## Friday, October 24, 2014

### The Experimental Determination of Reaction Rates

By the end of the 19th century chemists had succeeded in formulating rules for determining the rates of reactions and their dependence on temperature. Arrhenius discussed the dissociation of a substance into its component atoms. As an example of how one can determine the rates of a reaction and their dependence on temperature we can study the simple case where a diatomic molecule is split up into its constituent atoms.

The first two highlighted equations give the rates for the formation of the molecule A and its atoms B. There are two ways that we can change the amount of A. The first term in the first equation is the rate at which it spontaneously breaks up into two atoms of B. It is negative and proportional to A because every molecule of A has an equal chance undergoing the dissociation. The second term takes gives the rate at which two atoms of B come together to form A. The rate is positive and proportional to B2 since each atom of B can potentially interact with all the other atoms of B. The two equations are difficult to solve simultaneously but we can use the fact that the total number of atoms is a constant. The result is a nonlinear rate law for A which can be solved using a standard integral formula. To simplify the calculation of A we can define some additional constants. The system converges to an equilibrium value Aeq and an equilibrium constant K can be defined that relates the amounts of A and B. The formula for Aeq is found by letting
t → ∞.

The solid curves in the figure below show the changes in the amount of A and B with time. Since initially B = 0 and A follows the rate law for simple decay with k0 playing the role of the decay constant. By determining the initial slope of the curve for A and the equilibrium constant we can experimentally determine the reaction rate constants, ki.

By repeating this experiment at different temperatures we can determine the dependence of the rate constants on temperature.

Supplemental (Oct 24): One can find papers by Deville, Wilhelmy, van't Hoff, Guldberg & Waage, Arrhenius & Oswald in Leicester, A Source Book in Chemistry, 1400 - 1900. Ludwig Wilhelmy studied the dissociation of sucrose into its simpler sugars and gave the rate equation and the formula for exponential decay in the amount of sucrose. Arrhenius won the Nobel Prize in 1903 for his work on conduction in solutions and is known for studying the influence of atmospheric CO2 on ground temperature.