Gary L. Bertrand
Professor Emeritus of Chemistry
Missouri University of Science and Technology
(formerly University of Missouri-Rolla)
A general reaction with n ≠ 1 is described mathematically as:
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:
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:
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:
Once n has been determined, k or k' can be calculated from the relationship above,
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:
For a First-Order Reaction:
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.