Rate equations and first-order reactions

Differential rate equations like these are not much use to the practising chemist, so it is usual to integrate the differential form of the rate equation, shown above, to obtain more useful expressions. This can be carried out as follows for a first-order reaction. In this reaction, compound A reacts to form products. At the start of the reaction (time 0) the concentration of A is equal to a mol IT1, while the concentration of products will be zero (since the reaction has not started). At some later time, t, the concentration of products has increased to % mol IT1 and as a result the concentration of A has fallen to (a — x) mol IT1. This can be represented mathematically as 

From the law of mass action, the rate of reaction is proportional to [A]. If we rewrite rate as dx/dt, i.e. the rate of production of x with respect to t, and substitute a — x) for [A], then 

The rate constant, k, is a very important measure of a reaction rate and has the dimension of time-1 for a first-order process. This can be shown 

On a practical point, the fact that the units of concentration cancel out for a first-order reaction means that any physical quantity that is proportional to the concentration may be used in the equation in place of concentration, e.g. light absorbance or titration volume. This is very useful, since it means data measured in the laboratory can be inserted directly into the integrated rate equation. 

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