The Kinetics of Isotopic Exchange Reactions S. R. Logan University of Ulster at Coleraine, N. Ireland BT52 1SA I t is fairly widely known that in an isotopic exchange reaction, progress toward equilibrium follows an equation comparable t o that applicable t o many reversible isomerization reactions. That is to say, regardless of the actual reaction order, the approach toward equilibrium follows what look like pseudofirstorder equations. Few kinetics texthooks consider the kinetics of these reactions, and of those that do (1.2). even fewer consider the questions of how one might determine the actual order of such reactions. I t is convenient to start with an isotopic exchange in which the isotope effect on rates and equilibria may be assumed negligible. Let us consider the reaction, AX+BX*+AXa+BX
(1)
where Xu denotes a distinctive isotope of element X. Assumine that initiallv lAXl . . = a and IBX'I = b, while IAX'I and [ ~ kare ] zero, we may deduce that sincein the absence of isotope effects the equilibrium constant is unity, the equilihrium concentrations of the two reactant species will he, respectively, a2/(a b) and b2/(a b ) , and those of the products will both he abl(a b). Let us assume that the reaction of AX with BX* is kmetically of orderp with respect t o the former and of order q with respect to the latter. On this basis, the initial rate of the where k. is the reaction will he eiven bv the ~ r o d u ck"aPb9. t , a appropriate racronst&t of.order ( p q ) . ~ h e r e a f t i rin sense. the rate of reaction between AX soecies and BX soecies rkmains constant, since the total concentration of the former remains a t a and of the latter a t b. However, when some exchange has occurred, there are now four different categories of A X B X reaction, namely: (i) That of AX with BX*, which alone is relevant to the forward exchange reaction. Using the variable x to denote the concentrations of AX* and of BX at time t, this makes up a fraction (a  x)(b  x)lab of the total. (ii) That of AX with BX, which has no relevance to the isotopic exchange reaction. (iii) That of AX* with BX*,which is also irrelevantto the exchange reaction. (iv) That of AX* with BX, which is the reverse isotopic exchange and makes up the fraction x2/abof the total.
+
+
+
+
Thus a t time t the rate of the exchange reaction may he expressed as:
The variation with temperature of k' will mirror that of the actual rate constant k,, so values of k' may be used to determine the activation energy associated with the exchange reaction. Rearranging eq 4 we may obtain
T o determine the actual order p of the reaction with respect to AX. a number of exoeriments should be performed in which the initial conceniration of AX is variedwhile that of BX' and the temoerarureare kept consrant. From earh run. the pseudofirst&der rate cons& k' is derived and, using eq (5), In [k'l(a b ) )is plotted against In a to yield the slope, which may he equated to @  1).Similarly, q may be found by varying the initial concentration of BX* while keeping constant that of AX. Some treatments of isotopic exchange reactions, designed to show how the reaction oroeresses toward eauilibrium in a . simple experiment, may express (1,2)the net rate of reaction, not as in eq 3 in terms of the nthorder rate constant, but rather in terms of the initial rate R achieved with concentration of AX equal to a and of BX* equal to b. Thus, instead of eq 2 we have
+
On integration, this leads to
In some instances in the literature a syg~bolother thanR has been used for this initial rate with specified concentrations of each reactant, which would tend to suggest tbat this parameter isa rateconstant. H does, of course, have thedimensions of a rate and it is important toappreciate that eq 7 will not serve t o discover the actual orderif the reaction. The foregoing discussion has been appreciably simplified by the assumption that isotope effects are negligible. Where this is not the case, these are, in general, kinetic isotope effecta on both the forwardand reverse reactions that do not exactly cancel so that there is also an isotope effect on the equilibrium. Consequently, x, is no longer given by ab/(a b), but by a new value that takes cognizance of the equilibrium constant not being unity. I t has been shown (3) that where an appreciable isotope effect exists, the progress of thereaction will stillapproximate closely to eqs 3 and 7 when the correct x, value is used.
+
Integrating and using the fact that x = 0 when t = 0,we have
Thus by plotting In ( x ,  x) against t one obtains k', a pseudofirstorder rate constant pertaining to the initial concentrations of that kinetic experiment. I t is actually equal to the function:
Multiple Exchange
In some instances the reacting molecule may contain more than one atom X tbat mavundereoexchanee with X*. Let us consider the reaction ofVAXa,wiere all tiree X atoms are Volume 67 Number 5
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37 1
equivalent, with BX*, taking the respective initial concentrations to be a and b. Exchange may successively yield the species AXzX*, AXX*,, and A x ' s If at time t these species have respectively attained the concentrations u, v, and w then the concentration of AX3 will have fallen to (a u  u w ) and that of BX* to ( b  u  2v  3w). For simplicity we will neglect isotope effeds. The extent of exchange is no longer represented by the concentration of any single AX3 species, but rather by those of the three product molecules, appropriately weighted, since AX*3 represents the exchange of three X atoms but AXzX* of only one. Thus we may introduce the parameter y to represent the extent of reaction, where y = ( u 20 3w). Moreover, y will represent, at time t, the concentration of

+ +
RY
It is convenient to represent the rate of exchange in terms of the initial rate, R, between AX3 and BX* at these initial concentrations. There are three contributions to the forward rate; those involving AX3, AXzX*, and AXX*,, with respective orobabilities in the ratio of 3:Z:l bvvirtue of the number of fatoms remaining to be exchangld. Likewise, there are three contributions to the reverse process of reaction with BX, namely those with AXzX*, AXX*,, and AX'S, with respective probabilities in the ratio of 1:2:3.Thus we have & R ( b  u  2 u  3 w ) a u  u  w 2u u dt b a ++I 3a 3a

+ + 3w)
(u 2u
b
{
{.+.+;I. u
2u
w
the species CkH,;Di, establishing the various equilibria such as do dz * 2d1. This does not in general occur among hydrocarbons, but, whereas CH4 and CDa do not readily react together (5),NH3 and ND3 readily undergo exchange at room temperature (61,presumably catalyzed by the acidic surface of the containing glass vessel. (iii) Those where the exchange of D for H on the alkane molecule may occur several atoms at a time and where ex; chanee amone the deuteroisomers does not occur. Thus. as the reaction proceeds, the concentrations of the various deuteroisomers need not satisfv the eouilibria such as dn d? = 2dl, though a11 such will eventualiy be satisfied when complete equilibrium is finally established. The derivation of eqs 8 and 9 assumed exchange in single steps, so the counterpart of these equations, using @ instead of y, will be applicable to category (i). With a slight generalization of the argument, the same will apply to category (ii) in which the buildup of the amounts of the dl, dn, isomers follows the same pattern.
+


+
...
Depletion of Orlglnal Substrate
Another equation has been used in isotopic exchangereactions, dealing with the rate of depletion of the original unsubstituted compound. In terms of the simplest exchange process, reaction 1, one can think of the progress of the reaction either as the increase of the AX* concentration or as the depletion of the AX concentration toward their respective equilibrium values, and the equations are identical. For the case of the reaction of AX2 with BX* to yield first AXX* and then AX*,, one may make a distinction. If the initial concentrations are, respectively, a and b and at time t those of AXX* and AX*, are u and u, then those of AX, BX, and BX* are ( a  u  v ) , ( u 2u) and ( b  u  2u). We may, as before, express the rate in terms of theinitial rate R obtained with concentrations [AXJ = a and [BX*] = b.
+
where the equilibrium value y , = 3abl(3a tion we obtain the equation,
+ b). On integra
which may be closely compared to eq 7.
Exchange reactions of the H atoms of simple alkanes with D2. catalyzed by the surfaces of transition metals. have been e&ensively studied over several decades. From'the occurrence of such a reaction on a catalyst surface, several different products are conceivable. From the alkane molecule CkH,, where all the H atoms are equivalent, we may have products CkH,,Di, where i is any integer from 1to m. We may denote the extent of exchange using the parameter @, defined (4) as d = x,
+ 2x, t 3x, + .. .mx,
Making use of the fact that the distribution of X* in species AX2 satisfies the equilibrium, AX, + AX', + 2AXX' that is, that u2/{u(a u  v ) ) = 4, and introducing the variable x = (I u, this equation reduces to
+
"
=
2 i5
(10)
where x i denotes the fraction of the total alkane present as CkH,iD;, termed the di isomer. Of such exchange reactions, three categories may be distinguished. (i) Those where the exchange process of H for D can occur only oneat a time just as was ksumed above for the reaction of AX? with BX*. This has the consequence that at all staaes thevarious deuteroisomers are in such amounts that they are in mutual equilibrium, i.e.,
(11)
Making use of the fact that the distribution of X* in species AX2 satisfies the equilibrium, AX, + AX,' =2AXX' that is, that u2/{v(a u  u)) = 4, and introducing the variable x = u + u, this equation reduces to
do + d, = 2d, d,+d3=2d2
and so on. (ii) Those where, under the conditions for the exchange of DP with CaH,, exchange of H and D readily occurs among 372
Journal of Chemical Education
where the final equilibrium values of x and y are respectively + b)/(2a+ b ) , and y, = 2ab/(2a b). This shows
x. = ab(4a
+
that thedecreaseof (a  x ) toward itsfinalvalueof4a1/(20 + bVis not of the form that could lead. in the present terminology, to the expression: . .
.
early stages of the reaction. However, mathematical simplification of eq 12 must acknowledge the differing values of x , andy,.
..
Clearly, the above equation is "without theoretical basis" (4) when applied to the depletion of A X 2 , and it would seem inconceivable that i t could be ooerative for the exchange  of any larger molecule, AX.. As remarked earlier, in its exchange reaction with BX* the depletion of AX follows a simple reversiblefirstordertype equation. The reason, as demonstrated by eq 12 is simply that the parameters x , measuring depletion of AX, concentration and y, measuring the extent of incorporation of X* in species AX,, are, in the unique case of AX, oneand the same. For other values of n, these two are almost the same in the
Acknowledgment
The author wishes to thank G. P. Shannon of the Department Mathematics for Literature Clted 1. Frosf,A. A.;Pearson, R. 0,Kineticsond Mechanism, 2nded.; Wiley: New York. 1951: p 192: Moore, J. W.: Pearson. R.G.KinelicaondMeehnniam,3rded.: Wiley: New York, .?.o...,..
.ZOL,"OLL.
Frbdlsndor, G.; Kennedy, J. W.; Miller, J. M. Nucleor m d Rod~orhemillry,2nd ed.; .7 . Wiley: New York. 1964: p 196. 3. Harris, G. M. Trans. foreday Soe. 1951.47. 715. 4. Kembsll,C.Bull.Soc. Chim.Belges 1958.67.373. 5. Morikawa, K.; Benedict, W. S.:Taylor. H.S. J.Am. Chem Soc. 1936.58.1445 6. Stedman,D. F.J. Chern.Phys. 1952,20,718.
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