I1
Fred Sicilio
Texas A & M College Colleoe Station and M. D. Peterson Argonne National Laboratory Argonne, Illinois
-
I
Ratio Errors in Pseudo First Order Reactions
Tm
rates of second order reactions are frequently studied by pseudo first order kinetics. A "large excess" of one reactant is used, and the assumption is made that the concentration of this excess reactant remains constant throughout the reaction. Tacitly, an error is present due to the fact that the concentration of the excess reactant does not remain constant; only when the ratio of reactant concentrations is very large may second order reactions be validly studied by pseudo first order kinet,ics. For the second order reaction
the relation between the second order velocity constant and the observed pseudo first order velocity "constant," k', is found
(1)
E always has a positive valueand it is the fractional error in the observed pseudo first order constant caused by the assumption that ( a - x ) / a is equal to unity in the derivation of equation ( 4 ) . It is independent of the absolute values of a and b, and depends only on their ratio, a/b, and on the extent of reaction, x/b, since
the general case, when a and b are not equal, (2) integrates to
Figure 1was constructed by simply assigning values to a, b, and x, and calculating per cent E. Curves for any ratios of a and b can be similarly constructed. Figure 1 shows that the ratio errors in the pseudo first order rate "constants" a t the start of the reaction are 20, 10, 4, 2, and 1% for initial concentration ratios of 5,10,25,50,and 100, respectively. That is, the product of the initial concentration ratio and the initial per cent error is always 100. The ratio error decreases
......
A+B-X+
the rate equation is 3-
"x
=
k(a - z ) ( b - x)
where a = initial caricentration of reactant A b = initial concentration of reactant B z = instantaneous concentration of product X k = second order specific reaction rate constant t = time elaptpsed
.---- .,,.,a ---.-.A.
Equation (3) simplifies into a first order rate equation, k(a
- b ) = k'
1
b
= - ln -
t
( b - x)
(4)
when a is much greater than b, since x cannot exceed b, and ( a - x ) / a is considered to be unity. When b is very small relative to a, k' is considered to be equal to ka. The aberration that keeps ( 4 ) from being a true equation is the absence of ( a - x ) / a in the log term, since the second order reaction is actually following equation (3) and the concentration terms are dependent variables. If equation (3) is divided by equation ( 4 ) ,
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Journal of Chemical Education
---./b.IO
g B
:
----___
O.?%
F I ~ U ,I ~.
Per cent r ~ t i o
----____ ---__
0.50 EXTENT OF IlEACTIO*
v
--..
21
as the reaction progresses, since the ratio of the concentrations of the unreacted reactants is then increasing. As the reaction approaches completion, the error approaches zero. If a second order reaction could be followed to completion, a plot of the instantaneous values of the pseudo first order velocity constants would have the general shape of one of the family of curves in Figure 1. Obviously, then, the graphical or averaged value of k' depends on the range in which the reactionis followed. The first part of a reaction is often stressed in kinetic studies, and unfortunately, it is during this part that the ratio error is greatest. During the latter stages of a reaction, the error decreases rapidly and becomes zero a t 100% completion. The limiting value of k', that a t the very end of the reaction, is not readily attainable by extrapolation of experimental k' values because of difficulties in experimental analyses when the concentration of B becomes very small and because of the rapidly changing slope of the k' curves during the late stages of reaction. However, this limiting value, which could be called a "true" pseudo first order velocity
constant, may be calculated directly from any individual value of k' by use of Figure 1 or similarly constructed curves; i.e., by multiplying the observed value of k' by (1 - E), the latter parenthetical term always being less than unity. In kinetic studies involving solutions consisting of a small concentration of one reactant dissolved in another reactant acting as a solvent, the ratio errors discussed above are frequently negligible, and the pseudo first order rate constants by t'hemselves have significance. Thus, even for a 1 M reactant in water, the maximum ratio error is only approximately two per cent, since the initial concentration ratio is about 55 to 1. In systems containing a third component as an inert solvent, however, the concentration of the excess reactant, and therefore, the initial reactant concentration ratio, may be so lowered that the maximum ratio error could greatly exceed the analytical errors. With such systems, Figure 1 may be used as a reference in choosing suitable ratios of reactant concentrations. The authors wish to thank D. E. Pearson for point,ing out the need for an analysis of this type.
Volume 38, Number 1 1 , November 1961
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