Stephen W. Tobey University of Wisconsin Modison
Determining Second Order Rate Constants
Although a variety of equations for evaluating second order rate constants from data obtained on reacting systems containing equivalent concentrations of reactants have already been described in the literature, a little consideration will show that these equations all suffer from one or the other of the following limitations. Either the actual chemical concentrations of the reactants in the system as a function of time must be known (1-3) or accurate zero or infinite-time readings must be known if the data are in terms of a physical property of the reacting system (4, 6). The simple equation described below eliminates these limitations, and allows second order rate constants to be calculated from physical measurements in the absence of zero and infinite-time readings. The derivation and application of the equation are straightforward. For a second order reaction A
+B
-
separated by a constant time interval (tz - t,), which may conveniently be approximately one half-life, are read from a plot of X t vs t . These (A, - X2) values are plotted against the corresponding values of (t2Xz - tlX1). This plot should be linear with a slope of (A)&. The only conditions which must be met for applicability of equation (10) are that equivalent initial concentrations of reactants be present in the reacting system and the time of initiation of the reaction be known. An inspection of the chemical literature (6-9) reveals numerous examples of second order rate data which could be evaluated to advantage using equation (10). However, the indispensability of (10) in a practical situation is most clearly demonstrated by the following experiment which involves determining /cz for the alkaline hydrolysis of phenyl-m-nitrobenzoate by a conductometric technique (10). The over-all equation for the reaction is
k,
products
and integration of equation ( 2 ) yields
provided (A)o = (B)o. When kz is determined by measuring the rate of change in the value of X, a physical property of the reacting system which is proportional to (A), the equations (A),
and
=
K(ht - A,)
(A), = K(A,
-Ad
(4) (5)
may be substituted into equation (3) to yield either hr = ha
- (A)~lczt(Xt- A,)
Note that two moles of hydroxide ion are consumed per mole of ester hydrolyzed. To maintain equivalent concentrations of base and ester in the reacting system during the reaction, (B)O,the initial ester concentration, is set equal to twice (A), the initrialester concentration. The data in Table 1 show clearly that an accurate determination of ROby an extrapolative method would be difficult because of the extremely rapid change in R with t i n the early stages of the reaction. This makes the use of (7) for determining ic2 undesirable. Furthcrmore, the stoichiometry of (11) shows that the pH
(6)
These equations permit the calculation of kl if either Xo or h,is known, and have found extensive use (4, 5). However, by taking advantage of the following simple manipulation of equation (6) or (7), an equation can be derived which permits the calculation of k2 when neither Xo nor h , is known. Considering (6) at two arbitrary times 1and 2, A, = ha - (A)okddA~- A,) (8) and A2 =
ho
- (A)~knts(Ar- A,)
(9)
Subtracting (9) from (8) yields To use eqnation (10) in determining kz,pairs of X I values
Plot of equation
I12) using doto from Table 1;
Volume 39, Number
kn = 3.33 M P set.-'.
9, September 1962
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473
Table 1. Data for the Alkaline Hydrolysis of Phenyl-mnitrobenzoaie in 50y0 Aqueous p-Dioxane a t 25.00°C t
(aec)
R (ohms)
[I;
- ;,]XIO (sec. ohm-1)
[$
L
eliminated and kz for the reaction can be readily calculated. The appropriate form is
-;I
2 X 10' (ohm-')
since
A, = 1 / R ,
I t is hoped that the calculational technique described herein may prove to be useful in a wide variety of second order kinetic studies. Literature Cited
Table 1 shows the appropriate calculations required to permit the construction of the figure, from n,hirh & is found to he 3.33 M-1 sec-1. Temperature was established rtO.02"C; resistances were iO.lJ7,; time measured +0.3 see. The time interval tz - 1, was chosen 100.0 sec. The concentration Ro = 2Ao = 6.275 X
nf.
in the reaction cell must fall as the reaction uears completion. Ultimately a point is reached at which phenoxide ion reverts to unionized phenol, thus making a true R, reading unobtainable and rendering (6) nwlrss for determining k2. However, by making use of (10) the necessity of knowing either Ro or R , is
474
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Journal of Chemicol Education
(1) DANIELRF., AND ALBERTY, R. A,, "Physical Chemistry," 2nd ed., John Wiloy & Sons, Inc., New York, 1961, p. 302. (2) ROSEVEARE, W. E., J. Am. Chem. Soc., 53, 1651 (1981). (3) PESSEN,H., Science, 134, 676 (1961). R. G., "Kinetics and Meeh(4) FROST,A. A,, AND PEARSON, anism," 2nd ed., John Wiley & Sons, Inc., New York, 1961, p. 38. (5) DANIELS,F., MATHEWS,J. H., ET AL., "Experimental Physicd Chemistry," 5th ed., McGraw-Hill Book Ca., Ine., New York, 1956, p. 132. (6) Data of MCGUIRE,W. J., M. S. Thesis, Northwestern University, Evitnston, Illinois, 1949, as quoted in Ref. (. 4. .) . D. 36. (7) WESTHEIMER, F. H., AND SHOOKHOFF, M. W., J . Am. Chent. Soc., 62, 269 (1940). J. H., J . C h e n ~ .Soe. (8) HEGAN,D. S., A N D WOLFENDEN, (LONDON), 1939.508. . SOC.(London), A78,157 (1906). (9) WALKER, J., P ~ cRoy. (10) TOBEY, S. W., M. S. Thesis, University of Wisconsin, Madison, 1959.
