Unknown-Order ReactionGary L. Bertrand Professor Emeritus of Chemistry Missouri University of Science and Technology (formerly University of Missouri-Rolla) background close A general reaction with n ≠ 1 is described mathematically as:_{t}^{n - 1} = 1/[A]_{t=0}^{n - 1} + ν_{A} (n - 1) ktFor a Pseudo-n ^{th}-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 ν_{A}, 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 t is defined as the time required for the initial concentration to be halved:_{1/2}_{t1/2} = [A]_{t=0}/2 ._{t=0}}^{n - 1} - {1/[A]_{t=0}}^{n - 1} = (2^{n - 1} - 1)/[A]_{t=0}^{n - 1} = ν_{A}(n - 1) kt_{1/2} .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:
_{1/2}[A]_{t=0}^{n - 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: _{1/2})_{1}/(t_{1/2})_{2} = 10^{n - 1}. Once n has been determined, k or k' can be calculated from the relationship above,^{n - 1} - 1)/[A]_{t=0}^{n - 1} = ν_{A}(n - 1) kt_{1/2}
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 t. If the time is measured for 10% of the reaction to occur, this time is referred to as _{f}t_{1/10}For any fraction of reaction, if the second initial concentration is equal to 10 times the first: _{f})_{1}/(t_{f})_{2} = 10^{n - 1}. For a First-Order Reaction: _{f} .^{n - 1} - 1}/[A]_{t=0}^{n - 1} = ν_{A}(n - 1) kt_{1/2}
The best values of the reaction rate constant ( k) can be obtained with data taken in the middle third of the reaction (from [A] to _{t} = (2/3)[A]_{t=0}[A]). Linear Least Squares regression with _{t} = (1/3)[A]_{t=0}Y = 1/[A] and _{t}^{n - 1}X = t gives ν or _{A}(n - 1)kν as the slope._{A}(n - 1)k' |