Anal. Chem. 1980, 52,365-367
365
Determination of Pseudo-First-Order Reaction Kinetics by Batch Microcalorimetry David F. Sargent” and Hans Joerg Moeschler’ Institut fur Molekularbiologie und Biophysik, ETH-Honggerberg, CH-8093 Zurich, Switzerland
T h e determination of reaction rate constants from microcalorimeter experiments has been described previously for various instrumental setups, and its validity is well established (1-6). T h e main problem is correcting the output signal for the large thermal inertia usually present. Bell et al. ( 1 ) derived rate data from t h e maximum change of temperature in an apparatus having virtually no time lag in the temperature sensing system. For systems showing larger and more complex time lags Thouvenin e t al. ( 3 )described an electronic analog circuit for on-line correction of the output signal while Camia ( 4 ) employed numerical integration. A recent report on the use of flow microcalorimeters has been given by Johnson e t al. (6). T h e limiting time constant has varied from fractions of a second ( 4 ) to several seconds ( 1 , 6 ) ,depending on, among other things, the computational complexity involved. In this report we describe a simple graphical method for extracting rate information for f i t - o r d e r (or pseudo-first-order) reactions using a commercially available batch microcalorimeter. T h e minimum reaction time constant that can be resolved is about 1 s.
EXPERIMENTAL The response of a batch microcalorimeter (e.g., LKB 10700 or similar device) can be modelled by a system of elements, each having a first-order response, in a series configuration (Figure 1). The number and response times of elements needed to describe the observed signals to within the experimental uncertainty can be found by analyzing the outputs to pulse inputs (electrical heating or fast chemical reactions). The time course for chemical reactions must, of course, be calculated from the moment mixing actually starts. To account for the time elapsed between activation of the “start mixing” switch and actual mixing, a delay element was introduced explicitly into the mathematical analysis ( A t = 4 s gave the best fit). For practical purposes it is not necessary to identify the time constants derived from such a “black box” analysis with physical components or processes. Nevertheless one can expect the results to reflect certain elements such as the equilibration times of the thermocouples with the reaction solutions. A further, easily identifiable process is the final reestablishment of thermal equilibrium between the reaction chamber and the thermostated reservoir. All these processes will depend on the physical parameters of the particular apparatus being used, and must therefore be determined for each experimental setup, as will now be described. The response to a pulse input of a system of fit-order elements in a series configuration, as shown in Figure 1, is described by n
R =
(1/71~2
. . . 7,) iC= 1 [ e x p ( - t / ~ J /1#1r l ( I / T ,
-
~ / T J ]
(1)
where n is the number of elements considered. Using standard exponential curve fitting techniques (e.g., 7 ) the minimum number of elements needed to fit the pulse response, and their corresponding time constants, are found. The response of the system to a given input can now be calculated using Laplace transforms and the experimental parameters found using curve fitting programs. For pseudo-first-order reactions the response is simple enough to be evaluated without constant recourse to a computer. The time at which the maximum occurs and the time course of the subsequent decay are sufficient to allow a reliable determination of the reaction time constant. ‘Present address: DBpartement de Biochimie, Case Postale 78 Jonction, CH-1211 Gen&ve8, Switzerland. 0003-2700/80/0352-0365$01.00/0
The time to maximum output can be found as follows. For an exponential input the output of the system described in the preceding paragraph is:
where T~ is the input (reaction) time constant, and T~ to T , are the previously determined system time constants. The time to maximum output is found by differentiating Equation 2 and equating to zero:
Solving for t, as a function of T~ with the constants T , (i = 1, n) as parameters, one can plot a curve from which the reaction time constant of a pseudo-first-order reaction may be read directly. As Equation 3 has no exact mathematical solution, numerical techniques must be used. A simple iteration program in BASIC is available from the authors. The final decay of the reaction curve is used to check the validity of the assumption of pseudo-first-order kinetics. If the decay constant is the same as that found with the heating pulses, then pseudo-first-order conditions prevail. If the decay is nonexponential or has an apparent time constant longer than that determined for a pulse input, then the reaction is either very slow or doesn’t follow first-order kinetics. In the former case the time constant of the reaction will be about equal to the time constant of the decay segment, and the peak will occur at a time consistent with this value. (For reaction time constants between 0.75 and 1.5 times the decay constant, the tail may appear to have a slightly longer time constant if it is not followed long enough.) If pseudo-first-order kinetics do not apply, the peak will come much earlier than predicted by Equation 3. A final practical point must be mentioned. The reaction in the microcalorimeter is started by a mechanical mixing of the reactants. The friction of the moving parts generates heat, causing a mixing transient unrelated to the heat of reaction. For determination of the exact position of the peak output this transient must be subtracted from the raw experimental curve. An example of this procedure is given in Figure 2.
RESULTS AND DISCUSSION T h e response time constants of a n L K B 10700 batch microcalorimeter were determined using both a n electrical heating pulse and the complexation reaction of valinomycin and potassium in ethanol: valinomycin
+ K+ + valinomycin.K+
(1)
T h e initial concentrations of the test solution were: valinomycin, 3.3 X M; K+, 3.3 X M (pseudo-first-order conditions). Using the rate constants k+ = 3.5 x lo7 M-ls-l, k- = 1.2 x lo3 s-l, found in methanol (9),this yields a reaction s, Le., virtually instantaneous time constant 70 = 8 X compared with the response of the microcalorimeter. Both the electrical pulse and the chemical reaction released about 10 mcal. Figure 3 shows the chart recorder trace for both tests. The response t o the electrical pulse rises earlier as there is no mixing delay. The trace from the chemical reaction is seen to rise more steeply, however, showing that the initial rate of heat production is higher once mixing occurs. This stimulus is therefore a better approximation t o a n ideal pulse input and was used in determining the system parameters. T h e observation also indicates that the rate of mixing is fast enough not to influence the overall dynamic response. In the analysis, 1980 American Chemical Society
366
ANALYTICAL CHEMISTRY, VOL. 52, NO. 2, FEBRUARY 1980 ll0,
1 -
Microcalorimeter
I
Figure 1. Model for the analysis of the response of the total measuring system. The unknown parameters, T , , of the microcalorimeter may stem from equilibration times for the reaction chamber/thermocouple and reaction chamberlthermal reservoir ,,
O..,,.''
.
.. , .
,
'.
'.
. . ,..... . .
........ ..
'.
....,.........
I I
--
t
oc-T---.IO
0
20
30
LO
50
60
70
time (sec) Figure 2. Correction of the raw data for the mixing transient (LKB 10700
Batch Microcalorimeter). The correction factor is found experimentally by repeating the mixing action until only a reproducible transient is obtained (upper curve). Lower section: original recording (-) and corrected curve (....)
20 I 0
20
10 6C 80 10C reaction time constant ( s e c )
120
110
'6C
Figure 4. Time of occmence of the peak in the microcalorimeter output (Equation 3) as a function of the reaction time constant for various values of the decay time constant, Other parameter values are the same as in Figure 3 5.
i 0
4
0
. 10
. 20
-.
30
..
~
LO
50
60
70
time (sec)
Microcalorimeter response to a 1-s electrical heating pulse "instantaneous" chemical reaction (see text). The line represents the response of a fourth-order system (time constants = 1.8 s (filter),4.5 s, 4.5 s, and 85 s)to a delta function input (Equation l), after allowing for a dead time of 4 s (mixing delay)
Figure 3. (-) and an
( 8 )
the voltmeter and chart recorder time constants could be neglected. In addition to the filter time constant of 1.8 s, a third-order microcalorimeter response was ndeded to fit the data ( T ~E 7 2 2 4.5 s, 7 3 = 85 s). The longest time constant, T~ 7 d (decay constant), represents the equilibration of the reaction chamber with the heat bath, and depends on the thermal capacity of the chamber contents. The value 7 d = 85 s was found with 6 mL of ethanol as solvent. The origin of the two shorter time constants is not certain, but they may reflect the response times of the thermoelements (one for each of the two chambers). No variation in the values of T~ and 72 with changes in reaction chamber contents was observed. The theoretical time to maximum calorimeter output was calculated according to Equation 3. This is plotted in Figure 4 as a function of the reaction time constant, with Td as a parameter. (As 7 d depends on the sample itself, it was determined after the experiment using an electrical pulse with the reactants still in the microcalorimeter). The minimum resolvable time constant is about 1 s.
O
L
0
20
. LO
t _ ,
80 time (sec) 60
100
120
Figure 5. Experimental points ( * ) and the calculated time course (-) of the microcalorimeter output for a slow pseudo-first-order reaction
(see text). Deviations in the rising portion come from uncertainties in the mixing transient correction and inadequate time resolution in the original recording. Best fit in the peak region was found with a reaction time constant of 9 f 2 s The response of the system to a reaction having pseudofirst-order kinetics and a suitable time constant is shown in Figure 5. In this case reaction I is again relevant, but the synthetic peptide S,S-bis(cyclo-glycyl-L-hemicystyl-sarcosyl-sarcosyl-L-prolyl) (IO)was used instead of valinomycin. The peptide concentration was 3.3 X lo4 M, K+ concentration 1.5 X M. These kinetic data, combined with the equilibrium binding constant determined separately, allowed the individual rate constants to be calculated (IO). When pseudo-first-order conditions are not fulfilled, an extended tail is encountered. An example is shown in Figure 6, where both peptide and K+ concentrations were 3 X M. The decay appears to be described by a single exponential of time constant 140 s. If the chemical reaction were really
Anal. Chern. 1980,52, 367-371
387
of the decaying portion of the curve and the time to maximum deflection, one can distinguish non-first-order kinetics.
LITERATURE CITED
1
541 n
11
-
0
1
2
21
0
4
6
8
1
0
time
’
L
0
2
------.
-
1
4
2
3
time
--_--5
6
7
8
.
_ _ ,
9
10
(min)
Flgure 6. Example of a non-firstader reaction The peak value occurs at 55 s. Inset: semi-logarithmic plot of the tail region showing an apparent exponential decay with a time constant of 140 s. See text
(1) Bell, R. P.; Clunie, J. C. Proc. R . SOC.London, Ser. A , 1952, 272, 16-32. (2) Lueck. C. H.; Beste. L. F.; Hall, H. K., Jr. J . Phys. Cbem. 1963, 67, 972-976. (3) Thouvenin, Y.; Hlnnen, C.; Rousseau, A. “Les d6veloppements recents de la microcalorlmetrle et de la thermogenhse”, Vol. 156; Editions du C.N.R.S.: Paris, 1967: pp 65-62. (4) Camia, F. M. Ref. 3, pp 63-94. (5) Mietes, Thelma: Mietes, L.; Jaitly, J. N. J . Phys. Chem. 1980, 73, 3801-3809. (6) Johnson, R. E.; Biltonen, R. L. J . Am. Chern. SOC. 1975, 97, 2349-2355. (7) Provencher, S. W. Biophys. J . 1976, 76,27-41. (8) James, F.; Roos, M. Comput. Phys. Commun. 1975, 70, 343-367. (9) Grell. E.; Funck. Th.; Eggers, F. “Molecular mechanisms of antibiotic action on protein biosynthesis and membranes”; Elsevier: Amsterdam, 1972; pp 646-685. (10) Moeschler, H. J.; Sargent. D. F.; Tun-Kyi, A,; Schwyzer, R. Helv. Chim. Acta, in press.
for details pseudo-first-order, then its time constant would also have a similar value. In this case, however, the calorimeter output would peak a t about 124 s (Figure 4), whereas the peak actually occurred a t about 55 s. Thus using the time constant
RECEIVED for review August 17, 1979. Accepted October 30, 1979. Finanical support of the Swiss National Science Foundation and the Swiss Federal Institute of Technology is gratefully acknowledged.
Inexpensive Microprocessor Controlled Programmable Function Generators for Use in Electrochemistry A.
M. Bond*
Division of Chemical and Physical Sciences, Deakin University, Waurn Ponds, Victoria 32 17, Australia
A. Norris Clanor Instruments, P.O. Box 75, Balwyn, Victoria 3 103, Australia
Several review articles on the use of microprocessors in the field of chemical instrumentation have been prepared by Dessy and co-workers (1-3). The major thrust toward their use in many laboratories would be expected to arise from their extremely high computing power provided a t minimal cost. However, in reality, extremely limited use has been made to date by chemists working in small, low budget laboratories who could actually take direct advantage of the very low cost. The reason for very few chemists directly exploiting the high quality low cost performance of microprocessors has been outlined by Dessy e t al. (1-3) and the main drawback cited is the difficulty of writing the software and the belief that access to a larger and rather expensive computer available to commercial manufacturers of instruments or larger institutions is almost mandatory for this task. It is certainly true that many manufacturers and chemists who have worked in the field have had access to rather expensive minicomputer systems with facilities for writing in high level languages, editors, cross-assemblers, etc., and an impression may easily be conveyed that the microprocessor is not readily developed on its own as an inexpensive computational component in instrumentation. However, it is worth taking time to consider that such aids, while undoubtedly very useful are by no means essential and certainly not for the carefully chosen simple tasks we wish t o demonstrate in the present article. 0003-2700/80/0352-0367$01 .OO/O
In our electrochemical research, we have seen considerable advantage in the use of microprocessors for performing an enormous range of tasks a t minimal cost, provided we could confine ourselves to using only the typical microprocessor “kit” and instruction set provided by the manufacturer and undertake all program development work with very restricted memory, etc., using accessories and aids that were less expensive than the microprocessor kit itself. Working for some time within the confines of these restrictions has now convinced us that outstanding results can be obtained relatively easily without access to larger laboratory computers. Furthermore, we now believe that despite some reports to the contrary it is eminently sensible to consider the use of microprocessor technology in any low-budget laboratory as a means of obtaining high quality performance and versatility a t extremely low cost. This theme in addition to the actual scientific data provided, constitutes one of the main implications of the present report. In the present paper we describe in detail the use of a standard MOS TECHNOLOGY KIM-1 microprocessor kit, as well as programming procedures and related approaches we have used to develop a programmable function generator for use in electrochemistry. The KIM-1 module uses an 8-bit microprocessor and is representative of the earlier generation kits available a t very low cost. Indeed our entire system including interfaces to a potentiostat can be built a t a cost 0 1980 American Chemical Society
