This can be used to estimate the order of reaction of each reactant. For example, the initial rate can be measured in a series of experiments at different initial concentrations of reactant A with all other concentrations [B], [C], … kept constant, so that
The tentative rate equation determined by the method of initial rates is therefore normally verified by comparing the concentrations measured over a longer time (several half-lives) with the integrated form of the rate equation; this assumes that the reaction goes to completion.
Although not affecting the above math, the majority of first order reactions proceed via intermolecular collisions. Such collisions, which contribute the energy to the reactant, are necessarily second order. The rate of these collisions is, however, masked by the fact that the rate determining step remains the unimolecular breakdown of the energized reactant.
For the rate proportional to a single concentration squared, the time dependence of the concentration is given by
The time dependence for a rate proportional to two unequal concentrations is
One way to obtain a pseudo-first order reaction is to use a large excess of one reactant (say, [B]≫[A]) so that, as the reaction progresses, only a small fraction of the reactant in excess (B) is consumed, and its concentration can be considered to stay constant. For example, the hydrolysis of esters by dilute mineral acids follows pseudo-first order kinetics, where the concentration of water is constant because it is present in large excess:
In the steady state, the rates of formation and destruction of methyl radicals are equal, so that
The reaction rate equals the rate of the propagation steps which form the main reaction products CH4 and CO:
This is zero-order with respect to hexacyanoferrate (III) at the onset of the reaction (when its concentration is high and the ruthenium catalyst is quickly regenerated), but changes to first-order when its concentration decreases and the regeneration of catalyst becomes rate-determining.
Notable mechanisms with mixed-order rate laws with two-term denominators include:
The reaction rate expression for the above reactions (assuming each one is elementary) can be written as:
This applies even when time t is at infinity; i.e., equilibrium has been reached:
These equations allow us to uncouple the system of differential equations, and allow us to solve for the concentration of A alone.
If the concentration at the time t = 0 is different from above, the simplifications above are invalid, and a system of differential equations must be solved. However, this system can also be solved exactly to yield the following generalized expressions:
With the individual concentrations scaled by the total population of reactants to become probabilities, linear systems of differential equations such as these can be formulated as a master equation. The differential equations can be solved analytically and the integrated rate equations are
The steady state approximation leads to very similar results in an easier way.
When a substance reacts simultaneously to give two different products, a parallel or competitive reaction is said to take place.
Each of these is discussed in detail below. One can define the stoichiometric matrix