First-Order/Pseudo-First-Order Reaction
Gary L. Bertrand
Professor Emeritus of Chemistry
Missouri University of Science and Technology
(formerly University of Missouri-Rolla)

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A First-Order Reaction is described mathematically as:

[A]t = [A]t=0eAkt

ln ([A]t) = ln ([A]t=0) - νAkt

For a Pseudo-First-Order Reaction, the reaction rate constant k is replaced by the apparent reaction rate constant k'. If the reaction is not written out specifically to show a value of νA, the value is assumed to be 1 and is not shown in these equations.

The simplest way to confirm that these are the proper equations to describe the reaction and to get an approximate value for k or k', is with the method of half-lives. The half-life t1/2 is defined as the time required for the initial concentration to be halved:

[A]t1/2 = [A]t=0/2 .
At this point in time,
ln(2) = kt1/2 = 0.693 .

The time required for the concentration to be reduced to 1/4 of the initial concentration (t3/4) is

ln(4) = kt3/4 = 2(0.693) .

In a First-Order Reaction, the concentration of the reactant remaining after x half-lives is:

[A]x/[A]t=0 = (1/2)x .

This relationship is unique to First-Order and Pseudo-First-Order Reactions. This may be confirmed either by following concentration in time for a single series of measurements over three or more half-lives, or by comparing the half-lives of different series of measurements in which the initial concentration is varied by a factor of 4 or more. The reaction rate constant may then be calculated from the "best" value of the half-life:

k = 0.693/t1/2

NOTE:The "First-Order Kinetics" exercise is designed so that you can get an accurate value of the half-life. Scan the concentration column for a value slightly larger than half of the initial concentration. Adjust the value of "Start Time" and the value of "Interval" to get a better estimate of the half-life. The same procedure may be used to find the time for the second and third half-lives.

The best values of the reaction rate constant (k) can be obtained with data taken in the middle third of the reaction (from [A]t = (2/3)[A]t=0 to [A]t = (1/3)[A]t=0). Linear Least Squares regression with Y = ln([A]t) and X = t gives - k or - k' as the slope.