1134
JOSEPH mr. BRAUXER
AXD
DAVIDJ. TT’ILSON
Vol. 67
INTItA3POLECULAR ENERGY TRANSFER I N UNIJPOLECULAR REACTIOSS. 11. A JTEAI n”. We would more or less expect v to be the order of molecular vibration frequencies, and, since this is the microscopic decomposition rate of a single oscillator, it is reasonable that it be approximated as constant. Introduction of chemical reaction may be looked upon as a perturbation of the relaxing system: and the eigenvalues of the perturbed energy transfer matrix u ill be perturbed to new values less than or equal to the corresponding eigenvalues of the unperturbed matrix, since the perturbation is negative semidefinite. l4 I n particular, XI, which is zero (according to equation 4) in the absence of chemical reaction, will be perturbed to n small negative value. The results of hand calculation on simple cases indicate that the only eigenvalue for which AXj, the perturbation, is comparable to or larger than Xj in magnitude, is XI. We shall accept the unperturbed values of the Xj, j # 1, as adequate approximations and calculate improved values for the X1. First, we employ a perturbation method which, as wc shall see, yields Kassel’s result for the rate of decomposition of activated molecules. The principle of detailed balance applield to an isolated molecule yields
+
where Pn,n+l;m,m-l is the probability per unit time for 1, m .+ m - 1 for the two the joint transition n -+ n oscillators, and a is a proportionality constant. If we let A(&,iz, . , . is)be the concentration of reactant molecules having i1 quanta in the first oscillator, etc., then we can write out the equations goveriiing the time dependence of this concentration in the absence of collisions and chemical reaction. For even small values of s and n (the total number of quanta in each
+
a
c
molecule; n =
ik),
these equations are quite un-
Ic=l
wieldy. However, in a given chemical reaction it is often one particular reaction coordinate of the reactant molecule which attains a critical value; we let the reaction coordinate’s degree of freedom correspond to oscillator number 1. The remainder of the reactant molecule we represent as an (s - ])-fold degenerate oscillator containing (n - il) quanta; the reactive oscillator then undergoes joint transitjons with this ( s - 1)degenerate oscillator. We the transition probabilities of equations 1 to describe the joint transitions of a 1-oscillator and an (s - 1)-oscillator; the result is
+
+
P(i1, n - il; il 1, n - il - 1) = a(il l ) ( n - il) P(i1, n - il; il .- 1,n .- il 1) = ail(n + s - il - 1 ) (2) where P(il, n -. il; il 1, n - il - 1) is the probability per unit time that one quantum will be transferred to the reactive oscillator from the rest of the molecule, and n is the total number of quanta in the molecule. Let Ai, be the concentration of reactant molecules having a total of n quanta, i of which are in the reactive oscillator. Then, in the absence of external perturbations and terms corresponding to chemical reaction
+
+
+ l ) ( n + s - i - 2)Ai+*,, + ai(n - i + - ~ [ +(i
i(n
+ s .-
l)Ai-l,n
i - l)]Ain, i,
+
- i) 0 , 1, 2,. . .n (3) I)(%
where
gs(m) = (m
=
~ i j ( ~ ) g s (- jl )
+ s - l)!/[m!(s - I)!]
(5)
We perform a similar transformation on M(n) yji(n)
=
[gs-l(j)
I-’/* M,i[gs-l(i)
(6)
This converts M(”) into a symmetric matrix g(n) having the same eigenvalues. One can then readily show tk% X1(d [(gs-l(n))”*, (g,-l(n - I ) ) ~ ’ ~. ., . (gs-l(~))’/x~
(7)
dA_in_ _ - a(i dt
(i)
~ j i ( ~ ) ~ ’ - 1
(12) K. E. Shuler, J . Cham. Phya., 83, 1692 (1960); seg also A. I. Qsipov, Doklady Ahad. Nark SSBR, 130, 623 (1960).
is the eigenvector of g(n)corresponding to XI = 0 by direct multiplication (13) See also, J. Brauner, Ph.D. Thesis, The University of Rochester, 1962, p. 32. (14) R. Courant and D. Eilbert, “Methods of Mathematical Physics,” Interscience Publishers, New York, N. Y.,1953.
IZ(n)x2(n) =
0
n+ 1
Ain(t)
= k=I
The perturbation is approximately given by
hkCinl< exp(X14
(10)
where the Cink are the elements of the kth right eigenvector of the matrix M(”) (including the chemical sink TABLE I EFFECT OF CHANGIXG v s = 3; n* = 16; v = 10 (first set), 1 (second set) N o . of quanta
in molecule
16
where all the elements of the matrix vanish except those along the main diagonal which correspond to n* or more quanta in the reactive oscillator. Doing the multiplication indicated and simplifying yields Ah, =
A1
=
-V
+ +
n ! ( n - n* s - l)! (n - n*) !(n s - 1)!
(9)
17 18 19 20 21 16 17 18 19 20 21
Kassel’s m.d.r. hKassd(n)
Our m.d.r. Xi(n)
X K ~ B S ~ I / ~
6.536 E-02 1,754 E-01 3.158 E-01 4.762 E-01 6.493 E-01 8.300 E-01 6.536 E-03 1,754 E-02 3.158 E-02 4,762 E-02 6,494 E-02 8.300 E-02
4,105 E-02 9.263 E-02 1,477 E-01 2.038 E-01 2 . 601 E-01 3.162 E-01 G . 161 E-03 1.609 E-02 2,834 E-02 4.201 E-02 5,652 E-02 7,152 E-02
1.592 1.893 2.138 2.336 2.496 2,625 1.061 1.090 1.114 1.134 1.149 I , 161
Thus, if we assume an initial random distribution of TABLE I1 quanta among the oscillators of the molecule, we find a EFFECT O F INCREASING THE ACTIV.4TIOiK E N E R G Y single time constant, X1 for the exponentially decaying s = 3 ; v = 10; n* = 9 (first set), 16 (second set) probability that the molecule has not reacted by a time KO. of quanta Kassel’s m.d.r. Our m.d.r. t , and this time constant is exactly that calculated by in molecule XKassel(n) Xi(n) XIin9sel/X1 Kassel theory. This is due to the fact that ex9 1,818 E-01 9,216 E-02 1.972 actly corresponds to an initial random distribution of 10 4.545 E-01 1.928 E-01 2,357 quanta. Use of a simple modification6 of Slater’s 11 7,692 E-01 2.937 E-01 2,619 12 1,099 E-00 3.931 E-01 2,795 “new approach to unimolecular rate theory” then yields 13 1,429 E-00 4,905 E-01 2.912 Kassel’s results for the unimolecular rate constant as a 14 1 ,750 E-00 5.860 E-01 2,986 function of pressure. 16 6,536 E-02 4,105 E-02 1.592 This rederivation of Kassel’s theory does not give us 17 1.754 E-01 9,263 E-02 1.893 the further refinement in unimolecular rate theory that 1,477 E-01 2,138 18 3,158 E-01 we desired. We therefore calculated the exact values 2.038 E-01 2,336 19 4.762 E-01 of the perturbed XI’S; two starting values taken were 2.601 E-01 2.496 20 6,494 E-01 X1 = 0 and XI = X)xassel (equation 9)) and then a modi3,162 E-01 2.625 21 8.300 E-01 fication of Newton’s metshodwas used through as many TABLE I11 iterations as were necessary t o obtain precisely the root of the secular equation which was nearest to zero. The EFFECT OF CHANGIYG s computation v a s done on the IBM 7070 of the Univerv = 1; n* = 16; s = 3 (first set), 7 (second set) Our m.d.r. To. of quanta Kassel’s m.d.r. sity of Rochester’s Computing Center, and is described in molecule XKam 1(n) h(n) XRassel!h in detail in reference 13. The program also calculated 16 6.536 E-03 6.161 E-03 1.061 the Kassel values of the microscopic decomposition 17 1.754 E-02 I . 609 E-02 1,090 rate (m.d.r.) from equation 9. The results are given in 2.834 E-02 1,114 18 3.158 E-02 Tables I, 11, and 111. The parameter v is v / a , and v 4.201 E-02 1.134 19 4,762 E-02 in equation 9 has been replaced by v. The numbers 20 6.494 E-02 5.652 E-02 I , 149 following the E’s are exponents of 10. Table I shows 7.152 E-02 1.161 21 8,300 E-02 that increasing the magnitude of the chemical sink 16 1.340 E-05 1.293 E-05 1.037 perturbation Y increases the magnitude of XI, but not by 17 6.934 E-05 6.341 E-05 1.094 as much as one would expect from Kassel’s theory; a 18 2.080 E-04 2.019 E-01 1.030 particular h1 is always less than the corresponding 19 4,743 E-04 4,598 E-04 1.032 8.850 E-04 1.031 20 9,121 E-04 XKassel. We also observe that the departures of hl 1 ,513 E-03 1.031 21 1.561 E-03 from X K increase ~ ~ with ~ the ~ number ~ of quanta in the terms) aild the hk are determined by the initial condimolecule, for fixed a*, s, and u. Table I1 shows that tions a i n ( 0 ) . Note that this solution holds only in the increasing n* decreases the degree of departure of a of collisions, Then the fraction of reactant particular h1 from the corresponding h ~ ~ Table ~ ~ ~ absence l . molecules in the nth energy state a t time t is simply I11 shows that increasing s decreases both XI and hKasse1 and decreases the discrepancy between them. The n n last set of hl’s in this table shows the effects of machine P(n,L) = C a 4 i n ( t ) / C A i n ( 0 ) i=O i=O truncation error, but this does not affect the qualitative n+ 1 conclusions. = C d,l, exp(hd (11) We now are in position to calculate the unimolecular k=l rate constant as a function of pressure. We proceed as where fallows, The solution of equations 3 yields
11317
ISTRAMOLECULAR EKERGY TRANSFER IN UNIMOLECCLAR REACTIOSS
May, 1963 n f l
dnk
= 1
k=l
By following Slater,3bs15as indicated in our earlier paper,6 one finds that the probability that a molecule containing n vibrational quanta will react (in the absence of collisions) in the time interval t to t dt'is
+
-bP(n't) - dt dt
n+l
=
E
-dnkhk
exp(Xd)dt (12)
k=l
0.01
The fraction of molecules which react before suffering a collision is then given by
=c nfl
L
0.01
-dnkhk
k=i xk
fb U
koiM/k
0.01 0.01
0.1
(14)
If the sum over the index k in equation 14 includes only one term, we simply recover the general Lindemann expression. Properly at this point we should 1 eigenvalues and eigencompute exactly all of the n vectors for the nth level, compute the d n k by assuming initial random distributions of quanta among the oscillators, and do the indicated sumination. This would have been ruinous financially, so we simplified and approximated as follows. We took the exact values of the X1's as calculated previously, approximated the h2's by as, the value computed in the absence of the chemical perturbation from equation 4, and ignored the higher eigenvalues altogether. (We thereby lump together the effects of all of the higher eigenvalues and associate with them a single time constant as.) The approximation of Xz by as mas checked for some simple cases. For n = 3 quanta, n* = 3 , s = 4, v / a = 2 , the approximate value for X2,'a is -4. the exact value (-4.55) is some 147, larger. This Ti-as the largest discrepancy we observed. The values of k / k a are very insensitive to this type of error, as indicated by some simple model calculations. We must still calculate dnl and dPz. We know that d,l dnz = 1 from equation 11. Tie assume that a t
+
(15) X. B. Slater, J . Chem. Phys., 34, 1256 (1956),
1.0 k~M/k,.
10
Fig. 2.-Comparison of Kassel theory (curve I ) to the new theory (curve 3): n* = 30, nmax= 50, s = 6, v = 10, &.lb = 300"K., and 5" = 298'K.
,
0.1
0.01
+
*
Fig. 1.-Comparisoin of Kassel theory (curve 1) to the new theory (curve 3): n* = 30, nmax = 50, s = 3, V = 10, Orib = 300"K., and I' = 298°K.
O.Ol}/
kA
10
(13)
where b is the gas kinetic collision constant for A-SI collisions, and h is assumed to be present in trace quantities in comparison to the inert gas If. We now multiply this factor by the rate at which reactant molecules are being excited to the (degenerate) energy state having n quanta to obtain the rate of formation of products from this state. We here make the usual assumption of a strong collision mecha n i ~ m , so ~ ' that the rate of activation of molecules to the nth state is just AbMP, where P , is given by gs (n) exp(-nhv'/kT)/[l - exp(-hv'/kT)], Y' is the frequency of the degenerate oscillators, s and gs(n) have been previously defined, and the other symbols have their usual meanings. To obtain the total rate of reaction, we merely sum the rate of reaction from each activated state ( n > n*) over all activated states; the result is
=
I 1.0
0.1
,
,
1.0
10
ka;M/k,.
Fig. 3.-Comparison of Kassel theory (curve 1) to the new theory (curve 3): rll* = 30, nmnx= 50, s = 6, u = 50, o v i h := 300°K., and T = 298'K. 1.0
0.01
1.0
0.1
koM/k
.,
10
Fig. 4.-Comparison of Kassel theory (curve 1) to new theorv (curve 3): n* = 30, nmax= 50, S = 6, V = 100, Ovib = 300'K., and I' = 298°K
the instant of excitation a molecule having a total of n quanta has a probability of decomposing per unit time given by 1rassel; this yields
+
in addition to d , ~ dnz = 1. We solve these equations simultaneously for dnl and dna.
KENNETH A. ALLES AYD IT7.J. MCDOWELL
1138
The expression for the uiijmolecular first-order rate constant is then given by m
- P,
k = b:ll n=n*
(16)
where kz = as for all n, and the X1)s are computed for the various values of n as indicated above. Equation 16 was then evaluated by machine computation, and plots of log (k,/lc,) vs. log (koM/k,) were made. ( k , = iimlc; Ido = lim /~/~11.)Equation iM-
m
-11-0
15 implies that 12, calculated by this theory is equal to k , calculated by Kassel theory for a model having the same values of s, v, hv’/kT and v.*; the two theories also lead to the same value of Ita. The salient features of the results are illustrated in Fig. 1 through 4. A comparison of Fig. 1 and 2 shows that iiicreasilzg the number of degrees of freedom decreases the discrepancy between the Kassel theory curves and those obtained by our theory. Coniparison of Fig. 2, 3, and 4, in which v / a , the ratio of the cheniical sink rate factor to the parameter governing intramolecular energy transfer, takes on the values 10, 50, and 100, respectively, indicates that decreasing the cxffi(ieiicy of intramolecular energy transfer causes a
Vol. 67
very marked broadening of the transition region between the low and high pressure h i t s . 6vib ii? all cases is equal to kv‘/li. As we had predicted earlier,6 the weak intramolecular energy transfer theory always leads to curves having a broader transition region between the low and high pressure limits. Variation of &,b end n* produced relatively niinor effects; increasing n” tends to make curves calculated by the two theories more similar. Machine calculations on the vibration of highly 4c 9-11 indicate that energetic triatomic energy “scrambling” among the normal modes is extremely rapid; this casts doubt on the utility of the model discussed here. We mould like to mention a suggestion made by Prof. Slater that the effects of anharmonicities may be substantially less extreme in more complex niolecules; this possibility makes the model discussed here somewhat more reasonable. Konetheless, as pointed out by one of our referees, the effects of anharinonicity mill certainly be large juqt prior to the molecde’s decomposition, no matter how large the molecule may be; the effects of anharn3onicities in such highly distorted molecules may 11-ellbe quite important. TI7e are currentlv working on machine calculations to investigate Slater’s rather reasonable conjecture. Acknowledgment.-We are indebted to Prof. Frank Buff and Dr. Everett Thiele for helpful discussions.
THE THORIUM SULFATE CQlIPLEXES FROM DI-n-DECYLL4311SE SULFATE EXTRACTION EQCILIBRIA BY KENNETH A. X L L E NAND ~ W.J. MCDOTVELL Oak Ridge Xational Laboratory, Oak Ridge, Tennessee3 Received December 14, 1969
A general method is described for obtaining aqueous complex formation constants from solvent extraction equilibria, The method involves experimentally controlled constancy of the chemical potentials in the aqueous phase, constant composition of the equilibrium organic phase being used a3 the criterion. A single-parameter Debye-Huckel equation with an arbitrarily fixed distance term is used as an analvtic model for the ionic interactions of the aqueous species. At constant sulfuric acid activity, constant extra-tant concentration, and constant organic thorium molarity the distribution of thorium between di-n-decvlamine sulfate in benzene and aqueous phases of varying sulfate ion concentrations is shown to lead to the fo!lming values of the hithertounreported formation constants of the thorium tri- and tetra-sulfate complexes, at zero ionic strength, K23 = [Th(S04)a-31 = 5.7 f: 1.2 and KS4= [Th(S04)4-41 = 0.009 f:0.003. [Th(SOa)r-‘I [SO*-* I [Th(SOa)zl[SO,-’]
Introduction Formation constants and energy data for the aqueous mono- and disulfate complexes of thorium have been r e p ~ r t e d . ~A search of the literature has been unsuccessful in finding any published reference to species containing more than two sulfates per thorium. The existence of such negatively charged species has been inferred by Kraus and Nelson4 from the negative slopes of plots of thorium distribution ratios bet-cveen an anion-exchange resin and aqueous phases of varyinp sulfate ion concentration. Corresponding solvent extraction data, taken under experiniental conditions ensuring constant composition of the organic phases, became available in the di-n-decylamine sulfate ex(1) Deceased. (2) operated for the U.S.A.E.C. by Union Carbide Nuclear Company. (3) (a) E. L. Zebroski, 1% UT.Alter, and F. K. Heumann, J . A m . Chem. Sac., 7 8 , 5646 (1951): (b) A. J. Zielen, %bad.,81,5022 (1939). (4) K. A. Kraus and F. Kelson, ORNL, private communloation, 1959.
traction system5 The present paper describes a general method, incorporating Debye-Huckel activity corrections, for obtaining complex formation constants from such data. The method is used in the computation of constants for the formation of the thorium triand tetrasulfate complexes. Experimental The materials and procedure used in making the thorium distribution measurements have been described previously.6 Solutions (Sa2SOI-H2S04) of varying sulfate ion concentration and constant sulfuric acid activity (6.4 x 10-5 31) were prepared according t o data compiled by B a e ~ . While ~ these tests extended t o higher ionic strengths than were covered in Baes’ treatment, titration of the equilibrated organic phases for total acid confirmed the constancy of aqueous sulfuric acid activity to within the precision of the titration, f:O.5%. Other parameters were held as constant as practicable: total amine molarity 0.1 f:0.0005, temperature (5) W. J. XlcDowell and K. A. Allen, J . P l y s . Chem., 66, 1358 (1961). (6) C. F. Baes, J r . , J . A m . Chem. Sac., 79,5611 (1957).
