4039
ANALYSIS OF SOLUTION KINETICSDATA
The Analysis of Solution Kinetics Data Coupled with Thermal Transients in an Adiabatic Calorimeter. 11. First-Order Reactions by W. J. Svirbely,* University of Maryland, College Park, Maryland 2074.2
E. D. West, The National Bureau of Standards, Boulder, Colorado 8080.2
and F. A. Kundell Salisburu State College, Salisbury, Maryland 21801
(Received M a y 24, 1971)
Publication costs borne completely by The 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 a calorimeter. Recently we presented a theory of adiabatic calorimetry which predicts the behavior of the observed temperature as a function of time due to the combined effects of a heat of reaction, a heat of solution, and the thermal transients of the calorimeter. This paper demonstrates the utility of the theory for a pseudo-first-order reaction through use of the data obtained in the saponification of ethyl acetate in the presence of a large excess of hydroxide ion. The results are quite satisfactory, thus substantiating the calorimetric theory of kinetic measurements for first-order reactions.
I. Introduction Calorimetric measurements of reaction rates in solution have generally suffered from an inadequate treatment of the effect of the temperature gradient in a calorimeter. A theory of measurement' has been recently developed for adiabatic calorimetry which predicts the behavior of the observed temperature as a function of time due to the combined effects of a heat of reaction, a heat of solution, and the thermal transients in the calorimeter. Appropriate equations were derived from the theory for first- and second-order reactions. The predictions of the theory for second-order reactions were tested by a study of the saponification of ethyl acetate in ethanol-water solutions. Calculations of the rate constants for that second-order reaction were self-consistent throughout the experiment, I n this paper we will test the predictions of the theory for first-order reactions. 11. Mathematical Procedure The equation, which was developed' for first-order reactions, is shown by A0
-A
.~
A0
( T , - Ti) -
- ( T , - Ti)(l - e--bl) (Tm- T4)
+
k (e-ac - e-b2) (1) b - IC
where A0 and A are starting and existing concentrations, respectively, of the reactant a t time t; T,, Ti, Tq, and T , are observed temperature, initial temperature, temperature resulting from a heat of solution effect
but prior to the beginning of the reaction, and temperature a t infinite time, respectively; b is a constant of the calorimeter; and k is the first-order rate constant. For a first-order reaction, the integrated form of the rate equation relating concentration to time is given by
A
=
Aoe-kz
(2)
On substituting eq 2 into eq 1 and rearranging the resulting equation, one obtains eq 3
(T, - T,)
+
(7'4
-
Ti)e-*'
=
ke -lJt (Tm - T"[-
-
be-kt
]
(3)
On defining (T1 - Ti)and ( T , - T4) by P and r , respectively, and on substituting these definitions in eq 3, one obtains eq 4
A procedure for evaluating the rate constant, k, through direct use of eq 4 is already in the literature.2 It is based on an estimate of IC, expansion of eq 4 via Taylor's expansion, least-squares solution for the correction term to the first estimate, and repetition of the process using the corrected k as the new first estimate. A program was written to accomplish the calculations. (1) E.D.West and W. J. Svirbely, J . Phys. Chem., 75,4029 (1971) (2) W. J. Svirbely and J. A. Blauer, J . A m e r . Chem. SOC.,83, 4116 (1961).
T h e Journal of Physical Chemistry, Vol. 76, N o . 26,1971
W. J. SVIRBELY, E. D. WEST, AND F. A. KUNDELL
4040 111. Discussion and Results
If a reaction takes place between two reactants, first order in each reactant, and if one of the reactants, namely B, is present in large excess (at least tenfold), then the pseudo-first-order rate constant, kl, is usually set equal t,o the product of the second-order rate constant, ICz, and the concentration of the reactant in excess, i. e., kl = kz(B). Some of the data we had obtained' for the saponification reaction of ethyl acetate was of this type and in those cases the reaction can be considered to be a pseudofirst-order reaction. To save computer time, we selected for our first estimate of kl a value based on the kz value previously determined. Since values of 2'4, Ti, T,, b, and timetemperature data were available, a corrected value of kl was obtained by the least-squares method based on eq 4. Values of T,(calcd) were obtained next for each of the experimental times in an experiment through use of eq 4 and the first corrected value of kl for the experiment. The T,(calcd) values were then compared with the T,(obsd) values and the quadratic mean error of fit was obtained. The procedure was repeated using this first corrected value of k1 as a new first estimate. The final results for one of the experiments are shown in Table I. The third column of Table I lists values of T,(calcd) obtained through use of the value of kl = 0.0974 min-'. It is gratifying to see how well eq 4 duplicates the experimental temperatures, including the decrease at the start of the experiment due to the endothermic heat of solution and then the upswing in temperature due to the exothermic heat of reaction. The agreement demonstrates the adequacy of eq 1 to describe first-order kinetics when the temperature gradient is taken into account. This fact supports our calorimetric theory of kinetic measurements. Table I1 summarizes the results for several experiments. The first-order rate constants in column 2 were obtained by the procedure just described. Column 5 represents the second-order rate constants which were calculated from the first-order rate constants in column 2 by means of the relation
~
Time, min
Column 7 lists the second-order rate constants of the original paper.l The comparison of the values listed in columns 5 and 7 indicates quite good agreement and thus serves as a check on the overall procedure. Acknowledgment. We wish to express our appreciation to the Computer Science Center of the University of Maryland and the National Aeronautics and
The Journal of Physical Chemistry, Vol. '76, No. 86, 1972
Temp,
O C
(exptl)
25.0216 25.0202 25.0204 25,0211 25.0217 25,0228 25.0238 25 * 0252 25.0261 25.0282 25.0294 25.0304 25.0309 25,0334 25,0344 25.0351 25.0357 25.0375 25.0383 25.0388 25,0391 25.0402 25.0409 25.0413 25.0415 25.0432 25.0437 25.0441 25.0441 25.0461 25.0476 25.0490 25.0496
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 5.00 5.50 6.00 6.50 8.00 8.50 9.00 9.50 11.00 11.50 12.00 12.50 14.00 14.50 15 * 00 15.50 18.00 18.50 19 I O 0 19.50 23.75 28.75 38.7;i 48.75
Temp,
O C
(calcd)
25.0216 25.0206 25.0205 25,0209 25,0217 25.0226 25,0236 25,0247 25,0257 25.0278 25,0289 25.0298 25,0308 25.0334 25.0341 25.0349 25,0356 25.0375 25.0381 25,0387 25,0392 25.0407 25.041 1 25.0415 25.0419 25,0436 25.0439 25,0442 25.0445 25.0463 25.0477 25.0491 25.0496
b = 0.84 min-1 T , = 25,0499' T4 = 25.0180" Ti = 25.0216'' (OH)- (concn) = 0.19200 mol/l. Ester concn = 0.00401 mol/l. kl = 0.0974 min-l Quadratic mean error of fit = 0.03%
Table 11: Summary of Data
(OH)-,
kl,a
kl k ' - 60(OH)-
~~~~
Table I: Time-Temperature Data for a Run
min-1
0.1002 0.1030 0.1166 0.0974
min-1
mol/l.
0.0983 0.1031 0.1173 0.0974
0.19174 0.17398 0.17421 0.19200
Eater, mol/l.
&ME kz x 108, of kz x lo', I./molfit, I./molaeo % seaC
8.54 0.13 8 . 5 5 0.01546 9.88 0.12 9.74 0.01748 0.004337 11.22 0.04 10.35 0.004011 8.43 0.03 8.07
First corrected value. bSecond corrected value. ence 1.
Refer-
Space Administration for Grant NsG-398 applicable to comput,er time.
