Calculations for Colorimetry

Along with operating the instruments, colorimetry also involves calculations to actually figure out the concentration of a solution from the absorbance measurements made by using the colorimeters. There are three methods that can be used depending on what information is available. They involve using proportionality, graphing and Beer's Law.

One example of each is shown in the following problems (which are also shown in example 10 in your workbook). After you look at those examples try your hand with the practice problems that follow (also given in example 11 in your workbook).

Proportionality Example

The proportionality approach to these kinds of problems focuses on the idea that the absorbance of a solution is directly proportional to its concentration. When using this approach it is necessary to be sure that the values given are for different concentrations of the same chemical measured under the same conditions (wavelength and path length).

Question: A solution with a concentration of 0.14M is measured to have an absorbance of 0.43. Another solution of the same chemical is measured under the same conditions and has an absorbance of 0.37. What is its concentration?

The solution to this problem can be set up using the equation shown in the top left box shown here, which simply says that the ratio of the concentrations is proportional to the ratio of absorbances. We can use c₁ to represent the unknown concentration.

For the next steps you have your choice of rearranging the equation to solve for c₁ and then substituting the known values (down then right) or substituting the known values and then rearranging the equation to solve for c₁ (right then down). Then finish up by carrying out the calculations.

c₁
^_____
c₂

A₁
^_____
A₂

c₁
^_______
0.14M

0.37
^______
0.43

c₁ =

A₁
^_____
A₂

x c₂

c₁ =

0.37
^______
0.43

x 0.14M

c₁ = 0.12M

Graphing Example

The graphing method is called for when several sets of data are available for concentration and absorbance. Graphing the data allows you to check the assumption that Beer's Law is valid by looking for a straight-line relationship for the data.

Note that you could use the data here to do this calculation by the proportionality method but you would have to chose which of the four sets of data to use as the standard. There is also enough data here to use the Beer's Law Equation if you chose to do that.

Question: The following data were obtained for 1.00 cm samples of a particular chemical. What is the concentration of a 1.00 cm sample that has an absorbance of 0.60?

Conc.	Abs.
0.50	0.69
0.40	0.55
0.30	0.41
0.20	0.27

The solution to the problem here is to graph the data (red circles) and draw a straight line through the points. If the data points are on or close to the line, that will confirm that the absorbance and concentration are proportional and Beer's Law is valid for this situation.

Then find the point on the line that corresponds to the desired absorbance (0.60). Read the corresponding concentration off the scale below (0.437M).

Graph of Absorbance vs Concentration (Ex. 10-b)

Beer's Law Example

Using the Beer's Law Equation makes the most sense when you are given the molar absorptivity or have the necessary data to calculate it. (Note that if you do have to calculate the molar absorptivity it will probably be easier to use one of the other methods.)

Question: The absorptivity of a particular chemical is 1.5/M·cm. What is the concentration of a solution made from this chemical if a 2.0 cm sample has an absorbance of 1.20?

The way to solve this problem using Beer's Law is to first write down the equation. Then rearrange the equation to solve for concentration. Then substitute the known values. (Of course, these two steps can be reversed if you wish.) Finally, carry out the calculations to get the answer (0.40M).

A =

a · b · c

c =

A
a · b

c =	(1.20) (1.5/M·cm)(2.0cm)

c =

0.40M

Colorimetry Calculations - Practice

Use the following questions to test your ability to do colorimetry calculations. Use any or all of the methods above. The answers follow. (Tese questions are also given in example 11 in your workbook.)

Using the data from the graphing example above, what are the concentrations of solutions with absorbances of 0.20, 0.33, and 0.47?

A solution is prepared to be 0.200M. A sample of this solution 1.00 cm thick has an absorbance of 0.125 measured at 470nm and an absorbance of 0.070 measured at 550nm. Calculate the concentrations of the following solutions:

Sample	Absorbance	Wavelength	Path length
1	0.055	470nm	1.00cm
2	0.155	470nm	1.00cm
3	0.120	550nm	1.00cm
4	0.048	550nm	5.00cm

What assumptions did you make when answering the previous questions?

Colorimetry Calculations - Answers

Here are the answers to the questions above.

Using the data from the graphing example above, what are the concentrations of solutions with absorbances of 0.20, 0.33, and 0.47?

Absorbance	Concentration
0.20	0.15M (or 0.147M)
0.33	0.24M (or 0.240M)
0.47	0.34M (or 0.343M)

Sample	Absorbance	Wavelength	Path length	Concentration
1	0.055	470nm	1.00cm	0.088M
2	0.155	470nm	1.00cm	0.248M
3	0.120	550nm	1.00cm	0.343M
4	0.048	550nm	5.00cm	0.027M

What assumptions did you make when answering the previous questions? That Beer's Law is valid in each case (or that the concentration and absorbance are indeed proportional in each case).

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E-mail instructor: Sue Eggling