A Simple Method for Analyzing First-Order Kinetics B. Borderie, D. Lavabre, G. Levy, and J. C. Micheau UA 470 Laboratoire des IMRCP. Universitb Paul Sabatier. 110 route de Narbonne, 31062 Toulouse Cedex, France First-order reactions are of particular importance since many higher order reactions can be approximated to pseudo-first-order kinetics. We describe here a new and simple method for accurate determination of the first-order rate constant (k,). The method hinges on calculation of the ratio A. It of two reaction rates measured a t times t and t represents a combination of a differential and time interval approach (I).
+
Descrlptlon of the Method Phvsicochemical methods (UVIvisible soectroscoDv. DOlarimetry, conductimetry, etc.j used to follow reaction kinetics Droduce a signal that is a linear function of concentration. ~ l i s t u d i eof s f&-order reactions are based on the following relationship (2): Ye = (Yo- Y.1. mp (-h,t)
+ Y-
(1)
in which Yt, Yo, and Y , are the magnitudes of the signal a t and kl is the first-order rate constant. times t, 0, and Differentiating with respect to time, we obtain the rate expressions: at t(dYldt), = (Yo - Y.)(-k,) exp (-k,t) = R(t) at t
+A
(2)
(dY/dt),+A= (Yo - YJ(-kl) exp (-k,t)
.exp (-k,A)
= R(t
+ A)
WAELENGTH lnml
Figure 1. Series of spectra recordad for the mermai relaxation of mercury dithizonate in xyiene immediately after irradiation. Note the presence of two = 484 nm) and its bands corresponding to the orange starting compound A(., Mue photoisomer A (, = BOO nm), and two isosbestic points (395 and 530 nm). The spechai characteristics are in agreement with the mechanism shown in the insert. The interval between recordings was 6 s.
(3)
The ratio of eq 2leq 3 gives:
+
R(t)lR(t A) = exp (k,A) = r
(4)
RIR for this method. hence the desienation .. lick be seen that this ratio is constant for all values oft. It can thus be calculated at different times throuehout the course of the reaction, and a mean value can be obtained. The rate constant kl is obtained from this mean r,: A similar equation has been described by Wentzell and Crouch (3), although its application for determination of first-order rate constants was not discussed. Experimental Appllcatlon
We applied this method t o a chemical reaction that is readily followed by UVIvisible spectroscopy: the monomolecular isomerization of mercury dithizonate in solution in xylene (4). Under irradiation the orange mercury dithizon= 484 nm) undergoes a photoisomerization to a ate (A, = 600nm). In the dark, this blue complex blue complex (A, reverts soontaneouslv to the stable oranee isomer (Fie. 1). T o illustiate the appfication of the AIR &hod t o the Erstorder thermal return reaction. we studied the reaction a t a wavelength (484 nm) a t which the optical density changes greatly with time. This reaction course is analyzed graphically in Figure 2. Using this method, k, can he determined without linear regression or knowledge of Y , (Table 1). To compare the RIR method with classical methods (Table 2) we calculated kl for two different wavelengths (484 and 6043 nm) taken from the same recording. The values should be identical as they correspond to the same reaction.
Figure 2. Graphical analysis of the RIR relationship applied to the thermal relaxation of mercury dithizonate after irradiation (recorded at 484 nm). me tangents at regular intervals (20 s) on the C U N e were drawn by hand. me csiculation Is shown in Table 1.
It can be seen that the RIR method gives results of comparable accuracv to Guagenheim's method or the KMS method -(5-8).
Conclusion We have described a new method for the analysis of firstor pseudo-first-order kinetics, which would be particularly suitable for teaching purposes as, t o our knowledge, there
Volume 67 Number 6 June 1990
459
Table 1.
Determlnatlonot Rate Constant k, by Graphlcal Analysls of the Curve Shown In Flgure 2 Uslng the RIR Methoda
No
1
2
3
4
5
6
Rates (RJa.u.
1.35
0.871
0.581
0.365
0.246
0.162
1.55
Ratioof Rates
1.50
1.51
1.57
1.52
A=20s r, = 1.53 k, = 2.13 X 10PC'
'values d rates of reactinn (R,) and corresponding ratios R O k . The mean of these ratios is r,. The time interval A Is 20 s.
Table 2.
Comparlron of the PrfnclpalGraphical Methods for Analysls of Flrst-Order Klnetlcs8 linear
Method
Relationships
semilog Partial time Gusgenhelm
I. Y,- v 1. = -k,t+ In Y, k, = in 2ltm In1 Y e b - YfI =-k,t+lnl Yo- Y - ~ ( W - ~ ~1)Ylld = Y,. pA- Y&-*lA 1)
KMS RIR
regression
1 - Y,I
I
-
k, = (1"
(rm))lA
YES
no Yes Yes
no
Y. known
k,(s-'Hh = 464 nm)
kW'Hh = 600 nm)
yes yes no no no
2.44 x 2.34 X lo-' 2.07 X lo-' 2.00 X 2.13 X lo-'
1.64 x 1oP 1.95 X 10P 2.01 X 2.07 X 2.12 X 10V
~hevaluasofrats constant k , wereobtainedtrommesamerecordingattwoditferentwavelengths(484and600nm)uslngflvedifterentmeth~s.Note that me RlRmethod producesthe least difference bemeen the two valves ot k,
are no methods available that d o not require knowledge of Y . nor recourse t o linear regression. It is a combination of a differential and a time interval method a n d has the following advantages:
.
It can he applied directly to the spectraphotometer tracings using simple graphical techniques. It is as accurate as classical methods used under the same conditions; the RIR method can also be readily adapted to computer analysis in real time. With s slightly noisy recording, a small portion of the overall curve can be used. Results can thus be ohtained quickly. The validity of the pseudo-first-order approximation can he tested: the approximation is valid if RIR remains constant.
. .
Starting Compound Mercury dithizonate was prepared using the method of Meriwether (4) and recrystallized in ethanol (P-11). We prepared M inxylene (caw = 43,000 M-I. cm-I). Inview solutions of 2.1 X of the reversibility of the reaction, a single solution can he used several times. The nature of the solvent, the purity of the starting complex,and the state of the cuvette surfaces will all affect the value of kl. Recording Conditions The reaction was followed hy UVIvisihle spectroscopy using a HP 8451A d i d e m a y spectrophotometer or a Beekman UV 5260 instrument. The experiment was carried out in euvettes of l-cm optical path length, thermostated a t 25 OC. The mixture was stirred magnetically to produce a homogeneous medium. The kinetics are started by irradiating the solution until it turns blue (around 1min
460
Journal of Chemical Education
with a slide projector lamp; 15 s with a 200-W, unfiltered, highpressure mercury vapor lamp). The cuvette is then immediately inserted into the spectrophotometer. By changing the recording wavelength, different curves of different amplitude and slope (positive or negative) can be obtained. Graphical Analysis The following procedure should be respected strictly: Choose a time interval A to obtain around six equidistant points over the whole reaction curve Y ( t ) . Draw the tangents to the curve a t each of these points. Calculate the slopes Ri in arbitrary units. Caleulate the five ratios RilRitl for the six slopes. Take the mean. Determine k ifrom relationship 5.
..
We should point out that the ahove procedure should be used, rather than calculating the values of k l a t the different times and then taking the mean of these values. The latter method will Lead to a loss of information from the central region of the curve.
1. Schwartz.L. M. And. Chem. 1981.53.206-213. cuEgPnh~im, E. A.Phii.Mag.;. 1926,53%543.
2. 3. 4. 5. 6
Wenfrell, P. D.; Crouch,S. R A n o i . Cham. 1986,58,2351-2855. Meriwether,L.S.:Breitner,E.C.;Slaan,C.L. J A m Cham.Soc. 1986.87.4441-4448. Kerdy, F. J.: Jaz, J.: Bruyiants. A. Bul. Soc. Chim. Belg 1958.67.687. Manwlrdorf.P.C..II. J.Aool. Phus. 1959.30.442.
