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where x is the amount of A and B decomposed in time t. Integration of equation (4.15) by the method of partial fractions yields

Rearranging into a linear form suitable for plotting gives

k2 can then be obtained from the slope, 2.303/ k2(a - b), of a plot of t (as ordinate) against log[(a - x)/(b - x)] (as abscissa).

Examination of equation (4.17) shows that the second-order rate constant is dependent on the units used to express concentration; the units of k2 are concentrationtime_1.

For reactions in which both concentration terms refer to the same reactant we may write

Unlike t05 for the first-order reactions, the half-life of the second-order reaction is dependent on the initial concentration of reactants. It is not possible to derive a simple expression for the half-life of a second-order reaction with unequal initial concentrations.

### 4.2.5 Third-order reactions

Third-order reactions are only rarely encountered in drug stability studies involving, as they do, the simultaneous collision of three reactant molecules. The overall rate of ampi-cillin breakdown by simultaneous hydrolysis and polymerisation may be represented by an equation of the form

where ka, kb and kc are the pH-dependent apparent rate constant for hydrolysis, uncata-lysed polymerisation and the general acid-base-catalysed polymerisation of ampicillin, respectively.12 As seen from equation (4.22) the decomposition rate shows both second-order and third-order dependency on the total ampicillin concentration [A].

(4.19) 4.2.6 Determination of the order of reaction

A similar equation applies to second-order reactions in which the initial concentrations of the two reactants are the same.

Integration of equation (4.19) between the limits of t from 0 to t and of x from 0 to x

The most obvious method of determining the order of a reaction is to determine the amount of drug decomposed after various intervals and to substitute the data into the integrated equations for zero-, first- and second-order reactions. The equation giving the most consistent value of k for a series of time intervals is that corresponding most closely to the order of the reaction. Alternatively, the data may be displayed graphically according to the linear equations for the various orders of reactions until a straight-line plot is obtained. Thus, for example, if the data yield a linear graph when plotted as t against log(a - x) the reaction is then taken to be first-order.

Fitting data to the standard rate equations may, however, produce misleading results if a fractional order of reaction applies. An alternative method of determining the order of reaction, which avoids this problem, is based on equation (4.23):

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