J O U R N A L OF T H E AMERICAN CHEMICAL SOCIETY (Registered in U. S. Patent Office)
VOLUME
(0Copyright, 1961, by the American Chemical Society) SUMBER 20
SOVEMBER 6, 1961
83
PHYSICAL AND INORGANIC CHEMISTRY [COKTRIBUTION FROM THE CHEMISTRY
DEPARTNEXT, UNIVERSITY
OF MARYLAND, COLLEGE PARK, MARYLAND]
II','
The Kinetics of Three-step Competitive Consecutive Second-order Reactions. BY W. J. SVIRBELY AND
JAY
A. BLAUER
RECEIVED MAY4, 1961 ki The rate equations for a three-step competitive consecutive second-order reaction of the type A B +C E, A kz k3 Adt. When A0 = 3B0, the soluC +D E, A D +F E has been solved in terms of a variable X where X =
+
+ +
+
si
+ +
+
+
tion is A = GleckiX Gze-kzX G3e-kaX where GI, Gz and G3are constants involving various combinations of k1, kl, k3 and Bo. Taylor's theorem was used t o expand the function A about the first approximations of k1, kz and k3 ( i , e , , Ok,, Ok2 and Ok3). If all higher order partials are neglected, then the equation in A becomes; A = f(Ok1, Okz, Ok3) af Akl bf Akz aki , o bk? I O !?! i A k3. The problem of evaluating Akl, Akn and Aka was accomplished by a least squares solution utilizing all experi-
+
~~
+
,
+
%o
mental time-concentration data. An iterative procedure was developed for carrying out the operations on the I.B.M. 704 electronic computer. On multiplying the equation for A by ek$ differentiating the resulting equation and repeating the process using eksX and ekaX successively, a third order differential equation in A and X was obtained which, after several (kl kz k 3 ) A * (kl k3 k2k3 klkn)A** klk2k3A*** = ( l / p k l k z successive integrations, leads to: A - A .
+
klk3
+
+ +
+
+
+
+ 3/2k2k3)X2B~+ (2k1 + 3kz + 3ka)XBo where A * = S,hAdX, A** =soX.4*dX,A*** = SoAA**dh.
+
Taylor's theorem
was used to expand this function of A about first approximations of k1, kz and ks and an iterative procedure was developed t o solve for Akt, Akz and Ak3 utilizing all experimental time-concentration data.
Introduction Recently, the kinetics of three-step competitiveconsecutive second-order reactions was investigated mathematically in terms of general variables which in principle would apply to any reaction of that kinetic type. The resulting analysis was applied to the alkaline hydrolysis of 1,3,5-tri-(4carbomethoxypheny1)-benzene which, because of its size and corresponding absence of interaction between the carbomethoxy groups, led to the result that the rate constants are in the statistical ratio k1:k2:k3 = 3 : 2 :1. However, the above procedure becomes too involved to be practical when there is no criterion for determining beforehand what relationship, if any, exists between kz and k3 for a particular reaction. This is the situation which exists when interaction between groups does occur or when the k's for a given
reaction vary as the result of changing the solvent composition. Therefore, i t becomes necessary for us to reconsider the problem and see whether a more practical solution can be obtained. Mathematical Analysis The reactions t o be considered are kl A+B--+C+F kz A C -+ 1)
+
+F
k3
A + D + E + F The pertinent rate equations for the above steps in terms of the molar concentrations A, B, C and D are
(1) Abstracted from a thesis submitted by Jay A. Blauer to the Graduate School of the University of Maryland in partial fulfillment of the requirements for the degree of Doctor of Philosophy, (2) Presented in part a t the Xew York City Meeting of the American Chemical Society, September, 1960. (3) (a) W. J. Svirbely, J. A m . Chem. SOC.,81, 255 (1959); (b) W. J. Svirbely and H. E. Weisberg, ibid., 81, 267 (1959).
4115
_-
dD - k2AC dt dt
=
- k3AD
kaAD
41 16:
W. J. SVIRBELY AND
JAY
A. BLAUER
Vol. 83
Let us define a new variable, X, so that X =
sot
Adt
2
it follows then that dX = Adt (7) On substituting equation 7 into equations 1 through 5, we obtain equations 8, 9, 10, 11and 12.
aT= - k l A - k2C - kaD dA
-dB =
dX
(8)
~
bsJ
-klB
ah3 0'
dE -
KID dX Equations 9, 10, 11 and 8 are integrated in that order. Constants of integration are evaluated in each appropriate case from the boundary conditions, which are: X = C = D = 0, A = Ao, and B = Bo a t t = 0. If the initial concentrations of species A and B are adjusted so that A0 = 3B0, then one obtains
The problem now becomes one of the evaluation of A k l , Akz and Aka. Once these values have been obtained, the first estimates "1, Okz and Ok3 may be corrected for the error terms and the process repeated with new estimates of k l , kz and ko. As the process is repeated the Akl terms will become increasingly smaller. In the evaluation of equation 16, we are solving for the three unknowns Akl, Akz and Aka. Therefore, we need at least three simultaneous equations, ;.e., three different sets of values of A and A. The values of X are obtained from the graphical integration of an A vs. t plot. Actually a run may provide as many as 15 experimental points, $.e., 15 sets of values of A and X. The problem therefore becomes one of using all fifteen equations in the total solution for Akl, Akz and Aka. The solution of equation 16 for the correction terms is accomplished by the method of least squares using all fifteen equations. For simplification, we make the following definitions
J
While we now have a solution for A in terms of the rate constants, it is in an impossible form for the direct evaluation of the rate constants. Ultimately, three different procedures were developed (two of which are described). Method 1.-On rewriting equation 13 for simplification only, equation 14 is obtained S = Gie-krX G2e-ktX Gse-k~X (1-1) The definitions ot S,GI,GZand GI are obvious on reference to equation 13. If we let "1, Okz and Ok3 be a set of initial estimates of the actual rate constants kl, kz and k3, equation 13 may be expanded about these initial estimates via a Taylor's series expansion to give
+
+
SI0 is the value of A/Bo calculated via equation 13 when Okl, Okz and Ok3 are substituted for the actual rate constants k l , kz and k t . dS/dkl IO, d S / d k 2 1 0 and d S / d k a 10 are the partial derivatives of A/Bo with respect to the three rate constants. These partial derivatives are evaluated from equation 13 when the rate constants k l , k2 and k3 are replaced by the estimates Okl, Okz and Ok3. H represents all higher order partial On neglecting H and replacing (k1 Okl), derivatives. ( k - Ok2) and (ka Ok3) by Akl, Akz and Aka, equation 15 becomes
-
-
In the evaluation of d S / d k 1 / 0 , differentiation of equation 14 with respect to kl yields
On differentiating the definitions of GI, GZ and Ga, one obtains = [k2(ks - hi)2 kaka(kz k3 (kz - ki)'(ka - k t ) 2
2;
+
]
-
(20) [(kp ki)'(ka k2ka - kz) The value of d S l / d k l I ~for each experimental value of A is determined from equation 17 on using the corresponding experimentally determined value of X and the outstanding estimatesokl, ok2 and Okp of the rate constants k t , kp and k 3 . dS Similarly one obtains equations for evaluating - and bkz o =
+
-
=
s-
5-10
xz=as G Io On substituting these definitions into equation 16, we obtain
+
+
J = XiAki XzAk2 XaAka (25) The system of determinants arising from the solution of equation 25 is
The iterative procedure described above was programmed for the use of an IBM 704 electronic computer. Reference to equation 13 shows that if any two of the three rate constants are identical then equation 13 will fail to describe the data since two of the coefficients of the exponentials become indeterminate. As a result, the iterative procedure just described will fail. Such situations actually existed in some of our experimental work on the alkaline hydrolysis of 1,3,5-tricarbomethoxybenzenein low dielectric media. Accordingly, another procedure is required. This new procedure is described in method #2. Method 2.-Start with equation 13 and for simplification write it as A = G1*e+lA Gz*e+ZX Ga*e+aA (27)
+
+
In equation 27 we have redefined GIBo, G ~ B and o G3Bo as GI*, Gz* and Ga*, respectively. Upon multiplying equation 27 by eklX and then differentiating the resulting equation with respect to X, we obtain dA KIA = Gs*(k1 kp)e-baA G3*(k1 ka)e-k~A (28) dX Two repetitions of the above process of multiplication and subsequent differentiation using, however, ekzx and ekaX successively as multipliers leads to equation 29, namely
-+
-
+
-
Oct. 20, 1961
:z +
For simplification we write equation 29 as dzA dA JI Jz--JsA = 0 (30) dXz dA On multiplying equation 30 by dX and then integrating between limits, we obtain
+
+
The reduction of equation 31 leads to dZA dA d~~ J I -dx JzA J3 AdX =
+
4117
COMPETITIVE CONSECUTIVE SECOND-ORDER REACTIONS
+
+ Jo’
?: 10
d k -
sI is given, the other two are obtained in a
dki ,o similar fashion)
:: 2 /e
- !o,
and
tgio
have fif-
1
bJ, These constants must be teen constants of the form -’ dkj 0. evaluated. These constants are defined as
10 .
If the kinetic run is dedX signed so that a t X = 0; C = D = 0 and B = BO;then a X -= 0, equation 8 reduces to and
nitions (only
Three equations involving
The comDlete solution of eouation 32 reauires an evaluation of the iimits --
higher order partials are neglected. The solution is a threedimensional iteration. The solution of equation 37 is by the method of least squares. The partial derivatives must be evaluated for each datum point and have the following defi-
2 10
= 1.0000
oi ; ;
= 1.0000
aJ,I
ak3 o
= 1.0000
(33) The differentiation of equation 8 with respect to X and the substitution of equations 9, 10 and 1 1 into the resulting equation leads to equation 34 when X = 0.
(34) On the substitution of the limits defined by equations 33 and 34 into equation 32, one obtains an equation which in turn can be integrated. The process is repeated once more and equation ( 3 5 ) is obtained.
A
- A0
+ JiA* + J?A** + J3.4***
J4/2 X2
+ J5X
For simplification we made the following definitions
(35)
where
+ +
+
J4 = kik?Ao PkikaBo 3kzk8Ro J6 = 2kiRa 3kzBo 3ksBo Although equation 35 involves four graphical integrations, it does have an advantage over equation 13. I t will not break down if any two rate constants are equal. Equation 35 offers a means by which the rate constants may be evaluated by a method of least squares without the use of an iterative procedure. However, such a procedure is possible only with the aid of an electronic computer. Since we wished to use a method adaptable to a desk calculator, we developed a satisfactory solution for equation 35 through use of an iterative procedure. For simplification, equation 35 is rewritten as equation 36, namely S = JiXl where S A0
+
+ J Z X Z+ J3X3 - $ X4 - JbX,
(36)
- A;
X i = A*; X Z = A**; Xa = A***; X 4 = ),z; X5 =: Equation 36 may be expanded about first estimates of the three rate constants by means of Taylor’s theorem. The starting equation is thus equation 37, namely
H=S--So (42) Equation 37 now becomes H = RiAki RzAkz RsAka (43) The system of determinants arising from the solution of equation 43 is ZHRiI ZRiz Z R I R ~ ZRiR3 ZHRZ = ZRZRI ZRs2 ZRzR3 (44) CHR3 ZRSRi ZRsRz ZRa2
+
1
+
1
Discussion The application of both methods to the alkalir!e hydrolysis of 1,3,5-tricarbomethoxybenzene is given in another paper. However, it is appropriate to show in this paper the validity of the analysis leading t o equation 13 and the subsequent testing of the programming developed for Method 1. In the special case where the relationship between the three rate constants is purely statistical, i.e., in which k1 = 3/2 kZ = 3k3, the rate equation3 reduces to (45)
On solving equation 45 for A , one obtains SI0 is the value of S evaluated by substituting valuesof
XZ,X S , X4, X 5and the estimates Okl, Oka and Ok3 into equaas tion 36. , bs and qs 1 are the values of as - - and
!
I
A
XI,
dki o dkz o ah3 o dkl’ dki %! evaluated from equation 36 using the estimates Okt, Ok2, aka Oka in place of the actual rate constants. Akl, Akz and Aka are the correction terms to be added to the estimates Okl, Okl and Okpto obtain an improved set of estimates. All
A
E
1
It follows that
+kJd
(46)
Through use of equation 6, equation 47 becomes 1 X =ka In [l
+ kaAot)
(48)
w.J. SVIRBELY AND JAY -4. BLAUER
41 1s
A combination of equations 46 and 48 leads to eq. 49 k3X
=
In
A A
-'
(49)
On assigning values to the constants i10 and ka, we may arbitrarily generate a test set of A values for various times t through use of equation 46. This test set of A values can be used to generate a test set of X values through use of equation 49. Both sets of test values are independent of any equations derived in this paper and apply to the situation where the rate constants are in the ratio of k i : k i : kB = 3 : 2 :1. On assigning a value of 6.51 to kB and 0.03000 to -40)values of X and A were generated and are listed in Table I. Also listed in Table I, column 3, are the values of A later calculated using the values of k l , k2 and ka generated by the iterative procedure. The initial estimates of the three rate constants and the values given by the iterative procedure are summarized below Table I. The iteration was stopped when Akl became smaller than lY0 of the outstanding value of k l . Three cycles through the iterative procedure were required in this test case. Close estimates of the rate constants were chosen as a measure of economy. The calculations were done by an IBM-704 electronic computer. In the application to experimental data, initial estimates of k1 were found by plotting A LIS. X and estimating the slope a t X = 0 in accordance with the equation 33. If k3 is much smaller than both kl and k p , equation 14 reduces to equation 50 for large values of X lim A I B O = A+
G3e-k3h
(50)
m
The logarithmic form of equation 50 is =
k log G~ - 3x 2.303
(51)
The initial estimates of k B were found by plotting log A us. X and estimating the slope a t large values of X. The initial estimates of ki were found by
[CONTRIBUTION FROM
THE
FIT OF
THE
Vol. 8.7
TABLE I TESTDATAAS CALCULATED BY THE ~ I A C I I I N K ITER.4TION
A, mole min./l.
0.01437 ,03909 ,06226 .OS034 ,09959 ,11689 ,12590 ,15239 .20914 ,26716 ,33698
ki ki k? k3
4,
A
Acaicd,
mole/].
mole/l.
0.02733 .0232
