11.4: 0th order Rate Law

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If the reaction follows a zeroth order rate law, it can be expressed in terms of the time-rate of change of [A] (which will be negative since A is a reactant):

\[-\dfrac{d[A]}{dt} = k \nonumber \]

In this case, it is straightforward to separate the variables. Placing time variables on the right and [A] on the left

\[ d[A] = - k \,dt \nonumber \]

In this form, it is easy to integrate. If the concentration of A is [A]₀ at time t = 0, and the concentration of A is [A] at some arbitrary time later, the form of the integral is

\[ \int _{[A]_o}^{[A]} d[A] = - k \int _{t_o}^{t}\,dt \nonumber \]

which yields

\[ [A] - [A]_o = -kt \nonumber \]

\[ [A] = [A]_o -kt \nonumber \]

This suggests that a plot of concentration as a function of time will produce a straight line, the slope of which is –k, and the intercept of which is [A]₀. If such a plot is linear, then the data are consistent with 0^th order kinetics. If they are not, other possibilities must be considered.