When we study rates of chemical reactions, we look for an equation that describes the reaction rate. The equation can be in one of two forms, known as the differential rate law and the integrated rate law.
If you have taken calculus, you’ll recognize these names and you even know how to derive the integrated law from the differential law. If you haven’t taken calculus, don’t worry! You probably won’t be asked to derive the equations. However, you should recognize each form of the law for zeroth, first and second order reactions.
The differential rate law gives the reaction rate as a function of the concentration of the reactants. The integrated rate law gives the concentration as a function of time. Both equations are useful, and you will probably use both in your chemistry class.
The table below shows the two rate laws for each reaction order in terms of a single reactant A.
| Order | Differential rate law | Integrated rate law |
| Zeroth | Rate = k | ![[A] = -kt + [A]_0](http://s0.wp.com/latex.php?latex=%5BA%5D+%3D+-kt+%2B+%5BA%5D_0+&bg=ffffff&fg=000&s=0 "[A] = -kt + [A]_0") |
| First | Rate = k[A] | ![\ln{[A]} = -kt + \ln{[A]_0}](http://s0.wp.com/latex.php?latex=%5Cln%7B%5BA%5D%7D+%3D+-kt+%2B+%5Cln%7B%5BA%5D_0%7D+&bg=ffffff&fg=000&s=0 "\ln{[A]} = -kt + \ln{[A]_0}") |
| Second | Rate = k[A]2 | ![\dfrac{1}{[A]} = kt + \dfrac{1}{[A]_0}](http://s0.wp.com/latex.php?latex=%5Cdfrac%7B1%7D%7B%5BA%5D%7D+%3D+kt+%2B+%5Cdfrac%7B1%7D%7B%5BA%5D_0%7D+&bg=ffffff&fg=000&s=0 "\dfrac{1}{[A]} = kt + \dfrac{1}{[A]_0}") |
You might have noticed that each of the integrated law equations is in the form y = mx + b, the equation for a line. In the old days (before graphing calculators), this is the method that chemists used to determine the reaction order. If the graph of concentration vs. time is a straight line, the reaction is zeroth order. If the graph of ln(concentration) vs. time is a straight line, the reaction is first order. And if the graph of 1/concentration vs. time is a straight line, the reaction is second order.