W. E. WALLES
A. E. PLATT
Autocatalysis Analyzed BY ANALOG COMPUTER AND BY EXPERIMENTATION A simple three-step mechanism i s proposed for autocatalysis and this mechanism is analyzed on an analog computer f o r various ualues o f reaction rates.
A n experimental procedure is pro-
posed f o r determining which material is the autocatalytic agent.
Thus, proper ualues
of the re-
action rates constants are determined f r o m this procedure utocatalysis is a term commonly used to describe of a homogeneous chemical reaction which shows a marked increase in rate with time, reaches its peak at about 50y0conversidn, and then drops off. The temperature has to remain constant and all ingredients must be mixed at the start for proper observation. This definition excludes those exothermic reactions which show an increase in rate with time (like explosions) caused by the rapidly rising temperature. It is the purpose of this paper to present a simple case of autocatalysis involving only three consecutive kinetic steps with rate constants k l , k z , and k3. By using an analog computer, the concentration of the reaction components and their rates of formation or disappearance have been determined for a wide range of values for k l , kz, and k3. After studying the peculiarities and regularities of these cases, answers to the following questions have been attempted: Which experiments have to be made to prove autocatalysis; to determine k l , k2, and k ~ ;to determine which of the reactants is the autocatalytic
A the experimentally observable phenomenon
agent? And, should the particular case of autocatalysis be kinetically described by a more complicated set of equations, rather than by the present one with just three steps? The autocatalytic thermal decomposition of an oxazolidinone will be used as a practical example. DISCUSSION OF LITERATURE The general kinetic treatment of consecutive processes has been discussed by R. M. Noyes ( 3 ) . H e pointed out that the differential equations are nonlinear and that no general solutions have been developed. This difficulty has been avoided in this study by using an analog computer (EA1 Model 231R). The general procedures for preparing kinetic problems for analysis by analog computer have been discussed by T. J. Williams (5). The usual theoretical treatment of autocatalysis in textbooks (7, 2 ) considers the reaction A + B .. . and postulates for the rate expression :
+
-d(A) dt
= k(A)(B)
From the integrated equation one can plot per cent conversion us. time, which shows the characteristic Sshape curve. Weaknesses in this treatment, however, are the lack of actual equations showing how B catalyzes the reaction and the need for a certain starting concentration of B. In addition, the above equation does not have much flexibility, as the maximum in d(A)/dt occurs at 50y0 conversion of A. KINETIC STEPS FOR SIMPLE AUTOCATALYSIS Selected for mathematical analysis was the autocatalytic thermal decomposition of a single compound A into two products B and C, of which B is the autocatalytic agent. Thus, A can decompose via two routes, a slow uncatalyzed one ( k l ) and another catalyzed by B (k3). VOL. 5 9
NO. 6
JUNE 1 9 6 7
41
Cmtcrnhatioru of reactants us. time
t
-t
b
-+ + -
The three essential kinetic steps are (4): h
A
B
C Start or background reaction
B + AB Complex formation
A
ha
AB
(2)
hr
2B
+C
Autocatalytic step
(3) (4)
These equations show that the autocatalytic agent B forms a complex AB with rate k p . Next, the complex AB decomposes with rate ks thereby releasing B in addition to forming B C . Reactions ka and k8 together form the path by which most of A decomposes. Reaction kl is the starter, but continues concurrently with ka and kr as long as there is any A. The rates of formation of A, B, C , and AB, in accordance with the three kinetic steps above are:
+
at
=
-kl(A)
- kp(A)(B)
h
h
The analog computer was programmed to plot (A), (B), 0 ,(AB),d(A)/dt, d(B)ldt, d(C)/dt, and d(AB)/dt as functions of time. Of the numerous possible combinations of values for kl, ka,and kr, a selection was made after two new parameters were defined:
_ ---and
kr
ki -
ka ki -
-=@
@ = 10' was selected 80 that the complex formation would not become rate controlling and a was varied from 0 to 2000 in small steps:
a=O a = 1 10 100 1000
2 20
200
2000
30 300 4 40 400
(5)
50
500
6 60 600 70 700 8 80 800
(7)
9
The concentrations of these components as a function of time, starting with pure A and with no B or C , are: (A), = (Ah
-
1'
dt
90 900
When @ is sufficientlylarge, a can be called the degree of autocatalysis and various autocatalytic reactions can be classified as : a=ltolO
Barely noticeable autocatalysis
a = 10 to 100
Mildly autocatalytic
a = 100 to 1000 Strongly autocatalytic
Figures 2, 3, and 4 show that during a not too detailed kinetic investigation cases with a = 1 to 10 would tend to escape being detected as autocatalytic, and in cases with a = 100 to 1000, the background reaction would tend to go by unnoticed. Thus, autocatalysis may turn out to be much more common than presently assumed. 42
I N D U S T R I A L A N D ENGINEERING C H E M I S T R Y
Figws 2.
0.0 11
&/di vs. tiww fUflC&Wi
( 2 ~
Of LI
f
0.01c
.L
$ 0.m O.W( 0.W
0.W (
RESULTS FROM COMPUTER RUNS 1.
Concentrations of Reactants vs. Time
Figure 1 shows the result of a typical run, produced directly by the analog computer for the following values:
kl = IO-' see.-' kr = 10" liters moles-' sec.-l
kr = 8.0 X
sec.-l
Therefore, a = 80 and 0 = lo' liter mole-I. The starting concentration of A was selected at 1 gram mole/liter. The analog computer has plotted (A),, (B)t, (C)t, and (AB), us. time. Under these conditions,
TABLE 1.
RELATION BETWEEN a AND
Run No.
1
2 4
B4
6
a/a
-0.99 1.95
-0.50 0.49
4.73 9.3 18.2
0.47 0.47 0.46
lo-' 10"
3.60 X 103.60 X 103.60 X 10-
26.7 35.0 43.0
0.45 0.44 0.43
153 531
X 10" X 10-
3.60 X 103.60 X 10"
42.5 147.7
0.43 0.37
746 945
X 10'
3.60 X 103.60 X 10-
207 262.5
0.35 0.33
X 10-
C6 c4
100 400
c3 c2
600 800 1000 7~Wwunrnu*&
a
3.60 X lo-' 3.60 X 103.60 X 10-
96 126 155
-
-,
TIME.
17.0 X lo-' 33.5 x 1065.5 X
60 80 100
IO&=
d(C)/ot Skmt
3.6 X 10-Ohr. 3.45 x 103.5 x 10-1
W.2 VS.
0.48 0.47 0.47
A3 A2 A1
-
FROM COMPUTER-PLOTTED CURVES FOR
2.90 3.74 4.71
20 40
c1
(13)
3.6 X 10' 3.6 X 103.6 X 10-
A6 A5 A4
10
( N o c4'
10.4 X 1013.45 X 1016.95 X lo-'
10
8
=
As kz >> kl we have k = 2 kl. A will have gone down to 0.90 mole/liter after 14.7 hours. So the influence of
none 3.4 x lO'hr.-' 6.8 X lo-'
B3 B2 Bl
* Ruu f m =
(A),
d(C)/df.,
n
B6 B5
LI
A would be gone in 14 hours, while B and C would be practically 1 gram mole/liter after 28 hours. The complex A 3 reaches a maximum concentration of 0.44 gram mole at t = 13.4 hours. By varying a from 0 to 2000 and keeping 0 constant at lo', a series of 26 curves of the same general type as shown in Figure 1 was obtained. The dotted line in Figure 1 represents the case where ka is 0 so that only kl and kz would use up A. The firstorder disappearance of A can be calculated from
X
x
x
10-
? I d@rml&
1Fn Pi-
2,3.d 43.
V O L 5 9 NO. 6 J U N E 1 9 6 7
43
Figure 3. dc/dt us. tim as function of a
k8 is pronounced even at concentrations for B of the order of 0.01 mole at f = 11 hours. II. Formation Rate of C w. Time
In Figures 2, 3, and 4 the computer was made to plot d(C)/dt us. time for a varying from 1 to 1000. As d(C)/dt = kl(A) kl(ALL) and (ALL) = 0 at 6 = 0 we see that the starting rate is equal to kl(A)o. In Figure 2 the rate is plotted in hour-'. So kl = IO-# set.? corresponds to 0.0036 hour-'. Therefore, the d(C)/df curves for various values of a (with kl constant) all start at the same point. For a = 1 the curve is only slightly different from that for only kl (with k p = 0 and k , = 0). For a = 2 the rate remains constant until about 50% conversion to C, then drops off. For a = 4 up to a = 10 the characteristic sharp rise in d(C)/dt illustrates autocatalysis. (According to the conventional definition of autocatalysis, the cases with a 5 2 would not be called autocatalytic.) Figure 3 shows d ( C ) / d t for a from 10 to 100 on a different time and rate scale than Figure 1. The maximum rate for a = 100 is 0.155 hour-', in contrast to a starting rate of 0.0036 hour-'. Figure 4 shows d(C)/dt for a varying from 100 to 1000 in steps of 200, again on a time and rate scale different from Figure 2. The maximum rate for h = 800 was determined as 0.945 hour-', which is 257 times as fast as the starting rate. This supports the contention that, in cases with a = 100 to 1000, the background reaction would tend to go by unnoticed. The varying shapes of the curves of d(C)/dt us. f can be typified by introducing the term "degree of autocatalysis."
+
111.
Datamindion of D.(lraa of Autocmhlysis
I t would be important if one could determine a G k d k l from experimentally accessible data. When fls kr/kl is sufficiently large, a characterizes the degree of 44
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
autocatalysis. Another way of characterizing the degree of autocatalysis can be derived from Figures 2, 3, and 4. If one defines u (peak rate of C production)/ (start rate of C production), u becomes another practical yardstick for the degree of autocatalysis.
TABLE II. ( I
RELATIONSHIP BETWEEN a/u AND (AB), (AB)a b MolrfLilU
2 4 6 8 10
0.50 0.49 0.48 0.47 0.47
10 20 30 40 50
0.47 0.47
60 70
0.44'
...
0.456
...
... 0.44 ...
80 90 100
0.43
,
... ... ... ...
... ... ... 0.46 0.45' 0.45
0.446 0.44 0.44 0.43 0.43
100 200 300 400 500
0.43
... ...
0.38' 0.37 0.356
600 700 800 900 1000
0.34'
0.341 0.33' 0.32'
Cmdwia: ./a
0.37
...
0.33
-
... ...
0.43
0.40'
... ...
1.o
0.9
Figure 4.
&/df us. time
0.8
a(
function "f n
0.7
i
0.6
.E 0.5
$ 0.4 0.3 0.2 0.1 0 0
2
The curves of d(C)/dt can be directly determined experimentally. In the case where C is a gas ( C O Sin our example, see below), a fully automated apparatus measures and plots d(C)/dt us. time. An empirical relationship has been derived between a and a from our analog wmputer runs by measuring a. The data are listed in Table I and plotted in Figure 5. The relationship between a and a can be derived by calculation (Section IV) which results in:
To check this theoretical relationship, both the quotient (AB),- as derived from the wmputer-plotted curves in Table I1 have been listed. The values for (AB),; derived from the computer-plotted curves for (AB), us. t are plotted in Figure 6 against the values for a/a derived from the wmputer-plotted for d ( c ) / d t us, t. The is the limit of accuracy. u/a and the value for
IV. Mdhematical Derivation of a/.
=
(AB)-
After finding empirically the relation between u/a and (AB),-, the exact equation relating the two has to be derived. By definition: a=
[d(C) 7
peak rate of C production -starting rate of C production
a ],x
TABLE 111.
10
12
20
(A)Mm
1
- (AIM m u
u
(.
0.005 0.02 0.035 0.035 0.07
10 20 50 100 500 1000
0.18
r9]
There-
24
26
-
(AB),,
0.45 0.43 0.35, 0.31
+
= ~ ~ ( A ), AB ks(AB),,
m x
(16)
This latter equation infen that d(C)/dt reaches its maximum at the same time at which (AB) reaches its maximum. This could be verified by observing the analog computer plots in the cases where both d(C)/dt and (AB) have been plotted as functions of time. Therefore:
-
-
-
--x.
a = k~ _ -
kl
22
0.0005 0.001 0.0007 0.00035 0.00014 0.0M)lS
ks
-+
kr -
+ kr(AB).
18
16
SIZE OF CORRECTION TERM (l/a)(A)*s
(1
and
14
a- - [ ~ ~ ( A ) A B ka(AB),,I a ki(A) . ka
and
As listed earlier, d(C)/dt = kl(A) fore, when starting with only A:
8
6
5)
["-I0
a=
4
a
ks
(NAB
+
ki
(AB)mu
which can be rewritten as
VOL 59
NO. 6 JUNE 1 9 6 7
45
This derivation does not give us exactly the simpler empirical relationship a / a = (AB),,. However, the term (l/a)(A)*= is insignificantly small (about in relation to (AB),-, as can be seen from Table 111 where the term was derived from the computer plots covering a wide range of values for a . Therefore, it can be concluded that the empirical relationship a
-a =
(19)
(AB)mu
has been supported by a theoretical derivation.
Maximum concentration of AB, (AB),- comes at the same time as [d(C)/dtl, a / a = (AB),, (20)
In addition to those, we observed: 1wo
If per cent formation of C as the reaction parameter
8w
is used, d(C)/dt peaks between 52 to 53% formation of C,
.""
reeardless of a
[TI,.) d(C)
yoformation of C
a
I
of
c
1000 52 100 53 10 52.5 /dt],, comes invariably at 42 to 43% forma-
r91,;
yoformation of C 42 42 42 43 42 42 is considerable and varies only a little over
200 100 70
(AB),,
m
J lM) .@I I1
20
10
0
0.1
0.2 a/*
0.3
ml IMI,
0.4
0.5
0.6
0.7
in u*/l
Figurd ' 6. Tha r~l&'onshipb c t w m a and 01 whnc a / a
(AB)--
When starting at 1 gram mole/liter concentration of A we get for (AB)a (AB),, mole/litcr
1000 100 10 1
illustrates further the impfiance
0.31 0.43 0.47 0.48
of the complwr
AB. PROCEDURE FOR DETERMINING AUTOCATALYSIS AND ITS PARAMETERS Based on the examination of the data gathered with the analog computer, a procedure can be charted to prove autocatalysis; to determine kl, kn, and ki; and 46
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
AUTHORS W.E. walles is an Associatc Research Scimtist and A . E. Plat1 is a Project h a d m in thc Chemicals Research Labmatory, Thc Dow Chemical Go. Thc authors wish to acknowledge hlp gum by M . J. Gross in opmating th analog computer, and by J . J . Davis and E. D . Prutnin thechcmical cxpniments. Thcy also thank R. G. Pearson, R. M . Stiles, J. W . C m p , T . Hmrhnan, and E. V. Luoma fm helpful discussions.
to learn whether the case under study can be satisfactorily described by the equations for simple autocatalysis, or whether a more complicated scheme should be used., Procdurn
(1) If B is the suspected autocatalytic agent in the reaction A + B C, measure d(C)/dt as a function of time until the maximum t, has been passed. The starting rate of t = 0 is ~ I ( A ) and ~ , Ao thus gives ki. Next, determine (2 (peak rate)/(start rate), and with Table I or Figure 5 find a out of u. Determine ks from k, = kia. (2) A small amount of pure B is prepared. A new reaction is started in which d(C)/dt is measured us. time, but immediately after the start a small amount of pure B is added to the reaction. If an increase in formation of C is noted, it confirms that B is the autocatalytic agent. A quantitative check can be made by the following reasoning: Call C' the extra C made by the addition of B(B'). From ths, we calculate a value for k3 (called k8'). If kr = ks', it has been quantitatively shown that B is the autocatalytic agent. But if ks' # k3, there can be various complications beyond the simple autocatalytic system. The time it takes for the added B' to become fully effective can be used to calculate kz.
+
c
Twe:ZA-C+B A+B-AB 2AB-3B
+C
I
100:
Ii
THERMAL DECOMPOSITION OF AN AS AN EXAMPLE OF OXAZOLIDINONE -.. ..-- -. -
AUTOEATALYSIS
When heated above about ZOOo C., the pure compound 5-methyl-2-oxazolidinone(OM) decomposes into two products, COz gas and N-(Z-hydroxypropyl)imidazolidinone (HPMI). This decomposition will be discussed in a later paper. The over-all reaction is:
CHr \H
CHs
\
6c'
/\252-
N
*
Ha H/ HG-N--CC
I
H z ~
Figure 8. T h a i decmposifion of pwt 5-mctyll-2-oxazolidinoidinotll at 252O lo C . No s o l d
CHs
\
c--o I
2
lime in h u l a
O
I
I
Hd
C
\/\
H
N
\ +con OH
1
O
H
Generaltype: 2 A + B + C
As both OM and HPMI are liquids at that temperature, the reaction proceeds visibly by Cot bubbling away, giving an easy and natural separation of the reaction products. The evolution of COa gas proceeds autocatalytically (Figure 7). During this reaction the temperature was held constant at 252' f 1" C. In Figure 7 the cumulative COzgas production has been plotted as a function of time. The S-shaped curve indicates autocatalysis. In Figure 1 a typical analog computer run shows a similar S-shape for the total production of C us. time. A more useful and accurate picture of autocatalysis is obtained by plotting d(C)/dt us. time (Figure 8). It demomtrates the rate increase with time, and the
ri,Iold
li
==--lo
8-
4li %=
10
m k h Fq. a
Figwa 9. Analog computn Tun for d(C)/dt us. t
V O L 5 9 NO. 6 JUNE 1 9 6 7
~~~
47
It is necessary to determine which compound is the autocatalytic agent maximum rate at about 50% formation of C. The rate data for Figure 8 were obtained with a fully automatic reaction rate apparatus, fully described in a later paper, which collected 85 Cot rate values in 850 min. Although the starting rate [d(C)/dtl0is somewhat uncertain in Figure 8 (see Figure 10 for more accurate data), we can estimate a value for a E (peak rate)/ (start rate) of between 58/(5 & 2) = 14 6. Using Figure 5, we find for a:
We will now try to derive values quantitatively for kl, k8, and ks' from the data in Figure 10. The three basic reactions to which the various k's apply could be chosen slightly differently from simple autocatalysis, so that A can represent one oxazolidinone ring, B the HPMI, and C CO,:
*
a = 30
* 14
To compare the experimental d(C)/dt us. f curve of Figure 8 with one calculated by computer for simple autocatalysis, we have plotted in Figure 9 the case for a = 20, which appears in Figure 2, but now rescaled to match the top and starting rates of Figure 8. Notice that the 01 = 20 curve has a much sharper maximum and a lower acceleration than the experimental curve. Cases with a = 40 and a = 10 are similar. Before deriving k ~ k,,, and kr, however, it is necessary to determine which compound is the autocatalytic agent. Datermindon of Aufosatolylis Agent
The CO, concentration in the reaction medium is constant as it escapes continuously; thus, it cannot be the autocatalytic agent. This was further verified experimentally by collecting the data in Figure 8 under two conditions: first, while bubbling COS through the reaction medium (fitted with reflux condenser) at about 10 times the peak CO, production rate, and second, while bubbling Nn at about 10 times the peak Con production rate. No experimentally observable difference in CO, production was found. Next, we have to determine whether HPMI (or B) or even some unknown intermediate, is the autocatalytic agent. Figure 10 shows the COI production rate d(C)/dt and shows the effect of the addition of a small amount of pure HPMI. The jump in CO, production after addition of HPMI makes it the suspected autocatalytic agent.
ZA%B+C
(21)
A+B%AB
(22)
2A B 23B
+C
(23)
These equations are each second order, so the Ps should be expressed in liter mole-' set.-' However, there are various reasons to work with a simpler set of equations. OM is even at a dilution of 1% in chloroform as a solvent present as a dimer. Therefore, 2 A An is mostly to the right, which would change the first equation to A, 3 B C and make this reaction first order. If the second reaction is written as An 2 B 2 2 AB we would get third-order kinetics, but there is evidence that B is also present as the dimer BE,leading to An Bp 2 AB, which would reduce the reaction to second order. Finally the third reaction can also be represented as AB A 4 2 B C, which would be pseudofirst order under excess of A.
+
+
+
+
-
+
Dotemination of k1
The rate of COt production via k1 (Figure 10) started at 8.5 liines/236 sec. When we use the appropriate calculations, this rate corresponds to
8.5 - x -10x 236 36
- 273 x-= 300
1 22.4
4.06 X lo-' mmole COr/sec./l99 m o l e O M This is a rate of (4.06 X 10-')/199 = 2.04 X 10mole CO,/mole OM/sec. at 245" C. To bring this rate on a basis of 2 A or A,, we divide by 2:
+
kl = 1.02 X lod set.-' (for the reaction An
B
C at 245" C.)
of ka Assuming, as before, that kr is faster than ka, we can calculate kr from the part of the curve prior to addition of extra HPMI. At t = 140 min. the reaction has proceeded 100 min. at an average rate of 17.5 lines per 236 sec. Total Cot produced : 17.5 10 273 1 X 100 X 60 X - X - X - = 236 36 300 22.4 D.l.rmindon
-
Tim in yhutll
Figurc 70. Test of HPMZ as surpccrcd aurocafalytic agmi in thmnd dccompoEirion of O M at 245O C. 48
INDUSTRIAL AND ENGINEERING CHEMISTRY
5.02 mmole COe As for every mole of COn there is produced 1 mole of HPMI (B), and all of B is in the form of AB (kt very fast),
it can be determined that at t = 140 min. we have made 5.02 mmoles of Cot, and 5.02 mmoles of AB (or B), and have lost 15.06 mmoles of A of the 199 mmoles present at start. Thus, at t = 140 min. there are still 199 15 = 184 mmoles of O M left (Figure lo), so that the COZ production via kl is 184 -X 8.5 = 7.9 lines 199 Of the total 26 lines in the COS production rate 7.9 came from kl and the other 18.1 from k3. The COz production rate via reaction k3 at t = 140 min. then is 18.1 - x -10x 236 36
- 273 x-= 300
1 22.4 8.65 X
mmole of COz/sec.
Because there were 5.02 mmoles of AB at time t = 140, the rate defined as mole COZproduced/mole AB present/ sec. becomes : 8.65 x 10-4 sec.-I 5.02 k3 = 1.72 X 10“‘ sec.-l (pseudofirst order)
If k3 is defined per 2 AB and the concentration of AB is 5.02 mmoles of AB/184 mmoles of O M or 0.273 mole/ liter, then for k3: 1.72
x
10-4
x
+
Estimation of kz
If the values for the COz production rate at t = 150 min. and at t = 160 min. are interpreted as indicating that 10 min. after addition of HPMI only half was autocatalytically active, a rough estimate of kz can be made. The results are about 184 mmoles of OM in which are mixed 1.26 mmoles of HPMI extra, B’. The reaction A B ’ 2 AB’ is almost first order in excess A. Thus:
+
B t t = Bo’~-W
or
1
the more complicated set of equations used now. We assign a = 169 as the degree of autocatalysis, indicating how many times as fast complex AB decomposes in comparison to AA. The observation that the maximum in the rate curve of Figure 8 is not so sharp as the various calculated curves for a = 20 up to a = 200 could be accounted for in various ways: if k2 were of the same order as k3, if B were taken away by a side reaction, or if formation of AB is an equilibrium A B AB, the COa production would be slowed down. These possibilities will be further examined in a later paper.
xJ -= 3.15 x 0.273
10-4
For B,’ = 0.5 BO’we get k2t = 0.69. For t = 10 min. this gives k2 = (0.69)/(600) = 1.5 X sec.-l If it is taken into account that A will disappear quickly, k2 can be expressed in liter mole-’ set.-' The original A is present in 10 gram moles/liter :
kz
= 1.5 X
liter mole-’ set.-'
Summary of Oxasolidinone Example
By representing the decomposition reactions of OM as
or k3
liter mole-’ set.-'
= 3.15 X
Determination of
(second order)
k3‘
From Figure 10 after addition of 0.20 gram of HPMI the rate of COz production jumped by 12.5 lines or 12.5 __ 236
x -10x - 273 x-= 36
300
x 10-4 1.26
-
4.75 X Comparison of k3 and k3’
+C
(25)
sec.-l
ka = 150 X
set.-' mole-’ liter
= 172 X
set.-'
and
p mole of COz/mole of HPMI/sec.
(24)
kl = 1.02 X
k3
5.98
A + B ~ A B
The values at 245’ C . for pure OM (no solvent) are:
5.98 X lov4 mmole of COz/sec.
ka’ =
(23)
2 AB 3 3 B
1 22.4
This 0.20 gram of HPMI represents 200/158 = 1.26 mmoles of HPMI which had formed 1.26 mmoles of AB. This gives a (pseudofirst order) value for k3’ of:
2A*B+C
=
k2
= -147
liter mole-’
These values and the three-step autocatalysis scheme are in good agreement with the experimental data, although further refinements are being worked out.
The values for kg’ and k3 are close enough together to indicate that HPMI is the autocatalytic agent. REFERENCES Determination of a
Returning to a = k3/kl, we find cy = (1.72 X (1.02 X = 169. This value, compared with a = 30 f 14 determined graphically from Figure 8, illustrates that the relationship a / a = (AB),a, holds no longer for
(1) Frost A. A Pearson R. G., “Kinetics and Mechanism,” 2nd ed., p. 19, McGrkw-Hill,’kew Yo&, 1961. (2) Laidler, K. J., “Chemical Kinetics,” p. 9, McGraw-Hill, New York, 1965. (3) Noyes, R. M., in “Progress in Reaction Kinetics,” Vol. 3, pp. 333-62, Macmillan, New York, 1964. (4) Wallen, W. E., i n “The Encyclopedia of Chemistry,” G. L. Clark, ed., p. 112, McGraw-Hill, New York, 1966. (5) Williams, T. J., Ch8m. Eng. Ncws40,88-96 (Jan. 8,1962).
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