## Graphing Example 2

The Beer-Lambert law is used in chemistry to relate the concentration of a solution to the amount of light it absorbs. A solution of Ni(NO_{3})_{2} will be colored green due to the nickel(II) ion. A spectrophotometer or a colorimeter can be used to measure the absorbance of the solution at different wavelengths of light. For the data below, a colorimeter was used since the solution has a visible green color and was set to 635 nanometers since the solution strongly absorbs this wavelength and is very sensitive to changes in concentration. As the concentration increases (darker green), the solution absorbs more light at 635 nm.

For dilute solutions, the Beer-Lambert law states that the absorbance of the solution will be directly proportional to its concentration. You can see for the data below, as the concentration increases, the absorbance also increases. And if you press the “Best-Fit Line" button, you will see that the best line that represents the data is not only linear but it comes close to going through the origin- (0, 0.018).

The equation of the best-fit line is often written as y = mx + b. The slope is “m”, the y-intercept is “b”, and for this data the absorbance is “y” and the solution concentration is “x”. Once you know the slope and the y-intercept for a linear fit, you now know how y depends upon x. In other words, if I were to give you a value for the concentration of a nickel(II) nitrate solution, you would be able to use that concentration and the equation of the best-fit line to tell me the expected value for the absorbance.

But that’s not all you would be able to do with the equation.

Data collected 11/15/2008.

What if I were to give you the absorbance instead of the concentration for an unknown nickel(II) nitrate solution? You could still use the equation of the line along with the absorbance to determine the solution's concentration. Let's assume that the absorbance is 0.220. Press the "Find the Concentration" button so you can see how the graph could be used to determine the concentration. If you wanted to use the equation of the best-fit line, y = mx + b, you would instead solve for “x”, the concentration, using the given “y”, the absorbance.

So don’t get locked into thinking that you can only get the dependent variable, y, from the independent variable, x, when given the equation of the best-fit line for the data. You have so much more that you can do!