O N THE U S E O F T H E D I F F E R E K T I A L EQUATION I N CALCULATING THE R E S U L T S O F K I B E T I C MEASUREMEKTS ; T H E REACTION BET W E E N A R S E N I C ACID AND POTASSIUM IODIDE N E A R THE EQUILIBRIUM BY W. C. BRAY
Introduction 1. The Method According to the present view that the rate of a reaction is proportional to the concentration of the "reacting" substances, the fundamental equation' in kinetics is
I T o test the correctness of the formula assumed in any special case, it is customary to calculate K from the derived integral equation. Many rates have, however, been measured which cannot be expressed by any simple form of Equation I. (On account of the large number of reactions which do obey this law it has been concluded in such cases that complicated reactions take place, not that the fundamental equation has ceased to be of general application.) When, for example, two independent reactions take place at the same time, the differential equation is the sum of two simple terms ; and in the neighborhood of an equilibrium the rate measured is a difference of two rates. T h e differential equation will, therefore, have the following form, dx dx, dx,---+--K,(A,--x,)"l(B,-xx,)"l . . .t dt dt - dt K2(A,-~,)"2(Bz-- X , ) ~ S, . I1
In the majority of such cases' the integrated expression will * A, B , x,t , have their usual meaning, cf. p. 574. For methods of finding the law governing complicated reactions see Mellor : ' I Chemical Statics and Dynamics," p. 55 (1904).
574
W. C. B Y a y
be so complicated that it will be difficult and often impossible to obtain values for the constants of the single reactions, I n these cases it is convenient to use the differential equation directly. I n the following paper a method of doing so has been described, and examples of its application given. Especial attention has been paid to the calculation of the constant in the neighborhood of the equilibrium, when the rate of the reverse reaction is not negligible. A detailed example is given in the second part of this paper (see p. 578). dx
The determination of 7d
From the form of the fundamental equation (I), it is evidx
dent that K may be calculated if --- is known, assuming of course dt that A, B, . . the initial concentrations, and x the amount of change in time t are given. When the differential equation is more complicated, as in (11), it is only a question of obtaining a sufficient number of rate measurements for varying concentrations of the reacting substances to be able to calculate the different K's accurately. T h e problem then is to determine dx -- at any time. dt
dx
4
X,-X,
T h e formula - = ___ where I,, x,,and I,, x, are the redt t' - f, sults of two sticcessive measurements, is not of general application, as it is only in exceptional cases that the rate is practically constant between the measurements. But by taking a sufficiently large quantity of the reacting mixture for each analysis, so that the actual change of concentration is small while the total change is large enough to measure accurately, it is often possible to obtain an accurate measure of the rate with this formula. From a series of such measurements it is easy to determine the order of a reaction.' T h e name '' Method of constant rates " which I used in a former paper in describing this method' was suggested by Professor W. Lash Miller. See Mellor : LOC.cit. Van't Hoff was the first to point out how this could be done. Bray. Jour. Phys. Chem. 7, 92 (1903).
575
Kiizetic Measurements dx dt
T h e rate - can always be obtained graphically. When dx
the results are plotted in the ( t , x) diagram, the ratio - may dt
be obtained by drawing the tangent at the point (t, x). T h e accuracy of the determination depends upon the number of measurements and on the care with which the diagram is constructed. This method is very simple, but has seldom been used. T h e corresponding analytical method is to obtain by trial m y simple function of A, B, x,t, which represents the course of the curve in the neighborhood of the point ( I , xj and to obtain dx - by differentiation. dt
T h e simplest case is when this function
is closely related to the more complicated final function, e. g., when it is the integral of a differential equation of the form dx
- _- k(A-x)”, where n is a small integer. dt
The method. A simple example Let the course of a reaction be expressed by the formula dx
- =z K(A-x) (B-x)’. (111) dt I n the simple case, when B is large compared with A, the
equation
dx - = K,(A-xj dt
may also be used to represent the
course of the reaction; k, may be easily calculated from the equation, ; and K = 4 B’ ’
T h e method used in the following calculations is merely an extension of the principles used in this well-known example. 14preliminary “constant” K is calculated from as simple a n equation as possible, and the final constant is derived from it by means of the differential equation. I t is not necessary that this preliminary k be constant ; in general i t will show a decided tendency to decrease or incTease.
dX
But the rate, 27 at a given
576
W. C. B r a y
time can always be accurately calculated when the value of K at this time is accurately known. It is evident that k must be calculated from pairs of consecutive measurements, and not, as is the usual way, by comparing each measurement with the ‘initial measurement. On account of the increased error in K resulting from this calculation it is advisable to plot the values of k and f, and to replace the zigzag curve by a continuous one. T o illustrate the method let us consider that B in the above example is only three times as large as A. T h e “constant” K will decrease rapidly as the reaction progresses. Let K , be the value of the constant at the point (t,, ,x),K , at the point dX = K , (A - xi),and = (t2,xJ,. . . . Then at the time t,,
dX But at these times - = K (A - xl) dt (B - x1)2,and K(A - xJ (B - x,),, respectively. Hence
K , ( A - x2)at the time t,.
Thus K is obtained in a much simpler manner than if the values of t and x had been substituted in the integral of Equation 111.
I t is evident that practically the same results for K would have been obtained if the preliminary constants had been calculated from the integral of the formula
dx
-=
dt ~(A-x) case K would have been equal to --____ ‘ (B-x)’
K(A-xx)”. I n this
A n application of the method T h e method may be used to test if the course of the reaction can be better expressed by another formula than the one used in calculating the results. As an example, let a set of values of K, calculated on the assumption that dx - = K(A - .x) (B - x ) ~ , dt
Kiizetic Measurements be given. If we wish to test the formula
577
dX - = K’(A - x ) ~ ( B -x ) , dt
then the values of K’ may be obtained from the values of K by means of the equation K(B -x) K’ = ____ (A-x)
‘
I n general the use of this method in such cases will lessen the labor of calculation very considerably. I t is scarcely necessary to repeat that the values of K must be calculated from pairs of consecutive measurements. The calculation of t h e constant in t h e neighborhood of an equilibrium T h e rate of a reaction approaching an equilibrium is equal to the difference of two rates. I n simple cases, as in the reaction between alcohol and acetic acid, where -
dt
and
= k( I
- x)Z - k’X2
k - - 4, or in the same reaction in ai1 alcohol-water mixk’ -
dx
k,, (I - x)- A,’%, the equation may be intedt grated.’ But when the order of the reactions is greater than the second, the integratioii is either impossible or gives a very complicated expression. T h e method described in the following paragraph is of general application, and as the example in Part I1 shows, is capable of giving accurate results. I n general at least one of the reactions may be measured at such a distance from the equilibrium that the reverse rate is negligible ; the order of the reaction and its constant may then be determined in the usual way. T h e order of the other reaction may be derived from this result and the equilibrium formula, or may itself be determined at a distance from the equilibrium. But when it is necessary to make measurements where the reverse rate is not negligible, the constants, which are calculated
ture, where
Nernst.
-=
Theor. Chernie, 4th ed. p. 560 (1903).
W C. BYay
578
on this assumption, will decrease more or less rapidly as the reaction proceeds and the error due to the reverse reaction increases. As i n the above examples, however, the actual rate at any time, dX
dt measured, may be calculated
from the preliminary constants.
And when the reverse reaction has already been measured, the dX
correction for the reverse rate may be calculated. d t correction, T h e sum of these two rates gives the theoretical rate of the direct reaction, from which the desired constant K may be obtained. dx ~
dx
( z measured + dt correction dX dt measured
Here as before the preliminary R's must be calculated from pairs of consecutive measurements.
11. The Reaction between AYsenie Acid and Potassium Iodide N e a r the Equilibrium T h e method has been tested by means of the velocity measurements of the reaction between arsenic acid and potassium iodide in acid solution.' T h e reverse rate, the reaction between arsenious acid and iodine (see Part I of the original paper), is accurately expressed by the equation K(arsenious acid) (tri-iodion) d( Iod i ne) - _ _ __ -- ~ - _ _ ~- (IV) dt (sulphuric acid) (potassium iodide)z
K in the unit chosen = 0.28. The measurements were made under such conditions that the rate of the arsenic acid reaction was practically negligible throughout the whole series of experiments. I found on calculating the error in the last measurements of each table that it was very small except in the last two measurements of Tables IV, XIII, XIV, and XVI, and that eyen in these cases it was not large enough to change the results in any way. * Roebuck. Jour. Phys. Chem. 6, 365 (1902).
Kip2et ic Meas zwem e n ts
579
T h e concentrations in the above equation (IV) are the analytical, not the ionic concentrations. Under the conditions of the experiments, the concentrations of the ions 13’,Ha, and I’ are practically proportional to the corresponding analytical concentrations, and it is clear that these ions niay be considered the “reacting” substances. On the other hand it is the concentration of the undissociated’ arsenious acid that is proportional to the analytical concentration ; and it seems probable that the undissociated acid takes part in the reaction, and not an ion as AsO,”’ or H,AsO,’. T h e formula found for the equilibrium and the theory of the intermediate formation of hypoiodite, advanced to explain the results, required the reaction between arsenic acid and iodine ion to follow the law: d(iodine) d‘f
= K,(arsenic
acid) (potassium iodide) (sulphuric acid) V.
Although the agreement was a remarkable one, the results left much to be desired : the rate was not strictly proportional to the potassium iodide or sulphuric acid concentrations, and the value found for R,, 0.326 X IO-^, was smaller than the theoretical value, 1.9 x IO-^. T h e decrease in the constants in Tables XVIII, X I X , XXVIII, and X X I X , which became much more marked when the calculations were made with pairs of consecutive results, led to the discovery that the rate of the arsenious acid reaction in these experiments was not negligible. T h e results of the recalculation (see following tables), showed that a higher value for K , must be assumed, and that in dilute potassium iodide solution the rate is strictly proportional to the first power of the iodine ion concentration. T h e numbers of the tables refer to the original paper, and the same nomenclature is used. V is the volume in liters, C, D, E, represent the number of units of potassium iodide, sulThe undissociated arsenious acid molecule contains only one combining weight of arsenic. Zawidzki. Ber. cheni. Ges. Berlin, 36, I427 (1903) ; Brunner and St. Tolloczko : Zeit. anorg. Chem. 37, 455 (1903).
W. C. B a y
580
phuric acid, and arsenic acid used in each experiment. One unit arsenic acid in I liter gives a 0.01molar H3As0, solution ; and the other units are chosen so that I unit potassium iodide and I unit sulphuric acid react with I unit arsenic acid according to the following equation, 3
I’ + 2H‘ f H34s0, = H,AsO,
+ Is’+ H,O.
T h e number of units of iodine x formed in the time f, given in the first column, is taken from the original tables. T h e second column contains the values for the constant kl’, recalculated from pairs of consecutive measurements. T h e fact that these values may be used without further correction to obtain the rate,
dX
-
dt measured
’ is a proof
of the accurate nature of
the experimental work. dX (in third column), = 2.3k1’(C - x) in Tables dt measured ’ XVIII and XIX. = 2 . 3 k l f ( E - x) in the other Tables. X 2 2v 7 _ _k 3 _ _ - ~ d t correction ’ (in fourth column), = (C -- x ) (D ~ -X ) where K, = 0.28. h1”)(in the fifth column), =
dX -
-
df measured
hj,(in the last column), =
_______ 2 ‘ 3 ‘‘’I v 3
(E--)
(D-X)
,in Tables XVIII
and XIX.
- _ _2‘3 __ _ _‘l”VS ___ (C-X)
ing tables.
(D-~x)
, in
the remain-
Xiizetic Measuvements
____
TABLEXVIII C, 0.1385; D, 22.85; E, 1.16; V,
I
k,’ X 103 recalculated ’
X
s x 104
1 measured !
I
0.0095
-
0.0172
11.1
0.0255 0.036 I 0.05 15 0.0594 0.0648
11.9 I 0.3 937 3.91 3.08
1
correction
I
30.9 30.9 24.2 19.8 7. IO 5..22
dx
k,’ X
103
-
-
-
11.2 12.2 11.0
1.71 I .87
12.5 8.4 9.8
1.96 1.33 1.93
_
0.0066 0.0125 0.022 6 0.0361 0.0427 0.0523 0.057 5
1.71
v, 0.12
x IO4
measured _
k,!’ X 103 corrected
0.36 0.80 I .80 5.26 8.2 I 1.4
TABLEXIX C, 0.1385; D, 22.85; E, 0.580; X
0.12
correction
~
-
-
-
-
-
6.31 5.52 6.91 339 4.28 2.79
14.6 14.7 14.I 8.6 8.7 5.2 3.1
0.24
6.42 5.71 6.77
I .56
0.50
1-80 2 *9 5.5 7.5 9.3
TABLEXXIII C, 0.644; D, 22.85; E, 0.136; V,
5.20
7.15 6.82 6.98
1.78 2.17 I .69 2.38 2.27 2.34
0.12
X
measured
correction
-
-
-
-
7.80 5.75 8.00 7.16 6.38 5.77 6.17 5.06 4.5‘
2.23 I .67 2.38 2.19
0.00386
-
-
0.0326 0.0460 0.0614 0.075 I 0.087 I 0.0989 0.1095 0.1173 0. I 206
7.80 5.75 8.00 7.00 6.06 5.21 5.04 3.37
-
2.28
9.70 6.93 4.20 3-07 I .48 0.80
-
0.22
0.36 0.48
0.62 0.74 0.79
2.00
1.86 2.02 I .69 1.51
W. C. B a y
~-
TABLE XXVI C, 1.88; D, 9.15; E, 0.290; V, 0.13
x
X
measured
0.0085 0.0140 0.0309
I
104
correction
0.05 17
0.0689
2.14
0.123 0.141 0.166 0.191
2.06 1.98
-
I .go
2.17 1.95
k,” X loy
k , X IO^
-
-
I .go 2.14
0.55 0.64 0.64 0.66
I
I
I -90 2.14 2.11
0.212
1
-
2.1 I
10.9
-
I
0.058
7.9‘
-
0. I 98
2.12 2.05
0.26 0.38
6.78 5.42 4.94 3.24
2.16
-
0.52
0.65
-_
2.03
0.67 0.66 0.67
2.40 2.34
0.80 0.80
TABLEXXVIII C , 2.270 D, 2.293; E, 0.310; V, 0.155 X
k,’ X
104
measured ,
1
correction
o.00161
-
-
0.0107 0.0122
2.30 2.53 2.16
0.38 0.43 0.40 0.j 6 0.49
0.0434 0.113 0.122
I .88 1.11
0.137
I .42
0. I 64
0.171
1.17 0.57
13.2
8.42 4.50 5.65
3.70 1.70
0.73 5.46
6.45 8.30 12.36 13.56
0.65
0.97 0.98
Kin e!i ic Measurem e z ts
____
C,
2.270;
____
x
0.00092 0.00551 0.0170 0.0602
0.0680 0.0776 0.0989 0. I 0 2 0
1
TABLEXXIX D, 1.147; E, 0.310; V, 0.155
nieasured
- I -_
0.82 0.99 0.77 0.69 0.40 0.20
‘, I
I
I 1
583
correction
I
-
-
5.75 5.67 4.40 3.70
0.21
2.00
8.60 9.37
0.95
I
2.96 334 5.10
1 ~
-
-
0.85
0.29
1.50
I 1
1 1
1.44
1.64 2.14 2.13
0.53
0.52 0.60 0.80 0.81
Tables XVIII and X I X , in which the concentration of the potassium iodide is small compared with the sulphuric and arsenic acids, show that the constant does not decyease as the reaction progresses, and that under these conditions the rate is proportional to the first power of the iodine ion concentration. T h e values for Kb, between 1.7 and 2.1 X X O - ~ , are in good agreement with the theoretical value calculated from K, and the equilibrium constant. In Table X X I I I the initially high value of k, = 2.1 x IO-^ appears to decrease as the reaction progresses. T h e initial value of k, in Table XXVI, 0.6 X IO-^, is very small. It, however, appears to increase as the reaction proceeds and the equilibrium is approached. In Tables XXVIII and X X I X , where the initial value of k, is still smaller, and the error due to the reverse reaction greater, this increase of R, is very marked. In the remaining Tables, XX-XXX, the error due to the reverse rate at the last measurements never exceeds I O percent, and the constants are practically constant in each table, although different values are obtained in the different tables. T h e following table gives a summary of the resiilts of the recalculations. T h e average initial values of R, and R,, and the average final values of K5 are given. T h e “percent error” at the last measure-
W. C. Bray
584 dX
ment = IOO
dt measured
d X -
is given in the last column.
T h e last
dt correction column refers to the change in the value of 12, during the reaction, and - mean an increase, a great increase, and a decrease, respectively. I n the remaining cases, K , is constant.
+, + +,
SUMMARY, TABLES XVIII Table
i
1
xx
'
XXI XXII XXIII XXIV XXV XXVI XXVII XXVIII XXIX
xxx
XXX.
Initial values
C
x 10
I
XVIII XIX
TO
io.138; i0.138; 1 5.64 i3.76 ,1.288 i0.644 11-88 1.88 I .88
3.76 2.27 2.27 1.88
9.4 0.29 10.13 9.4 0.29 i0.13 I2 2 2 . 8 5 0.13610. 22.85 0.136~0.12 18.25 0.29 10.13
~1.7
1.8
6 .o
I .8
0.4 5.6
I .04
18.1 7.0 9.7 5 .o
1.7 2.3
-
218 300
I ,
'
2 i
-
I .o
IO
2.10 I .52
I .6 I .8
98
0.8
2I O 01
1.01
-
0.7 0.98 0.81 -
21
IO
+ +
2.63
0.85
0.64 1.74 0.5 I 0.238 0.40 0.085 0.29 0.96 1 . 1 2 2.08
Final Per- Recent marks value k, X 106 error k ,
j
-
+
+
+ 8fJo )++ 2 1
965 IS+ 31
T h e values obtained for K , are practically constant in each experiment as long as the error due to the arsenious acid reaction is negligible. This proves that the rate of the reaction is proportional to the first power of the arsenic acid concentration. Owing to the presence of excess of hydrogen ions, the dissociation of the arsenic acid, although very small, remains constant during each experiment ; and the reacting substance may be an ion (as H,AsO,'), or the undissociated acid. T h e evidence seems to be in favor of the latter view, although for the present the question must remain an open one. T h e initial values of K5 vary in the different experiments between 0.3 X IO-^ and 2.6 >(: IO-^, which shows that the rate is not proportional to the concentration of the sulphuric acid or the potassium iodide in the somewhat concentrated solutions
.
Kziaetz'c Measuvenze?ats
'
585
used. (Sulphuric acid is nearly four times normal in Table XXIII.) T h e change in the rate produced by varying the initial concentrations is most easily determined by comparing the initial values of R, in the different experiments in the above table. As these constants are practically the same as the constants given in the original paper, the relationships given in that paper remain unaltered. T h e agreement with the theoretical order of the reaction is much better in dilute solutions than in concentrated, and Roebuck concluded that the reaction corresponding to Equation V was a limiting case which could only be realized in dilute solutions. T h e new result that in a dilute solution of potassium iodide the rate is proportional to the first power of the potassium iodide concentration agrees with this view. T h e constant k, i n dilute solutions is, however, smaller than the theoretical value. No satisfactory explanation why the simple reaction should not show its theoretical order has been found. It is true that these solutions are often fairly concentrated and contain large quantities of neutral salts, but the departure from the theoretical order appears to be too great to be explained by this cause alone. I n such cases it is customary to assume that a second independent reaction takes place. It is quite possible that a formula containing two terms might be found to express the results. T h e second reaction would have to be of the second or higher order with respect to the potassium iodide and sulphuric acid concentrations. Attempts to construct such a formula and speculations on the nature of the second reaction seem at present useless. Here again the small value of the constant does not agree with the explanation. If two independent reactions take place, the constant ought to be greater than the theoretical constant, because the equilibrium measurements were not made under conditions where the second reaction could have much influence. T h e possibility suggests itself that the equilibrium constant, and perhaps the function itself used to express the equilibrium, may not be correct. Although this is not probable, further ex-
586
W. C. B a y
periniental work along this line seems necessary. T h e effect produced upon the constant by changing the concentrations of the hydrogen and iodine ions ought also to be measured. T h e possibility that an error has been made in determining the equilibrium constant is, however, very much lessened by the following result. As the reaction approaches the equilibrium, K , in each of the experiments tends to assume a limiting value between 0.8 X IO-^ and 2.3 X IO-^. This means that K , in the neighborhood of the equilibrium seems to agree with the thoeretical value calculated from the equilibrium constant, and the constant k, of the arsenious acid reaction. T h e fact that K , remains constant in some experiments, in others increases, and in another decreases, is strong evidence that the correction applied for the reverse rate is accurate; in other words, we have no evidence that the formula which was obtained for the arsenious acid reaction at a distance from the equilibrium does not apply near the equilibrium. I t will be interesting to see if further experimental work K , tends towards the theoretical will confirm this result-that value. If it proves to be true,' then there is a much better agreement with the theory. T h e theoretical reaction takes place near the equilibrium, and probably at a distance from the equilibrium in dilute solutions. I n the latter case it must be assumed that the rate of the reaction .is retarded in some yet unknown way. T h e higher values of the constants in the concentrated solution may mean that this unknown influence ceases to have an effect in the concentrated solutions or that a second reaction is taking place, as explained above. T h e assumptions that, as the equilibrium is approached, the constant of the reaction can change, or that a coniplete change can take place in the nature of the reaction, are neither impossible nor improbable. T h e final result then of the recalculations is that there is a better agreement with the theory than was claimed in the As the equilibrium constant is itself not perfect, it is not to be expected that the limiting value for k j will be identically the same in the different experiments.
Kinetic Measuremeizls
587
original paper. T h e rate is proportional to the first power of the potassium iodide concentration in dilute solution ; the theoretical reaction appears to take place in the neighborhood of the equilibrium ; and a higher value for K , than 0.326 X IO-^ must be assumed. Considering the complicated nature of the reactions, and the fact that the solutions are fairly concentrated and contain large quantities of neutral salts, the agreement is remarkable. I t is, however, not a perfect one, and there are a nnmber of difficult problems which can only be solved by further experimental work. Mr. Roebuck has informed me that he has continued his work upon this subject during the past winter and that he expects to publish his results very shortly.
Summary I. A method had been described for using the differential equation in calculating the results of kinetic measurements. I t is of value when the integration is impossible or leads to a very complicated expression.
2.
I t depends on the determination of
action at any time.
T h e methods of
dX z, the rate of a redX determining a were
described. 3. T h e method was used to calculate the constant in the neighborhood of the equilibrium for the reaction between arsenic acid and potassium iodide in acid solution. T h e results obtained show that the method is a practical one. 4. I t was also pointed out that, when the constants for a set of experimental results have been calculated for one formula, another formula may be tested without integrating the second differential equation. This generally lessens the labor of calculation very considerably. T h e calculatioiis on which the above conclusions are based mere made in April, 1904. Leipzig, April, 1905.
