2508 Beer’s Law
 Page ID
 440574
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left#1\right}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)BEER’S LAW
1.0 INTRODUCTION
Many compounds are colored due to their absorption of visible light. Our eyes are naturally capable of detecting many different wavelengths or energies of light, and we perceive these different wavelengths as different colors.
When white light passes through a colored liquid, some of the wavelengths of light are absorbed and some are transmitted or passes through the liquid. The observed color of the liquid is the sum of the wavelengths of light transmitted. For example, a substance that absorbs in the wavelength range 500 750 nm will appear bluepurple to our eyes, because all of the red, orange, yellow and green light have been absorbed.
If a sample absorbs no light within the visible spectral range, then it will appear as a clear, colorless liquid. Water, for instance, absorbs such a small amount of visible light that our eyes generally cannot perceive the absorbance; unless you have a very large volume of water, it appears completely colorless.
Absorbance spectrophotometry is a commonly used laboratory technique for determining the concentration of substances in solutions.
A spectrophotometer is an instrument used to measure the amount of light absorbed by a sample. A light source directs light into the sample and the amount of light absorbed is measured. The power of the light source, Po, will be larger than P, the power of the light that is transmitted, because the molecules in the sample will absorb some of the light.
Most spectrophotometer provide measurements of % transmittance and/or absorbance, both are unitless (dimensionless). Percent transmittance, %T, a measurement of the light transmitted, is the ratio of the powers of the light source and transmitted light.
% T=100 PPo (equation 1)
Absorbance, A, a measurement of the light absorbed, is the log of the reciprocal ratio.
A= log10PoP (equation 2)
If all the light passes through the sample without any absorption, then the absorbance reading on the spectrophotometer would be ‘zero’, and % transmittance is 100 and the sample appears colorless. Conversely, if all the light is absorbed, then the absorbance would be infinite and % transmittance would be ‘zero’.
The following equation allows for interconversion between absorbance, A and % transmittance, %T.
A = log10 %T100 (equation 3)
A linear relationship exists between the absorbance of the solutes in solution and the concentration of the solution; the more solute molecules in the path of the light source, the greater will be the absorbance of the light. Low concentrations will transmit more light (and the color will appear lighter or less intense), and high concentrations will transmit less light (and the color will appear darker or more intense). This relationship is known as Beer’s law and is given by the equation:
A= εbC (equation 4)
where A is the absorbance of the solution, ε is the molar absorptivity of the solute, b is the path length of light passing through the sample of solution, and C is the concentration of the solution in molarity (moles/L). The constant ε is specific to each solute at each wavelength and the units are M1cm1. If b and ε are constant, then the absorbance, A, is directly proportional to the concentration, C.
Since the molar absorptivity, ε, of the solute differs with wavelength, the optimal wavelength for each solution must be determined prior to absorbance spectrophotometry experiments. When working with solutions with only one solute absorbing visible light the optimal wavelength is the wavelength at which the solute is absorbing the strongest, λmax. To find λmax, a plot of absorbance vs. wavelength for a given solute is obtained. This plot is called an absorption spectrum.
At a given wavelength, plot of absorbance on the yaxis vs. concentration on the xaxis is predicted to be a linear plot, with a slope equal to εb, and an intercept of zero. In absorbance experiments, this is referred to as the standard (or calibration) curve. Once the calibration curve is complete, you will be able to interconvert between absorbance, A and concentration, C.
Notice that the yintercept of the line is zero – this makes sense, because if the concentration the lightabsorbing compound is zero, then the absorbance should also be zero. In practice, the intercept will be a very small number, close to zero, but may not be exactly equal to zero.
In this experiment, students will prepare several dilute solutions from a standard stock solution of a dye. Using a spectrophotometer, students will determine the wavelength at which the solution is absorbing the strongest, λmax. Using this wavelength, students will measure the absorbance of each of the dilute solutions and construct a standard curve. Students will be given a sports drink sample and measure its absorbance. Using this absorbance and the standard curve, students will determine the concentration of the dye in the sports drink.
Unfortunately, Beer’s Law only holds true at absorbance values between 0.1 and 1. Above or below these absorbances readings no longer maintain a linear relationship. The original sports drink samples will be too concentrated. To ensure that the absorbance reading of the sports drink sample is within the usable range, the sample will need to be diluted. When determining which dilution to prepare, keep it simple such as a 1:2 or a 1:5 dilution. A 1:2 dilution indicates that the original solution has been diluted by adding enough solvent to double the volume. The diluted solution would therefore be ½ the concentration of the original. For a 1:5 dilution, you would add water to 1 mL of sports drink sample to reach a total of 5 mL. The concentration of the dilute sports drink sample (determined from the absorbance and standard curve) can then be used to calculate the concentration of the original sports drink sample using the following equation.
M1V1= M2V2 (equation 5)
References and further reading
Technique I: Use of Spectrophotometer of the laboratory manual
Experiment 2501 Using Excel for Graphical Analysis of Data of the laboratory manual
2.0 SAFETY PRECAUTIONS AND WASTE DISPOSAL
3.0 CHEMICALS AND SolutionS
4.0 GLASSWARE AND APPARATUS
5.0 PROCEDURE
Part A: Preparation of Standard Solutions

Obtain about 25 mL of the assigned stock dye solution(s) in a clean, dry beaker. Please do not take more than what you need! Any excess will be wasted. Record the name, chemical formula, observed color and concentration of each stock dye solution (approximately 10 ppm) in Data Table 1 in Section 6.0.

Prepare the following standard solutions using test tubes and graduated pipets to withdraw the specified volumes. Make sure to record the actual volumes to two (2) decimal places. For example, if you must withdraw 2 mL of water, the water level can go beyond the 2 mL mark but record the exact volume, such as 2.01 mL in Data Table 2 in Section 6.0.
To use the graduated pipet, first use the bulb to draw solution into the pipet above the zero mark. Cap the end of the pipet with your finger. Slowly allow air into the top of the pipet until the meniscus is just touching the ‘0.00’ calibration mark. Move the end of the pipet over to your receiving test tube. Drain the pipet until the meniscus is just touching the ‘2.00’ mark. Remove the pipet and return the additional liquid back to the stock you have in your beaker. 2.00 mL has now been transferred. Adjust these instructions for other volumes as appropriate.
 Cover and securely close each test tube with a small piece of parafilm. Shake each solution well to mix.
Part B: Dye Scan
Follow instructions on the preparation and proper use of a Spectronic 200 spectrophotometer (see Technique I: Use of Spectrophotometer of the laboratory manual for additional information). Turn on the spectrophotometer and allow it to warm up for 30 minutes.
 Add laboratory water to a square plastic cuvette or glass test tube cuvette until there is about 4 cm of water from the bottom and place in the sample stage.
 Blank the instrument by pressing the autozero button.
 Empty the cuvette and shake out as much of the water as possible.
 Run a scan of one of each of the standard dye solutions from 400 nm to 700 nm.
 Record the maximum wavelength (λmax) and absorbance at the peak in the spectrum in Data Table 1 in Section 6.0.
Part C: Absorbance Measurements
The following steps can be performed on a Spectronic 20 or Spectronic 200 spectrophotometer (see Technique I: Use of Spectrophotometer of the laboratory manual for additional information). If switching to a Spectronic 20 ensure that the instrument has warmed up, stabilized, and blanked with laboratory water. Absorbance measurements should all be done on the same day, using the same spectrophotometer.
Spectronic 20
 Measure the % transmittance of the standard solutions at λmax and record the data in Data Table 2 in Section 6.0. Note: Absorbance will be calculated in Section 7.0 Calculations and Data Analysis.
 For each subsequent measurement, empty and rinse the cuvette with the next solution. Rinse the cuvette by adding a small amount of the next solution to be measured, swirling to coat the cuvette and empty before filling the cuvette until there is about 4 cm of the solution from the bottom.
Spectronic 200
 Either in Scan mode, or in Live Display mode, measure the absorbance of the standard solutions at λmax and record the data in Data Table 2 in Section 6.0.
 For each subsequent measurement, empty and rinse the cuvette with the next solution. Rinse the cuvette by adding a small amount of the next solution to be measured, swirling to coat the cuvette and empty before filling the cuvette until there is about 4 cm of the solution from the bottom.
Part D: Analysis of Dye in Sports Drink
 Obtain about 25 mL of the assigned sports drink sample in a clean, dry beaker.
 Measure the absorbance or % transmittance of the undiluted sports drink sample at the appropriate wavelength(s) and record in Data Table 3 in Section 6.0.
 Depending on how high the reading is, predict three dilutions to prepare to ensure that they all fall into the proper range. For example, if the undiluted sample has an absorbance of 1.200, and then you make a 1:1 dilution (5.00 mL undiluted + 5.00 mL water), you would expect the absorbance to be 0.600, which would be in the usable range.
 Record how you prepared each dilution in Section 6.0 (be mindful of glassware and significant figures) as well as the % transmittance or absorbance of each of your three sports drink samples. Note: If % transmittance measured, absorbance will be calculated in Section 7.0 Calculations and Data Analysis.
 If the absorbance reading of one of the dilutions does not fall within the usable range, prepare a fourth dilution and measure the % transmittance or absorbance.
6.0 DATA RECORDING SHEET
Briefly describe how the dilutions were prepared.
(e.g. 4.02 mL Gatorade + 6.00 mL water measure with graduated pipets)
Sports Drink Sample B:
Sports Drink Sample C:
Sports Drink Sample D:
Sports Drink Sample E (if needed):
7.0 CALCULATIONS AND DATA ANALYSIS
 Calculate the molarity of the stock dye solution that was provided. Assume that the solution is dilute enough that its density is the same as that of water (1.00 g/mL). (Hint: Using the concentration provided and molar mass of the dye, treat this as a unit conversion problem. Convert g solute/g solution to moles solute/liter solution.)

Using the dilution equation, M1V1 = M2V2, calculate the concentration of dye (M2) in each of the diluted solutions, where M1 is the stock concentration of dye calculated above, V1 is the volume of stock dye recorded in Data Table 2, and V2 is the total volume of the standard (sum of dye and water volumes). Record the values in Data Table 2. Remember to record the value in the Data Table with the appropriate number of significant figures and units. Show one sample calculation here.

If a Spectronic 20 was used to measure the % transmittance, the absorbance of standard solutions and sports drink samples will need to be calculated. Using equation 3 (A = log10 %T100), calculate the absorbances. Remember to record the value in the Data Table with the appropriate number of significant figures. Show one sample calculation here.
 After completing Data Table 2, make a graph of absorbance (yaxis) vs. the concentration of dye (xaxis). This is the standard curve. Notice that the data is linear. Obtain a best fit line or linear regression. Attach the graph when submitting and provide the equation here.

Based on the absorbance of the sports drink samples in Data Table 3, determine the concentration in units of molarity of dye in each sports drink sample using the equation of the standard curve. Record the value in the Analysis Table below with the appropriate number of significant figures and units. Show one sample calculation here.
For example, if the equation of your standard curve is y = 2.1033E4x + 0.0039, and the measured absorbance for a sports drink sample is 0.3110. The concentration of dye is 1.46E5 M using the algebra shown.
(0.31100.0039)2.1033×104=[dye]
 Using the dilution equation, M1V1 = M2V2, calculate the concentration of dye (M2) in the original sports drink sample, where M1 is the concentration of dye calculated for each dilution, V1 is the total volume of the dilution in Data Table 3 (sum of sports drink and water volumes), and V2 is the volume of sports drink used in each dilution. These values should be very similar to each other as you are working with dilutions of the same original sports drink. Record the values in the Analysis Table. Remember to record the value in the Analysis Table with the appropriate number of significant figures and units. Show one sample calculation here.
7. Calculate the average concentration of dye in the original sports drink sample.
8.0 POSTLAB QUESTIONS AND CONCLUSIONS
1. Molar absorptivity, ε
a. Calculate ε based on Beer’s Law (equation 4) and the absorbance and concentration of one of your standards. The pathlength of the cuvette used is 1.0 cm. Choose wisely. Do not choose a standard that might have an error.
b. The slope of the line of your standard curve is derived from the molar absorptivity (ε) of the dye and the pathlength (b) of the sample in the spectrophotometer. The pathlength of the cuvette used is 1.0 cm, calculate ε based on the trendline of your standard curve.
c. Compare the two molar absorptivity values (from Beer’s Law and from the trendline). Which do you think is more accurate? Offer an explanation for your choice.
2. Locate and label the undiluted and diluted sports drink sample data points on your standard curve. Do all of these points fall within the usable range of the spectrophotometer? Do all of these points fall within the data points of the standard solutions?
3. How is the precision of your calculated concentrations of the original, undiluted sports drink sample? Calculate the standard deviation and relative standard deviation for the three dilutions. What does your relative standard deviation tell you about your precision?
4. Based on your average concentration of dye in the original sports drink sample and the molar mass of the dye, calculate how many grams of dye are present in 946 mL of sports drink.