The differential rate law describes how the rate of reaction varies with the concentrations of various species, usually reactants, in the system. The rate of reaction is proportional to the rates of change in concentrations of the reactants and products; that is, the rate is proportional to a derivative of a concentration. To illustrate this point, consider the reaction

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(3.5) |

The rate of reaction, *r*, from equation (2) (using that the
stoichiometric coefficient for species A is –1) is given by

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(3.6) |

Suppose this reaction obeys a first-order rate law, i.e. *r* =
*k* [A]. This rate law can also be written as

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(3.7) |

This equation is a first order ordinary differential equation that relates the
rate of change in a concentration to the concentration itself. Integration of this
equation produces the corresponding *integrated rate law*, which
relates the concentration to time. Let’s solve this (3.7) simply differential
equation with the following initial condition: . After the separation of variables we get the following
equation:

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(3.8) |

Integrating both sides we provide the integrated rate law for a first order reaction (Figure 3.1):

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(3.9) |