Introduction. The amount of radiation absorbed may be measured in a number of ways: Transmittance, T = P / P 0 % Transmittance, %T = 100 T

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1 Introduction Many compounds absorb ultraviolet (UV) or visible (Vis.) light. The diagram below shows a beam of monochromatic radiation of radiant power P 0, directed at a sample solution. Absorption takes place and the beam of radiation leaving the sample has radiant power P. The amount of radiation absorbed may be measured in a number of ways: Transmittance, T = P / P 0 % Transmittance, %T = 100 T Absorbance, A = log 10 P 0 / P A = log 10 1 / T A = log / %T A = 2 - log 10 %T The last equation, A = 2 - log 10 %T, is worth remembering because it allows you to easily calculate absorbance from percentage transmittance data. The relationship between absorbance and transmittance is illustrated in the following diagram: So, if all the light passes through a solution without any absorption, then absorbance is zero, and percent transmittance is 100%. If all the light is absorbed, then percent transmittance is zero, and absorption is infinite.

2 The Beer-Lambert Law Now let us look at the Beer-Lambert law and explore its significance. This is important because people who use the law often don't understand it - even though the equation representing the law is so straightforward: A= bc Where A is absorbance (no units, since A = log 10 P 0 / P ) is the molar absorptivity with units of L mol -1 cm -1 (sometimes lowercase a is used instead of ) b is the path length of the sample - that is, the path length of the cuvette in which the sample is contained. We will express this measurement in centimeters. c is the concentration of the compound in solution, expressed in mol L -1 The reason why we prefer to express the law with this equation is because absorbance is directly proportional to the other parameters, as long as the law is obeyed. We are not going to deal with deviations from the law. Let's have a look at a few questions... Question : Why do we prefer to express the Beer-Lambert law using absorbance as a measure of the absorption rather than %T? Answer : To begin, let's think about the equations... A= bc %T = 100 P/P 0 = e - bc Now, suppose we have a solution of copper sulfate (which appears blue because it has an absorption maximum at 600 nm). We look at the way in which the intensity of the light (radiant power) changes as it passes through the solution in a 1 cm cuvette. We will look at the reduction every 0.2 cm as shown in the diagram below. The Law says that the fraction of the light absorbed by each layer of solution is the same. For our illustration, we will suppose that this fraction is 0.5 for each 0.2 cm "layer" and calculate the following data: Path length / cm %T Absorbance

3 A = bc tells us that absorbance depends on the total quantity of the absorbing compound in the light path through the cuvette. If we plot absorbance against concentration, we get a straight line passing through the origin (0,0). Note that the Law is not obeyed at high concentrations. This deviation from the Law is not dealt with here. The linear relationship between concentration and absorbance is both simple and straightforward, which is why we prefer to express the Beer-Lambert law using absorbance as a measure of the absorption rather than %T. Question : What is the significance of the molar absorptivity,? Answer : To begin we will rearrange the equation A = bc : = A / bc In words, this relationship can be stated as " is a measure of the amount of light absorbed per unit concentration". Molar absorptivity is a constant for a particular substance, so if the concentration of the solution is halved so is the absorbance, which is exactly what you would expect.

4 Let us take a compound with a very high value of molar absorptivity, say 100,000 L mol -1 cm -1, which is in a solution in a 1 cm path length cuvette and gives an absorbance of 1. Therefore, c = 1 / 100,000 = 1-5 mol L -1 = 1 / 1 c Now let us take a compound with a very low value of, say 20 L mol -1 cm -1 which is in solution in a 1 cm path length cuvette and gives an absorbance of 1. Therefore, c = 1 / 20 = 0.05 mol L -1 = 1 / 1 c The answer is now obvious - a compound with a high molar absorptivity is very effective at absorbing light (of the appropriate wavelength), and hence low concentrations of a compound with a high molar absorptivity can be easily detected. Question : What is the molar absorptivity of Cu 2+ ions in an aqueous solution of CuSO 4? It is either 20 or 100,000 L mol -1 cm -1 Answer : I am guessing that you think the higher value is correct, because copper sulfate solutions you have seen are usually a beautiful bright blue color. However, the actual molar absorptivity value is 20 L mol -1 cm -1! The bright blue color is seen because the concentration of the solution is very high. -carotene is an organic compound found in vegetables and is responsible for the color of carrots. It is found at exceedingly low concentrations. You may not be surprised to learn that the molar absorptivity of -carotene is 100,000 L mol -1 cm -1! Review your learning You should now have a good understanding of the Beer-Lambert Law; the different ways in which we can report absorption, and how they relate to each other. You should also understand the importance of molar absorptivity, and how this affects the limit of detection of a particular compound.

5 Colorimetric Analysis (Beer's law or Spectrophotometric Analysis) Along with operating the instruments, Beer's law also involves calculations to actually figure out the concentration of a solution from the absorbance measurements made by using the colorimeter (or spectrophotometer). There are three methods that can be used depending on what information is available. They involve using proportionality, graphing and Beer's Law. 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 (BOTH wavelength and the path length). Question: A solution with a concentration of 0.14M is measured to have an absorbance of Another solution of the same chemical is measured under the same conditions and has an absorbance of What is its concentration? The solution to this problem can be set up using the equation shown below, which simply says that the ratio of the concentrations is proportional to the ratio of absorbances. We can use C 1 to represent the unknown concentration. You can derive this equation from Beer's law (Absorbance = b c) C 1 / C 2 = A 1 / A 2 (ONLY for absorbances that are measured/predicted at the SAME Wavelength) Therefore, C 1 = (A 1 / A 2 ) * C 2 Substitute all the values as follow: Thus, C 1 = 0.12M A 1 = 0.37; A 2 = 0.43 & C 2 =0.14M

6 Graphing Example The graphing method is called for when several sets of data involving STANDARD SOLUTIONS are available for concentration and absorbance. This is probably the most common way of Beer's law analysis based on experimental data collected in the laboratory. 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. Question: What is the concentration of a 1.00 cm (path length) sample that has an absorbance of 0.60? Concentration (M) Absorbances The solution to the problem here is to graph the data 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. Recall that Beer's law is expressed as Absorbance = b c. To find the concentration for a solution that has an absorbance of 0.60, you will first need to find the slope of the BEST-FIT line. From the slope of the best-fit line together with the absorbance, you can now calculate the concentration for that solution (i.e. Concentration = Absorbance / Slope) **Notice that the SLOPE of the best-fit line in this case is actually the PRODUCT of the molar absorptivity constant and the path length (1.00cm). Beer's Law Example Here is an example of directly using the Beer's Law Equation (Absorbance = b c) when you were given the molar absorptivity constant. In this equation, is the molar absorptivity constant. b is the path length of the cuvette. c is the concentration of the solution.

7 Note: In reality, molar absorptivity constant is normally not given. The common method of working with Beer's law is in fact the graphing method (see above). Question: The molar absorptivity constant of a particular chemical is 1.5/M cm. What is the concentration of a solution made from this chemical that has an absorbance of 0.72 with a cell path length of 1.1cm? To find the concentration, simply plug in the values into the Beer's law equation.