Rapid Method of Determining First-Order Rate Constants from Experimental Data Richard 0.Vale Department of Biochemistry, School of Medicine, University of Pennsylvania, Philadelphia, Pa. I9104
IN INVESTIGATING METHODS of estimating decay constants for multiple exponential kinetic curves, a tabulation or graph of tf(t) us. time, t, was found to be useful. When f ( t ) is directly proportional to the percentage of reaction, the function tf(t) exhibits under some conditions one maximum for each exponential term; furthermore the time of each maximum is approximately equal to the reciprocal of the decay constant of t h e respective exponential term. When a single first-order reaction is being measured, t f ( t ) has one maximum when time equals the reciprocal of the rate constant. Consider the (possibly pseudo first-order) reaction C $ Product
7000
r
6000
-
5000
-
t f (t) (orbi trory units)
Here
0
t f ( t ) = t[C, - C,l exp [-kobatl
(1) where C, is the initial concentration of C (initial product concentration is zero), C , is the equilibrium concentration of C, and kobsis the observed rate constant. Equation 1 has a single maximum when t,,, = l/k,b, andf ( t ) = [C, - q / e . Figure 1 shows a graphical determination of kobs for a randomly chosen experiment. The same data fitted to the common ln[C - C,] us. t function (1) yielded 0.02299 secdl compared to 0.0213 sec-l for the graphical t f ( t ) method. If the data had been collected with the tf(t) method in mind, the difference in the rate constants would even be smaller. When the kinetic analysis yields N exponentials, i.e.
IO
20
40
30
60
70
80
90
Figure 1. A graphical determination of a rate constant using the t f ( t ) method The maximum data point (circles) yields f m , , = 47 sec.; the solid line was generated from the results of a least-squares log plot; the actual tmax = 43.3 sec 0.5 r
N
f(t) =
50
t i m e (sec.1
I
A n exp [-k%tI
\ t, m o x
(2)
n= 1
If
j i
the graph t f ( t ) us. t can have as many as N maxima. This is readily apparent when one notes that tf(t) from Equation 2 is a sum of functions like that in Equation 1 . When all N maxima are observed, the nth maximum occurs when tnmal= l / k n . This case has been observed when the decay constants are widely separated and a theoretical case is shown in Figure 2. Even when all of the maxima are not observed, the kn can be estimated. The broadened peaks indicate k , of similar magnitude; more commonly, the shoulders indicate the presence of at least one k,. This method is most useful for computer curve-fitting algorithms requiring initial decay constant estimates. In the case of unknown C, (Equation 1))the rate constant can be found in a manner similar to Guggenheim’s (2). Guggenheim’s method utilizes Equation 3
c - c‘ = [co - ca](1 - eXp [-kobsA]) eXp [-kobst]
(3)
where C is reactant concentration measured at time t and C’ is the reactant concentration measured at time t A ( A , a constant, is usually one or two half-lives in magnitude) (3). Since Equation 3 can be written in the form
+
(1) W. E. Roseveare, J . Amer. Chem. SOC.,53 1651 (1931). (2) E. A. Guggenheim,Phil. Mag., 2 538 (1926). (3) A. A. Frost and R. G. Pearson, “Kinetics and Mechanism,” 2nd ed., John Wileyand Sons, New York, 1961, p 50.
0
2
4
6
8
10
I
12
14
16
18
20
t i m e lunilless)
Figure 2. The theoretical t f ( t ) function when f ( t ) = 0.1 exp[-0.1~1 10 exp[-10 t ]
+
demonstrating one maximum for each exponential term. The earlier maximum yields a low estimate of the larger rate constant
of Equation 1, i.e. t[c-
c’]= Kt eXp [ -kobst]
(4)
where K = [C, - C,] (1 - exp [-kobsA]), a constant, a graph of t[C - C’] us. t will have a maximum at tms, = l/kobm The major advantages of this method are that it is simple and rapid and that under some conditions it can yield estimates of the decay constants of a multiple exponential kinetic curve. The tf(t) method with continuous measurements could provide an electronic analog method of automatically measuring rate constants by displaying t f ( t ) . This method weights the points in the region of tm,, heavily. In return the experimenter gets an objective estimate of the error in the rate constant by calculating the reciprocal of the time measurements adjacent to the assumed
ANALYTICAL CHEMISTRY, VOL. 42, NO. 14, DECEMBER 1970
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maximum time. The graphical log plot slope determination does not offer any error estimates, but the weight of the data is more evenly distributed. Since deviations from simple first-order behavior are mathematically equivalent to increasing the number of exponential terms, the if(t> method should not be used to quantitatively detect these deviations. Practically this graphical me-thod should be less sensitive to deviations; which usually occur near the end of the reaction, than the graphical log plot. The most important data for the tf(t)plot are found
around 37 completion and the data beyond the maximum can be ignored; this is not true for log graphs. ACKNOWLEDGMENT Data used in Figure 1 were kindly supplied by Kenneth Brown. RECEIVED for review May 8, 1970. Accepted September 8, 1970.
Microdetermination of Molybdenum by Anodic Stripping at stant Current Using the Hanging Mercury Drop Electrode Philippe Lagrange, and Jean-Paul Schwing
lnstitut de Chimie, I rue Blaise Pascal, 67-Strasbourg9 France ANODICSTRIPPING at constant current or with continuously varying potential has been widely used for the determination of small concentrations of metal ions plated into a mercury cathode at controlled cathode potential (1-3). Molybdenum cannot be plated into a mercury cathode and gives in acidic media soluble reduced species. However, we could show that at pH 5 the reduction of Mo(VI) at a mercury electrode (4, 5 ) gives a solid product (MoOz.2HzO) forming a thin film that shows the properties of a semiconductor. Our aim was to achieve a method for the microdetermination of molybdenum based upon the accumulation of Moog. 2 H z 0 on the mercury drop, through cathodic reduction, followed by anodic stripping a t constant current. Only a few examples of anodic stripping of a precipitate have been studied until now ( I , 2) as a method for microanalytical determinations. The following factors have been shown to be of importance in this study: the concentration of molybdenum; for a given concentration, the preelectrolysis time, Le., the mass of M o O z . 2 H z 0 deposited; and the intensity of the anodic stripping current. The stripping time is measured as shown in Figure 1, which gives the potential of the working electrode GS. the time at constant stripping current. EXPERIMENTAL The cell comprises nitrogen inlet and outlet tubes, a reference electrode (Ag-AgCl, 3M NaCl), an auxiliary electrode and a working electrode (hanging mercury drop) prepared as indicated by Gerischer (6), and Ross, De Mars and Shain (7, 8). The weight of each mercury drop is 0.00760 gram. (1) G. Charlot, J. Badoz-Lambling and B. Tremillon, “Les reactions
Clectrochimiques,” Masson et Cie, Paris, 1959.
(2) P.Delahay, “New Instrumental Methods in Electrochemistry,” Interscience Publishers, New York, 1954. (3) J. J. Lingane, “ElectroanalyticalChemistry,” Interscience Publishers, New York, 1958. (4) P. Lagrange and J. P. Schwing, C. R. Acad. Sci., Ser. C,263,
848 (1966). (5) P. Lagrange and J. P. Schwing, Bull. Soe. Chim. Fr.,536 (1968). (6) H. Gerischer, Z . Plzys. Chem. (Leiprig),202, 302 (1953). (7) J. W. Ross, R. De Mars and I. Shain, ANAL.CHEM., 28, 1768 (1956). (8) R. De Mars and I. Shain, ibid., 29, 1825(1957).
1844
e
Thermoregulation at 25 =t 0.1 “C is achieved by a water circulation device. During the cathodic deposition of MoOz.2Hz0(as well as during the anodic stripping), the solution is reproducibly stirred with a magnetic stirrer, the stirring rate of which is controlled. The cell is coated with a silicone coating in order to avoid the adsorption of molybdic anions on glass, a process which is quite strong as we have previously shown (9, IO). The electric circuit allows the working electrode to be maintained at -0.80 volt us. the reference electrode during a preset preelectrolysis time and, immediately thereafter, to apply to this electrode a constant anodic stripping current. During the anodic stripping, the potential of the working electrode is recorded, leading to the curve shown in Figure 1. The solutions contain variable amounts of molybdenum (2 x to 10-3 mole/liter), introduced as NazMoOd solution (Merck, p.a.); they are 0.02M in CH3COOH, 0.02M in CHaCOONa9and 3M in NaCl. The pH of these solutions is near p H 5 where the predominant species is probably the paramolybdic anion Moi02d6-. RESULTS AND DISCUSSION Preelestrolysis time. Solutions containing 10-4 mole/liter of Mo(V1) have been electrolyzed at -0.80 volt us. Ag-AgCl, 3M NaCl electrode during increasing lengths of preelectrolysis time and the stripping times have been measured for a current intensity of 10 PA. (Figure 2). The quantity of electricity required for the stripping is very nearly proportional to the preelectrolysis time as long as the preelectrolysis time remains less than 260 seconds for lO-4M solutions. This result means that, in order to observe direct proportionality between the preelectrolysis time and the quantity of electricity necessary for complete stripping, the thickness of the deposit has to be limited. Integration of the preelectrolysis current corresponding to the reaction Mo(V1) 2e --I.Mo(IV) has shown that the allowable maximum thickness corresponds to about 100 monomolecular layers of MOO:! 2H20. The
+
-
(9) P. Jost, Institut de Chimie de Strasbourg (France), personal communication,1970. (10) G. Goldstein and J. P. Schwing, Bull. SOC.Chihim. Fr., 728 (1967).
ANALYTICAL CHEMISTRY, VOL. 42, NO. 14, DECEMBER 1970
