unaltered, so long as all the X atoms in a given molecule are, a t least in an exchange sense, equivalent and concentrations are expressed in gramatoms per unit volume of X (in either exchanging molecule). Nonequivalent X atoms in a given molecule will, in the general case, exchange at different rates. The observed total rate is then a superposition of two or more firstorder rates, and the present simple treatment cannot be applied, unless the different rates are of completely different orders of magnitude. For heterogeneous exchange reactions, the expressions in this paper have 6The terminology used by Davidson and Sullivan differs somewhat from t h a t used in the present paper
782
T. H. KORRIS
only limited validity. Zimens (15) has discussed the kinetics of such cases. His treatment has particular reference, but is not limited, to exchange hetn.een a solid and a liquid (or gas) phase. Expressions are derived, applicable to rate determination by three different alternative processes: (1) diffusion through a layer of the less dense phase adsorbed on the surface of the more dense; ( 2 ) chemical reaction a t the interface; (3) diffusion vithin the solid phase. The expressions for the first trvo cases, while more directly applicable to the study of heterogeneous systems, are equivalent to those in this paper; hence these remain valid. The present treatment, however, cannot be applied to Zimens’ third process. This is because the atoms in the interior of the solid are not equivalent to those on the surface and consequently exchange a t different rates. The rate law becomes a summation of several rate terms, as for the case in the previous paragraph.e Zimens (15) gives experimental criteria for distinguishing between the different ratedetermining processes. Here, however, our primary concern is whether the simple firstorder rate laiv is applicable. The test lies in a plot, for a given run, of log (1  F ) versus t , using several different values of F (and t ) . 4straight line indicates applicability. I n certain cases, this straight line may not run through the origin, as, for example, was observed by Prestwood and Wahl (11) (\There separation errors, or separationinduced exchange occurred). A curvedline plot indicates a lack of equivalence of exchanging atoms, heterogeneous exchange, or faulty technique. For heterogeneous systems the units in vhich a and b should be expressed tend to become confusing. Following Zimens (E),one may express them as amounts (e.g., gramatoms) ; R is then obtained in corresponding units (amounts1 time). 4lternatively, if concentrations are used, it should again be emphasized that the units of a and b must be identical. Thus, considering the exchange between two substances dissolved each in different, immiscible liquids, a and b cannot be expressed as concentration of each substance in its own phase; rather they might be given as amounts per unit volume of either one phase (or per unit volume of both phases taken together). From this it follows, as a corollary for such a case, that successive runs, to be comparable, must duplicate not only the concentration of each reactant in its o m phase but the phase volume ratio as well. The last statement, based entirely on the equations given for R , involves no assumption, other than the applicability of these equations, concerning the mechanism or rate law of the heterogeneous reaction. It can, however, be supported and extended by arguments involving the inherent nature of such reactions. Heterogeneous rate laws are dimensionally different from homogeneous ones. Thus it is commonly recognized that such geometrical factors as surfacevolume ratios will affect the rates. Fuzek and Smith ( G ) have studied certain catalytic hydrogenations and shown that the total volume of the system must be included as a term in the rate law. Evidently, then, in order for different runs in a heterogeneous kinetic study to be comparable, geometrical factors in general 6 For an experimental illustration of this phenomenon see, for example, the paper by Polessitsky (10).
KINETICS O F ISOTOPIC EXCHASGE REACTIOSS
783
must be maintained constant. This implies, for the example of the previous paragraph, that the phase volume ratio, as before, and, in addition, the total volume of the system must be kept constant. Or again, considering an exchange study between a gas and a liquid, runs being made in sealed bomb tubes: the total tube volume for comparable runs must alnays be the same. If this is impractical, assumptions about the dependence of R on the volume become necessary. In heterogeneous reactions it is sometimes inconvenient, difficult, or impossible to express the concentrations of both a and b in the same units. Here, though, it n 4 l often be obvious that one of the reactants is present in large excess (Le., a >> b or rice z’ersa). I n such a case the equation which will be used (equation l’, I”, i’, or 4” plus an appropriate expression for F from table 1) will contain only the concentration of the component present in small amounts and not the other concentration. Here any reasonable concentration units may be employed. SUMMART
An isotopic exchange reaction must proceed toviard equilibrium according to a firstorder rate law or according to concurrent firstorder l a w . X discussion is given of the implications of this concept for the experimental study of the kinetics of exchange reactions. Expressions are tabulated for the fraction of exchange equilibrium, F , attained a t any time, t , in terms of quantities ivhich are experimentally determinatile. REFERENCES (1) BEECK.O., OTVOS, J. W ,STEVESSOS,D. P . , AND WAGSER,C. D.: J. Chem. Phys. 16, 255 (1948). (2) BIGELEISES, J . , et a i . : Science 110, 14, 149 (1949); J. Chem. Phys. 17. 344, 425, 998 (1949). (3) DAVIDSON, XORMAS,A S D SLLLIVAS,JOHSH . : J. Am. Chem. SOC.71, i 3 9 (1949). (4) DUFFIELD, R . B., ASD CALVIN,M.: J. .4m. Chem. Soc. 68, 557 (1946).
( 5 ) FRIEDLASDER, G . , .4KD K E N N E D YJ., W.:Introduction to Radiochemistry, p . 285. John W l e y and Sons, Inc., S e w York (1949). (6) F ~ Z E J. K ,F., A S D SMITH, H. A , : J. Am. Chem. SOC.70, 3T43 (1948). (7) LISDSAY, J. G . , SICELCHERAN, D. E., A N D THODE, H . G. : J. Chem. Phys. 17,569 (1949). (8) AIcK.ku, H. A . C . : S a t u r e 142, 997 (1938). (9) PITZER,K . S.: J. Chem. Phys. 17, 1341 (1919). (10) POLESSITSKY, .I.:Compt. rend. acad. sci. C.R.S.S. 28, 441 (1940). (11) PRESTWOOD, R . J., ASD WAHL,A. C . : J. Am. Chem. Soc. 71, 3 1 3 i (1949). D . P., Fv.4GSER, c. D., BEECK,O . , ASD OTVOS,J. 17.:J. Chem. Phys. (12) STEVESSOK, 16, 993 (1948). (13) WEIGL,J. W . , A N D CALVIN, SI.: J. Chem. Phys. 17, 210 (1949). (14) YANKWICH, P . E . , ASD CILVIN, 51.: J. Chern. Phys. 17, 109 (1919). (15) ZIMEKS,K . E.: Arkiv Kemi, Mineral. Geol. AZO, S o . 18 (1945); A21, S o . 16 (1946).