Unknown-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 general reaction with n ≠ 1 is described mathematically as:

1/[A]tn - 1 = 1/[A]t=0n - 1 + νA (n - 1) kt

For a Pseudo-nth-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.

One way to determine the order n 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,
{2/[A]t=0}n - 1 - {1/[A]t=0}n - 1 = (2n - 1 - 1)/[A]t=0n - 1 = νA(n - 1) kt1/2 .

This shows that there is a general relationship for all values of n (including n = 1) between the initial concentration and the half-life for a reaction studied with different initial concentrations at the same temperature:
t1/2[A]t=0n - 1 = constant.

If the reaction is First-Order, the half-life will not change with concentration.
If the order is greater than one, the half-life will decrease as the iniital concentration is increased.
If the order is less than one, the half-life will increase as the initial concentration is increased.

The order of the reaction (n) may be found by determination of the half-lives for a reaction studied at two initial concentrations. If the second concentration is equal to 10 times the first:
(t1/2)1/(t1/2)2 = 10n - 1.

Once n has been determined, k or k' can be calculated from the relationship above,
(2n - 1 - 1)/[A]t=0n - 1 = νA(n - 1) kt1/2

NOTE:These relationships are not restricted to the time for 50% reaction. They apply to any fixed fraction of reaction (f). The time required for that fraction of the reactant to disappear is called tf. If the time is measured for 10% of the reaction to occur, this time is referred to as t1/10
For any fraction of reaction, if the second initial concentration is equal to 10 times the first:
(tf)1/(tf)2 = 10n - 1.

For a First-Order Reaction:
- ln(1 - f) = k tf .
For any other order (n),
{1/(1- f)n - 1 - 1}/[A]t=0n - 1 = νA(n - 1) kt1/2

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 = 1/[A]tn - 1 and X = t gives νA(n - 1)k or νA(n - 1)k' as the slope.