Tutorial Created by:
Mathematics, Science, and Technology Education
NC State University
Beer's Law states that there is a linear relationship between concentration of a colored compound in solution and the light absorption of the solution. This fact can be used to calculate the concentration of unknown solutions, given their absorption readings. First, a series of solutions of known concentration are tested for their absorption level. Next, a scatter plot is made of this empirical data and a linear regression line is fitted to the data. This regression line can be expressed as a formula and used to calculate the concentration of unknown solutions.
Open Excel. On Unity/Eos computers, the program will be located on the Application Launcher. On other computers, it will probably be located under the Start Menu.
Your data will go in the first two columns in the spreadsheet (see Figure 1a).
Begin by formatting the spreadsheet cells so the appropriate number of decimal places are displayed (see Figure 1a).
If you do not have your own data, you can copy the data seen in Figure 1b.
The concentration data is probably better expressed in scientific notation.
The last step before creating the graph is to choose the data you want to graph.
This is shown in Figure 2.
With the data you want graphed, start the chart wizard
The first dialogue of the wizard comes up
The Data Range box should reflect the data you highlighted in the spreadsheet. The Series option should be set to Columns, which is how your data is organized (see Figure 4b).
The next dialogue in the wizard is where you label your chart (Figure 4c)
Keep the chart as an object in the current sheet (Figure 4e). Note: Your current sheet is probably named with the default name of "Sheet 1".
The initial scatter plot is now finished and should appear on the same spreadsheet page as your original data. Your chart should look like Figure 5. A few items of note:
With your graph highlighted, you can click and drag the chart to a wherever you would like it located on the spreadsheet page. Grabbing one of the four corner handles allows you to resize the graph. Note: the graph will automatically adjust a number of chart properties as you resize the graph, including the font size of the text in the graph. You may need to go back and alter these properties. At the end of the first part of this tutorial, you will learn how to do this.
When the chart window is highlighted, besides having the chart floating palette appear, a Chart menu also appears. From the Chart menu, you can add a regression line to the chart.
A dialogue box appears (Figure 6a).
The chart now displays the regression line (Figure 7)
The linear equation shown on the chart represents the relationship between Concentration (x) and Absorbance (y) for the compound in solution. The regression line can be considered an acceptable estimation of the true relationship between concentration and absorbance. We have been given the absorbance readings for two solutions of unknown concentration.
Using the linear equation (labeled A in Figure 8), a spreadsheet cell can have an equation associated with it to do the calculation for us. We have a value for y (Absorbance) and need to solve for x (Concentration). Below are the algebraic equations working out this calculation:
y = 2071.9x + 0.111
y - 0.0111 = 2071.9x
(y - 0.0111) / 2071.9 = x
Now we have to convert this final equation into an equation in a spreadsheet cell. The equation associated with the spreadsheet cell will look like what is labeled C in Figure 8. 'B12' in the equation represents y (the absorbance of the unknown). The solution for x (Concentration) is then displayed in cell 'C12'.
Note: If your equation differs for the one in this example, use your equation
Duplicate your equation for the other unknown.
Note that if you highlight your new equation in C13, the reference to cell B12 has also incremented to cell B13.
The readability and display of the scatterplot can be further enhanced by modifying a number of the parameters and options for the chart. Many of these modifications can be accessed through the Chart menu, the Chart floating palette, and by double-clicking the element on the chart itself. Let's start by creating a better contrast between the data points and regression line and the background.
In the Chart Area Format dialogue, set the border and background colors (see Figure 9b)
Now, delete the horizontal grid lines
Now, adjust the color and line weight of the regression line and the color of the data points
Finally, you can move the regression equation to a more central location on the chart
If necessary, resize the font size for text elements in the graph.
The results can be seen in Figure 9c.
This is the end of the first half of the scatter plot tutorial.
In this next part of the tutorial, we will work with another set of data. In this case, it is of a strong acid-strong base titration (see Figure 10). With this titration, a strong base (NaOH) of known concentration is added to a strong acid (also of known concentration, in this case). As the strong base is added to solution, its OH- ions bind with the free H+ions of the acid. An equivalence point is reached when there are no free OH- nor H+ ions in the solution. This equivalence point can be found with a color indicator in the solution or through a pH titration curve. This part of the tutorial will show you how to do the latter.
Note that there should be two columns of data in your spreadsheet:
Column A: mL of 0.1 M NaOH added
Column B: pH of the 0.1 M HCl / 0.1M NaOH mixture
Now, create a scatter plot of titration data, just as you did with the Beer's Law plot.(Figure 11).
Continue through steps 2 through 4 of the Chart wizard:
The resulting plot should look like Figure 12:
The next logical question that you might ask is whether a linear regression line or a curved regression line might help us interpret the titration data. You may remember that our goal with this plot is to calculate the equivalence point, that is, what amount of NaOH is needed to change the pH of the mixture to 7 (neutral)?
Create a linear regression line:
Looking at the data (Figure 13a), it is clear that the first 45 ml of NaOH do little to alter the pH of the mixture. Then between 45 ml and 55 ml, there is a sharp rise in pH before leveling off again. The data trend does not seem linear at all and, in fact, a linear regression line does not fit the data well at all.
The next approach might be to choose a different type of trendline (Figure 13b):
You can see that a second order polynomial curve does not capture the steep rise of the data well. A higher order curve might be tried (Figure 13c):
Still, the third order polynomial does not capture the steep part of the curve where it passes through a pH of 7. Even higher order curves could be created to see if they fit the data better. Instead, a different approach will be taken for this data. Go ahead and delete the regression curve:
Instead of adding a curved regression line, all of the points of the titration data are connected with a smooth curve. With this approach, the curve is guaranteed to go through all of the data points. This is both good and bad. This option can be used if you have only one pH reading per amount of NaOH added. If you have multiple pH readings for each amount added on the scatter plot, you will not end up with a smooth curve. To change the scatter plot is a (smoothed) line graph (Figure 14a):
The result should look like Figure 14b:
This smooth, connected curve helps locate where the steep part of the curve passes through pH 7.
The chart can be enhanced by adding a reference line at pH 7. This clearly marks the point where the curve passes through this pH.
Further refinements in the chart can be made by (as you did with the Beer's law chart):
The result should look like Figure 15.
The above chart gives a good overview of the entire titration. If you would like to focus exclusively on the steep part of the curve between 45 and 55 ml of added NaOH, a new chart can be created which limits the X Axis range. Start by making a copy of the current chart:
With the new chart highlighted (Figure 16):
Next, both vertical and horizontal gridlines can be added to more accurately locate the equivalency point (Figure 17):
With enhancements similar to what you did to the other chart, the result will look like Figure 18.
Even with this smooth curve passing through all of the data points, it is still an estimation of what intermediate mL added/pH data points would be. A clear inaccuracy is where the curve moves in a negative X direction between the 50 and 51 mL data points. More data points collected between 49 and 51 mL would both better smooth the curve and give a more accurate estimation of the equivalency point.
This is the end of the second half of the tutorial.
Rev 2/00 EW