Treatment of Kinetic Data for Opposing Second-Order and Mixed First- and Second-Order Reactions J. R. Pladziewlcz, J. S. Lesniak, and A. J. A b r a h a m University of Wisconsin, Eau Claire. WI 54702 In kinetics and physical chemistry texts ( I ) , the treatment of integrated rate laws for opposing second-order, eqs 1and 2, andmixed first- and second-order, eq 3, reactions depends upon the initial concentration conditions. For example,
Integrated rate equations used when [Ale = [BIoare different from those used when [Ale z [Bl0and [CIo = [Dlo = 0. E. L. King has recently shown (2), in a paper eloquent for its conciseness. that the use of different inteerated eouations for different initial concentration conditiok for tcese and related svstems is unnecessarv. Kine exnresses reactant concentrations in terms of theirkquililbriim values and A, the A. displacement from equilibrium, such that [A] = [A]. When this is done eqs 1-3 can he integrated to yield eqs 4-6, respectively (2).
+
light, pressure of gases) that is linearly related to concentration, as in eq 7,
where Po, P,, and P, are the measured property a t time zero, t, and equilibrium, respectively. The logarithmic forms of the equations resulting from this substitution can also be used to obtain linear plots from which kl can be obtained. However, the application of nonlinear least-squares computer routines to data of this kind is made easier by replacing A in eqs 4 and 6, for example, and solving for the dependent variable, P,, as a function of independent variable, t, and the other parameters, yielding eqs 8 and 9, respectively. In deriving these equations the constant of integration has been evaluated a t time zero and factored to insure that the numerators of the left-hand side of eqs 4 and 6 are not undefined when A is negative. To apply these equations K must be known.
where
In
A
4 + [A], + [Bl, + (11K) = -k,[[Al, + [B], + 1 I N t + c
(6)
These equations can he applied to kineticdata by computing A from concentration data (or absorbance or some other property that can be related to concentration) and plotting the left-hand side of eq 4,5,or 6, as appropriate, versus time. I t is often convenient to express A in terms of a system property, P, (absorbance, conductivity, rotation of polarized
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Journal of Chemical Education
where a = [A],
+ [B], + 1IK
Figure 1. Observed (open circles) and fitted (solid squares) concentrations of reduced piastocyanin in me reaction described by eq 10 tilted using NUSO and ea 8.
The advantage of expressions like eqs 8 and 9 is that they can be used directly in nonlinear least-squares fitting routines, such as the commercial program NLLSQ (3) or routines constructed from noncommercial programs like those available through project Seraphim (4a) and other sources (46) to fit Pt; t data for any combination of the parameters P,, Po. kl. NLLSQ and eq 8 have been used (5)to fit kinetic data for the oxidation of reduced plastocyanin, (PCu(I)), a copper-containing protein, by 1,l'-dimethylferroceniumion (Fe(CpMe),?,
A plot of experimental and fitted protein-concentrationversus-time values based on eq 8 for this system is given in Fieure 1. Eouation 9 has also been used to fit absorbance: " time data from the supplement to the paper of Swaddle and Kine (6) run 1F for the reaction of Cr"+ with HF at constant [~+jdf0.123M at 94.7 OC, eq 11.At constant [H+],
+
CraC HF e CrF2++ H+
(11)
the kinetic data for the reaction described by eq 11conform to eqs 6 and 9. A plot of experimental and fitted absorbances based on eq 9 are given in Figure 2. The rate constant for CrF2+ equation (the pseudo-first-order rate constant for the reverse of eq 11at constant [Hf]) obtained from this fit is
Figure 2. Observed (open circles) and fitted (solid squares) absnbances fw me reaction described by eq 11 fitted by using NUSQ and eq 9.
l.58(f 0.03) X 10-5s-', in close agreement with thevalue 1.56 X 10-5s-1 reported by Swaddle and King (6). In summary, it is unnecessary to use different integrated equations for different initial conditions for treating kinetic data for several kinetic systems (2) including those to which eqs 1-3 apply. Moreover, it is possible to fit these data for parameters of interest using available nonlinear leastsquares computer routines. Acknowledgment Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for support of this work. We are also grateful to James H. Espenson for bringing E. L. King's work to our attention at a time when we were attempting to fit kinetic data for the protein redox reactions. (1) see for example: Bern". S. W. "The Foundations of Chemical Kinetics"; Mffimw: New York, 1960; pp 29-31: Frmf A . A,: Pearron, R. G. '"Kinetics and Mechanism", 2nd ed.: Wiley: New York, 1961,pp 186188;Moore,J. W.;Pesnon, R.0."Kinetics sod Mmhanisms". 3rded.;Wiley: N m York. 1931. p 53; Espenmn. J. H."Chemiesl Kinetics and Reaction Mechanisms"; Mffiraw: N m York. 1981. p 46; Cantellan, G. W."Phmical Chemistry", 2nd d.; Addkon-WedexResding, MA, 1971, p 749. (2) King,E.L.Int.J. Chem. Kinel. 1982,14,1285.
(3) Christian.S.D.;Tucker.E.E.Amar.Lob. 1962,14.31. (4) (a)Trindle,C..IChem.Educ. 1933.60,566;(b)Moom,R.H.;Zeigler,R.K.,ReportL* 2367 end addenda; Lor Alamoa Scientific Laboratory, Lae Alamas, NM,1959. (5) Pladzieuicz, J.R;Brenner, M.S.;Rdeberg, O.A.: Liklu.M.D.I n o m Chrm. 1985.24 1". ( 6 ) Swisddie,T.
W:. King, E. L. Inarg. Cbm. 1965.4.532
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