4029
ANALYSIS OF SOLUTION KINETICS DATA
The Analysis of Solution Kinetics Data Coupled with Thermal Transients in an Adiabatic Calorimeter. I by E. D. West* The National Bureau of Standards, Boulder, Colorado 80302
and W. J. Svirbely* Department of Chemistry, University of Maryland, College Park, Maryland
20742
(Received M a y 24, 1971)
Publication costs borne completely by T h e Journal of Physical Chemistry
Calorimetric measurements of reaction rates in solution have generally suffered from inadequate treatment of the effects of the temperature gradient in the calorimeter. This paper is based on a more adequate theory
of adiabatic calorimetry which predicts the effect of the thermal transients on the time-temperature observations from which rate constants must be calculated. The theory treats the calorimeter as a linear system with a characteristic response to an instantaneous energy input; in other words, the temperature output is related to an energy input through a kind of transfer function. The form of the characteristic response is predicted from the heat flow problem, and the necessary constants are determined experimentally. The forms of the time-temperature relationships are determined for first- and second-order reactions. The theory has been tested for the saponification reaction of ethyl acetate in aqueous ethanol. Rate constants calculated by simplified methods show large trends during the experiment. Calculations of rate constants by the relationship derived for second-order reactions are self-consistent throughout the experiment.
I. Introduction The advantages of kinetic measurements by calorimetry have been set down by Sturtevant.* “AIany physical methods require the use of relatively concentrated solutions in order that an appreciable change in the observed property be obtaincd. The great sensitivity attainable in calorimetric mcasurements would make i t possible to work with dilute solutions even with reactions involving very small heat changes, and in many cases to obtain an accuracy not easily reachpd by other methods.” One of the disadvantages has been that some of the data cannot be used in the analysis. For example, Sturtevant found i t necessary to discard data taken in the first 20 min. We attribute this disadvantage to an inadequate theory of calorimetry. We propose a mcthod of overcoming this disadvantage and other possible inadequacies of the method due to thermal transients in the calorimeter. Applications have been made of calorimetry to thermochemistry and to kinetic^.^ Yet, the theory of calorimetric measurements used can hardly be called highly developed in the sense of starting from basic physics and proceeding rigorously to the details of the mcasur ement A better theory might be expected to give some insight into the problem of evaluating the heat exchange due to thermal transients taking place during the reaction or some other heating process. For thermochemistry, the important question is whether or under what conditions the electrical calibration takes these
.
transients into account. For kinetics, the problem is to relate the obwrved temperatwe as a function of time to the internal energy and the heat exchange. Recently, West and Svirbely4 have developed for adiabatic calorimeters a theory of measurement based on the first law of thermodynamics and the equations for heat transfer by conduction and radiation. They analyze the problem of heat flow in an adiabatic calorimeter and its effect on the temperature as a state variable as well as on the heat transfer and on the evaluation of the electrical and the mechanical work done on the calorimeter. In this paper, we develop the theory insofar as it applies to kinetics. We also present an analysis of the data obtained from a study involving the saponification of ethyl acetate in ethanol-water mixtures to test the thcory.
11. Experimental Section A. Chemicals. Ethyl Acetate. Reagent grade ester, obtained from J. T. Baker Chemical Co., was frac(1) Abstracted from the Ph,D. thesis of E. D. West, University of Maryland, 1969. (2) J. M. Sturtevant, J . A m e r . Chem. Soc., 59, 1528 (1937). (3) (a) J. M. Sturtevant, Physics, 1, 232 (1936); (b) G. Laville, C. R. Acad. Sci., 240, 1195 (1955); 240, 1060 (1955); (c) E.Calvet and H. Prat, “~~icrocalometrie,” Masson et Cie, Paris, 1956; (d) E.Calvet and F. Camia, J . Chim. Phys., 55,818 (1958); (e) F.Becker and H. Hoffman, 2. Phys. Chem. (Frankfurt a n M a i n ) , 50, 162 (1966); (f) F. Becker and F. Spalink, ibid., 26, 1 (1960); (g) H. J. Borchardt and F. Daniels, J. A m e r . Chem. Soc., 79, 41 (1957); (h) R.P. Bell and J. C. Clunie, Proc. Roy. Soc., Ser. A , 212, 16 (1952); (i) C.H.Lueck, L. F. Beste, and H. K . Hall, J . Phys. Chem., 67, 972
(1963). (4) E. D. West and W. J. Svirbely, t o be published.
The Journal of Physical Chemistry, Vol. 7> k , as would be the case for a relatively slow reaction, eq 17 would decay to single exponential term. However, cven then the two equations would not be identical due to the factor b / ( b - k ) . However, if -In
I;( I 3 __-
is plotted against t, a straight line would be obtained at large values of t. The slope of the linear section would be k (curve 1, Figure 5). However, a similar plot would be linear, with a slope of k , over the complete time range of the experiment if the temperature gradicnt is negligible (curve 2, Figure 5 ) . The plots in Figure 5 arc based on the assumed values of k and b equal to 0.0648 and 1.27 min-’, respectively. The significant conclusion deduced from Figure 5 is that, although the heat of solution is zero, there is an “incubation” period. The commonly observed ‘(incubation” period can therefore be explained by the temperature gradient by itself. This is probably a general
Time, min. Figure 5 . Effect of the thermometer response function on first-order reactions.
result, applicable whenever the observed system contains a delaying element. D. The Thermometer Response Function f o T SecondThe Journal of Physical Chemistry, Vol. 76, N o . $6,197I
E. D. WESTAND W. J, SVIRBELY
4036 Order Reactions. To apply the analysis developed in section I I I B to the kinetics data obtaincd in this research, it is necessary to detcrmine the constants in the response function given by eq 10. To answer this question, two kinds of experiments were carried outa series of solution experiments and a serics of electrical heating experiments. I n the solution experiments, ethyl acetate was addcd to 55 wt % ethanol-water solution. A summary of the temperature measurements is recorded in Table 11. Table I1 : Data for Solution Experiments Wt, g
0.4143
Time, min
0 0.5 1 1.5 2 m
0.3082
0 0.5 1 1,5 2 m
0.4757
0 0.5
1 1.5 2 co
0.4134
0 0.5 1 1.5 2 m
The electrical heating experiments were devised to give the same information as the solution cxperimcnts discussed above. This can bc done provided the clectric heater immersed in the solution is the thermal cquivalent of a heat, of reaction, For this reason, the heater should have a very low heat capacity and should be in good thermal contact with the solution. The total energy supplied by an electrical heater operating a t constant power, p , is the product of the power and the time, t. To find T o - T , for constant power input, me perform the convolution of ( p t ) and the response to thc step function given by eq 10. We obtain
Temp, OC
25,3048 25.2781 25.2692 25,2634 25,2603 25.2562 25.2040 25,1848 25.1775 25.1733 25,1705 25.1677 25.0599 25,0288 25.0178 25.0116 25.0090 26,0044 25.1252 25,1009 25,0881 25.082 25,0786 25 0752
0.450 0.267 0.148 0,083
0.470 0.269 0.154 0.077 0,439 0.242 0.129 0.082
0.514 0.258 0.136 0.068
Because the constant p can be brought outside the integral, the integration can be performed to give eq 20
If the power is held constant for a long time and then shut off at time, t', which is so large that all e-*it are negligible, the corrected temperature, T,, for t > t' is obtained'" by adding to eq 20 the equivalent of eq 20 written for - p and starting at t'. For t > t' we obtain eq 21
Simplifying, we obtain
I
I n the ratio of temperature differences in the last column, T ois the observed temperature corrected for heat leak, T , is the ultimate value of T,, usually taken after about 10 min, and Ti is the initial value of T obefore the ester is added to the solution. Our first problem is to determine how many terms of eq 10 are required to represent the data of Table I1 adequately. Equation 10 can be transformed to eq 10a
To determine whether one exponential term is adequate, the logarithm of the ratio of the temperature differences is plotted against time. Reference to eq 10a shows that if one exponential term is adequate the plot should be a straight line with a slope equal to -b. Since such a linear relationship was obtained, the response of the thermometer function has been taken to include only one exponential term. A value of b = 0.75 min-I was obtained. The Journal of Physical Chemistry, Vol. 76, N o . 26, 1971
On defining T ' = T , in eq 20 at the time t', and subtracting T' from both sides of eq 22, we obtain eq 23
When t >> t', To approaches a constant value which we define as T,. On dividing eq 23 by its equivalent for t >> t ' , we obtain eq 24
For a single exponential term as was demonstrated to be valid in the solution experiments, the summations reduce to a single term and cancel. On subtracting the resulting equation from (10) B. M . Brown, "The Mathematical Theory of Linear Systems," Wiley, New York, N. Y . , 1981, p 82.
AXALYSIS OF SOLUTION KINETICS DATA
(T,
- T')
(T, - T')
4037
= 1
we obt'ain
Tm - T, T , - T'
- 1')
e-b(l
Evidently, a plot of
against t will have a slope of b. Experimentally, the power is turned onto a heater coil of fine manganin wire suspended in the solution just above the stirrer. After 5-10 min, the power is turned off and the temperature is followed as a function of time. This experiment can be repeated much more readily than the solution experiments and permits repetition for exactly the same amount of solution. The data are plotted in Figure 6. From the graph, the value of b = 0.84 min-I mas obtained. E . The Analysis of the Kinetics Data. On defining IC' = k(B0 - Ao), eq 7 for a second-order reaction can be rewritten as eq 26
A A.
-- -
dr
=
A0
(Bo- Ao) Bo
lc'e"''
(1
-
5
e-kfr)'
(27)
BO
On inserting this derivative into eq 13, along with the experimental condition that only one exponential term is required, w e obtain eq 28
T,- Ti
=
(T4 - Ti)(l - e-at)
+ ( T , - T4) X
On rearrangement and solving for (A0 - A)/Ao), we obtain eq 29
AO - A Ao _ I _
(T, = -
- Ti) - (T4 - Ti)(l ( T , - 7'4)
I
I
1.0
46
F
I.!
Time, rnin. Figure 6. Determination of the thermometer response function from electrical heating experiments (eq 2 5 ) .
(BO- Ao)e-k" (Bo- Aoe-"c)
where Ao, A , and Bo represent the concentrations of starting ester, existing ester, and starting hydroxide ion, respectively. On taking the derivative of A with respect to the time, 7,the ratc equation is given by eq 27
_-d A
I0
- e-bc) +
The temperature Td, which would be reached by solution alone, is obtained from the solution experiments, and the constant b is obtained from the heating ex-
periments. The integral is evaluated by numerical integration, using a value of k' obtained by successive approximation. The integral is in the nature of a correction term. For largr values of t, either e-k'r or e - b ' t -') is small, so the integral will contribute less toward the end of the experiment than it will in the early part. For a calorimeter which equilibrates very rapidly, b is large and the integral will have a small effect. As a check, we note that, if the effect of the temperature gradient in the calorimeter is neglected, eq 29 reduces to eq 6 which was developed from the simple theory. The ultimate corrected temperature, T,, is estimated from the rate equation. On rearranging eq 8, we obtain eq 30
T,
=
T,
+ [ ( T , - T4) - AoBo- ( T c
- T4)]e-"' (30)
It is usually adequate to use the last observed temperature for T, on the right-hand side, obtaining eq 31
The data of Table I have been treated by eq 29 to obtain values for the concentration, A . These values are then used in eq 7 to calculate the rate constant, IC. The results are tabulated in Table 111. The third column of Table I11 represents the contribution of the first The Journal of Physical Chemistry, Vol. 76, N o . 26, 1971
4038
E,. D. WESTAND W. J. SVIRBELY
term on thc right-hand side of cq 29 and the fourth column rcpresents the integral correction. Together they represent the fraction of thc estcr used up. All of our cxperiments werc carried out to a t least 96% completion.
Table I11 : Rate Constant Calculations Time, min
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
5.5 6 6.5 7 7.5 8 10 10.5 11 11.5 14 14.5 15 15.5 19 19.5 20 20.5 24.75 34.76
Rate oonstant x 108, 1. mol-sea-1
9.00 8.49 8.92 8,73 8.66 8.58 8.52 8.57 8.59 8-56 8.54 8.45 8.48 8.43 8.41 8.39 8.41 8.40 8.41 8,44 8.44 8.45 8.45 8.47 8.56 8.56 8.57 8.56 8.65 8.88
-
1 A/AQ, uncorrected
0.014 0.039 0.080 0.116 0.154 0.190 0.226 0.264 0.298 0,330 0.361 0,387 0.416 0,441 0.466 0.490 0 577 0.595 0.614 0.632 0.708 0.721 0.734 0.747 0.819 0.827 0.835 0.842 0.896 0,961 I
Time, mln.
Correotion
term
0.036 0.054 0.062 0 065 0.065 0.063 0.061 0,059 0.056 0.054 0.051 0.049 0.046 0,044 0.042 0.040 0.033 0.031 0.030 0.028 0.022 0.021 0.020 0,019 0.014 0.014 0.013 0.012 0.008 0.003 I
To illustrate the effect of the integral term, the rate constants for the data of Table I have been calculated with and without the integral correction term. The lower curve in Figure 7 is a plot of the rate constant calculated without the correction, but after allowance for the heat of solution. For the upper curve, the rate constants were calculated using the correction. For the first 2 min, both curves appear to suffer from difficulties in accounting for the heat of solution, but the corrected values are strikingly more consistent than the uncorrected ones. After the first 2 min, the lower curve obviously undergoes a trend which is still evident at 20 min when the reaction is 84Q/, complete. By contrast, the rate constants represented by the upper curve are quite consistent. We take this as evidence for the correctness of the formulation by which they were calculated.
The Journal of Physical Chemistry, Vol. 76, N o . $6, 1971
Figure 7. Effect of the correction term on the calculated rate constant.
Table IV : Summary of Rate Constants Initial ester concn, mmol/l.
17.48 43.00 39 28 31.47 4 . t537 15.46 4.011 I
Initial hydroxide
Temp range,
mmol/l.
OC
wt % ethanol
l./mol-sac
173.98 173.54 173.60 173.74 174.21 191.74 192.00
0.1183 0.2995 0.2786 0.2193 0,0326 0.1006 0,0280
83 83 83 83 83 85 85
9.74 9.85 9.66 9.75 10.35 8.55 8.07
k
x
10'
Table IV summarizes the rate constants for all of our experiments, By interpolation of the data in the literature6J~llfor 7.05 X and this reaction, IC values of 6.21 X 6.4 X 10-8 l./mol-sec in 85% ethanol-water mixtures l./mol-sec in 83% ethanol-water mixand 7.3 X tures are obtainable. It does not appear that additional terms in the thermometer response function could account for the differences between these literature values and the values listed in Table IV. The corrected rate constant must be greater than the uncorrected value because the effect of the calorimeter is to delay the observed temperature rise, which therefore indicates too large a concentration and too small a rate.
Conclusion When heat is evolved or absorbed during a reaction occurring in an adiabatic calorimeter, there is a delayed response in the observed temperature which should be a measure of the energy input or output. This delayed response can virtually destroy time-concentration data in kinetic studies unless recognition is taken of the temperature gradient. The treatment of the data by a linear theory of calorimetric measurements, which is developed in this paper for application to kinetic studies, gives rate constants which show very little trend with time during the entire experiment. (11) E. Tommila, A. Koivisto, J. P. Lyra, K. Antell, and S. Heimo, Suom. Tiedeakat. Toim., 47,3 (1952).
