September 2004:
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| September 2004: | |
| In chemical kinetics, the rate of change of a reaction is most often dependent on the amount of reactants present. Usually, the greater the concentration of the reactant, the faster the reaction takes place. Assume A is the reactant of a chemical reaction, and the rate of change of the reaction is given by -d[A]/dt = k [A]3. From this differential equation, derive an equation for the half-life of [A], which is the time required for the concentration of A to reach half of its original value. | |
| To solve such a differential equation, we simply separate the variables by rearranging the equation. The equatoin is rearranged to -d[A]/A3 = k dt. We integrate the left side from A0 to A and the right side from 0 to t. The integral results in 1/(2A2) - 1/(2A02) = kt. In order to find the half-life of A0, we substitute A0/2 for A. After some algebra, we solve for t and we obtain t = 3/(2A02k)
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