7: Quantitative Spectrophotometry and Beer's Law (Experiment)

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Introduction

The absorption of electromagnetic radiation is the basis of numerous modern analytical methods. These methods can provide, for example, qualitative information concerning molecular formulas and bonding, and quantitative information regarding the energy levels and the concen­trations of the absorbing species. In this experiment, you will compare the visible absorption spectra in an aqueous solution of several transition metal cations, examine how their separate absorptions depend on the concentrations of the ions in the solution, and use the combined absorp­tions to determine the concentrations of these ions in an aqueous mixture.

The visible region represents only a small portion of the electromagnetic spectrum. The types of transitions (quantized energy levels) that are associated with the different regions of the electromagnetic spectrum are summarized in Table \(\PageIndex{1}\). For example, the absorption of light in the visible region is typically due to the promotion of valence electrons to higher energy orbitals. In the visible spectra of transition metal cations, d-orbitals often are involved, as predicted from their electron configurations.

Table \(\PageIndex{1}\): Types of Spectroscopic Transitions

Type of Transitions	Portion of Spectrum	Approximate Wavelengths
Operations of Spins	Radio Waves	≥ 10 cm
Molecular Rotations	Microwaves and Radar	100 - 1 mm
Molecular Vibrations	Infrared	1000 - 1 µm
Valence Electrons	Near IR, Visible^* and Near UV	1000 - 200 nm
Core Electrons	Far UV and X-Rays	2000 - 1 Å
Nuclear Transitions	Gamma Rays	≤ 1 Å

* Visible Region: \( \lambda \, = \) 390 nm to 700 nm

The amount of light that a species absorbs in a spectroscopic transition can be related quantitatively to the number of absorbing species. This relationship is called the Beer-Lambert Law, or more simply Beer's Law. Consider monochromatic light of a given intensity incident on a sample, as shown in Figure \(\PageIndex{1}\). If this light can be absorbed by the sample, then the transmitted light will have a lower intensity than the incident light. The transmittance, \( T \), is defined as the ratio of the transmitted intensity, \( I \), to the incident intensity, \( I_0 \):

\[ T=I/{I_0} \]

The percent transmittance, \( %T \), is simply \( 10^2T \). The transmittance is decreased if either the concentration, \( c \), of absorbing species is increased or the path length, \( b \), of the sample is increased since both will increase the number of absorbing species in the path of the light.

Figure \(\PageIndex{1}\): Schematic Diagram of the Spectrophotometric Absorption of Light (CC BY-NC-SA;??????)

 A related quantity is the absorbance, A, which is given by

\[ A=-\log_{10}T \]

The absorbance is particularly important since it, and not the transmittance, is directly proportional to the concentration of the absorbing species and the path length. This proportionality constitutes Beer's Law and is commonly written as

\[ A= {\epsilon}bc \]

The path length, \( b \), is usually expressed in cm (usually 1 cm), and the concentration, \( c \), in molarity of moles per liter (M). The quantity \( \epsilon \) is called the molar absorptivity and has units \( {\textrm{M}}^{-1} {\textrm{cm}}^{-1} \) so that the product \( {\epsilon}bc \) is dimensionless, as are both the absorbance, \( A \), and the transmittance, \( T \). The molar absorptivity is characteristic of the species. It tells us how much light the species absorbs at a particular wavelength. A graphical plot of either the absorbance at constant path length or the molar absorptivity versus wavelength is called the absorption spectrum of the species.

The logarithmic relation between the transmittance and concentration requires justifica­tion. This arises since in each infinitesimal portion of the path the decrease in the intensity is proportional to the intensity incident on that portion. As the light travels through the sample, the intensity decreases in each succeeding portion are smaller since the intensity reaching that portion is less. The mathematical details are given at the end of this experiment’s instructions.

Beer's Law forms the basis for the analytical use of spectroscopy to determine concentra­tions. As indicated by equation \(\PageIndex{3}\), a plot of the absorbance at a given wavelength for a particular species versus the concentration of the species yields a straight line with a slope equal to \( {\epsilon}b \) and an intercept of zero. Since the path length of the cell used for the absorbance measure­ments is typically known, the molar absorptivity of the species at a chosen wavelength is readily determined from such a Beer's Law plot. The concentration of the species in an unknown sample can then be determined by measuring the absorbance of the sample at the same wavelength in any cell of known path length.

The wavelength for such an analysis should be chosen so that small changes in wavelength do not yield large changes in absorbance. Namely, the chosen wavelength should be in a relatively flat portion of the absorption spectrum. Typically, a wavelength associated with a maximum in the spectrum is chosen, since at a maximum the slope of the spectrum is zero or horizontal and simultaneously good sensitivity is obtained in the analysis (significant absorbance for a given concentration).

One should always establish, before its analytical use, that Beer's Law is followed by a species over the concentration range of interest since deviations from Beer's law often occur at high concentrations. Typically, these deviations can be traced to changes in the absorbing species or the bulk solution with concentration. For example, in concentrated solutions the solute molecules are closer together on average and interact with each other, changing their energy levels and spectroscopic properties from a dilute solution. The species of interest also may exist in equilibrium with other species that have different molar absorptivities. In such cases, a graph of absorbance versus concentration will appear to deviate from a straight line at high concentrations.

If two species are present, and neither affects the light-absorbing properties of the other, then the observed total absorbance is simply the sum of the absorbances of the individual species. When this is true, the individual concentrations can be determined from spectrophotometric measurements. Because interactions often do arise, sometimes when least expected, the absorption spectra of the species should be investigated when they are separate and when they are simultaneously present to determine whether the absorbances are indeed additive before any analytical spectrophotometric measurements.

Two cases can be differentiated. Consider the case where at some wavelength one of the species absorbs strongly, while the other does not absorb, and at some other wavelength, the converse is true. Then, these two wavelengths can be used separately to determine the concentrations of the two species, just as if the other component were not present since the absorbance at each of the wavelengths depends on the concentration of only one of the two species. The case where both species contribute to the absorption at the two wavelengths is somewhat more complicated, but the two concentrations can still be determined if the absorbances are additive.

Let's use the same path length \( b \) for all measurements. Then, the parameters ε and b in equation \(\PageIndex{3}\) can be combined into a single parameter \( k \) that depends only on the species and the wavelength at constant path length. The absorbance of species 1 at a particular wavelength, which we label by the subscript \( i \), can be written as

\[ A_{1i} = k_{1i} c_{1} \]

The total absorbance at this wavelength for a solution of the two components becomes

\[ A_{i} = A_{1i} + A_{2i} = k_{1i} c_{1} + k_{2i} c_{2} \]

A similar relation holds for any wavelength, so long as the absorbances are additive. Hence, the measurement of the total absorbance at two wavelengths (\( i \) = 1 and 2) provides equation \(\PageIndex{6}\) and equation \(\PageIndex{7}\), in the two unknown concentrations \( c_{1} \) and \( c_{2} \):

\[ A_{\textbf{1}} = A_{1{\textbf{1}}} + A_{2{\textbf{1}}} = k_{1{\textbf{1}}} c_{1} + k_{2{\textbf{1}}} c_{2} \]

\[ A_{\textbf{2}} = A_{1{\textbf{2}}} + A_{2{\textbf{2}}} = k_{1{\textbf{2}}} c_{1} + k_{2{\textbf{2}}} c_{2} \]

The various \( k \)s are determined from Beer's Law plots for the separate components at the two particular wavelengths chosen for analysis. Simultaneous solution of equation \(\PageIndex{6}\) and equation \(\PageIndex{7}\) then yields the concentrations of the two absorbing species, \( c_{1} \) and \( c_{2} \), in a mixture that contains both in terms of the absorbances of the mixture at two wavelengths, \( A_{1} \) and \( A_{2} \), and the \( k \)s.

Operation of Spectrometer

A spectrophotometer to measure absorption spectra combines a light source that emits a continuous band of radiation, a monochromator (usually a reflection grating with associated optics) to select a narrow range of wavelengths, and a photoelectric detector to measure the light intensity, as indicated in Figure \(\PageIndex{1}\). The optical path of a typical spectrophotometer is shown in Figure \(\PageIndex{2}\).

Figure \(\PageIndex{2}\): Schematic Diagram of the Optical Path of a Spectrometer (CC BY-NC-SA;??????)

In this experiment, you will be using an Ocean Optics USB2000 spectrophotometer that can measure light over a wide range of wavelengths simultaneously. The light path in this instrument is shown in Figure 3. A color picture of the light path is posted on the laboratory wall. (Note that instead of using optics to select a narrow range of wavelengths, all wavelengths are measured at the same time by the detector, a linear CCD array. CCD stands for charge-coupled device, which is the same kind of electronic device used in digital cameras and scanners.)

The light source (not shown in Figure \(\PageIndex{3}\)) is a violet LED-boosted tungsten bulb (390-900 nm) integrated into a package including the cell holder that attaches directly to the spectrometer. The light from the lamp passes through the cell and then enters the spectrometer through an entrance slit. The light is then reflected from a collimating mirror and is dispersed by a diffracting grating. The resulting dispersed light is reflected from a focusing mirror and then strikes the solid-state CCD detector that generates an electrical signal proportional to the radiant power (light intensity) at each wavelength. The signal from the detector is then transmitted to a computer where software plots the spectrum on the display in transmittance or absorbance units.

Figure \(\PageIndex{3}\): Drawing of Ocean Optics USB2000 Spectrometer (CC BY-NC-SA;??????)

The calibration procedure entails setting 100%T over the wavelength range with a cuvette containing a reference or blank sample. This is equivalent to setting absorbance to zero. This is required since the output of the lamp and the sensitivity of the detector vary with wavelength. The software automatically sets absorbance to zero (100%T) and stores the reference spectrum. The blank solution is missing the component of interest but is otherwise as identical as possible to the solution to be analyzed for the component of interest. Often the component of interest is simply dissolved in a solvent while the blank solution is just solvent. An identical cuvette containing the solution of interest is then inserted into the spectrometer, and the absorbance is shown over the wavelength range on the monitor. The reading for the solution then represents the absorbance at the chosen wavelength due to the component of interest. The calibration has accounted for any absorption (or reflection or scattering) of the light by the cuvette and other species in the reference solution.

The proper handling of a cuvette is important. Often one cuvette is used for reference and a second cuvette for the sample. Any variation in the two cuvettes (smudges or scratches on the surface, etc.) will cause errors. There are several important precautions to follow:

1. Do not handle the lower portion of the cuvette through which the light passes.

2. Wipe off any liquid drops or smudges on the lower portion of the cuvette with a clean, supplied Kimwipe. Do not use rough paper (which may scratch the cuvette) or cloth towels (which may leave lint).

3. When using two cuvettes simultaneously, always use one for the reference or blank and the other for various samples. Do not interchange the cuvette.

4. Use the cuvette that has the lower absorbance for the reference. This can be established by filling both cuvettes with deionized water and measuring the absorbance of each cuvette.

5. Insert the cuvette into the spectrometer carefully to avoid any possible scratching of the plastic surface.

6. Always rinse the cuvette with several small portions of the solution before filling it with the solution.

 

Procedure

Refer to the Logger Pro with Spectrophotometer section in Appendix 3 and calibrate your spectrometer with the blank solution (DI water). To measure the spectrum of a standard or unknown solution relative to the reference, rinse and fill the cuvette 3/4 full with the solution of interest and insert it into the sample holder of the spectrometer. Ensure that the frosted side of the cuvet is orthogonal to the direction of the incident light. 

Note: In the following parts, DO NOT dispose of any waste solutions of \( \ce{Cr^{3+}} \) or \( \ce{Co^{2+}} \) by pouring down the drain. Collect these wastes during the laboratory period in your 600 mL beaker. At the end of the laboratory period, pour the accumulated waste into the waste container provided in the hood. Rinse your beaker with a small amount of water and transfer the rinse liquid to the container. Your rinsed beaker can then be stored in your locker.

Absorption Spectra of Aqueous \( \ce{Cr^{3+}} \) and \( \ce{Co^{2+}} \)

• Obtain around 40 mL each of aqueous stock solutions that are 0.0500 M in \( \ce{Cr^{3+}} \) and 0.15 M in \( \ce{Co^{2+}} \), respectively.
• You will measure the absorption spectrum of each of these solutions from 400 to 900 nm.
• In each case, rinse the sample cuvette first with deionized water and then with a small amount of the solution you wish to measure, discarding the rinse into your waste beaker. Then, fill the cuvette 3/4 full of the solution. Store the remaining solution for later use.
• Record the maxima. These should be around 575 nm for \( \ce{Cr^{3+}} \) and 510 nm for \( \ce{Co^{2+}} \). These will be the wavelengths that are used for recording absorbance for each dilution. 

Dilution Prep for \( \ce{Cr^{3+}} \) 

Prepare solutions that are 0.0100, 0.0200, 0.0300, and 0.0400 M in \( \ce{Cr^{3+}} \) by dilution from the 0.0500 M stock solution. The dilutions are carried out using 2 and 4 mL volumetric pipettes and a single 10 mL volumetric flask. You should be able to perform each dilution with only one or two transfers with the pipets. Carefully rinse the pipettes and the flask with deionized water before their use to remove any residue from earlier use. Blow out the remaining water in each pipette, and rinse each with a small amount of the stock solution. Discard the rinse stock solution into your waste beaker. To minimize error, it is best to start by making the least concentrated dilution first.

• Prepare a table that shows the required volume of stock solution to prepare each dilution using \( \ce{M_1} \ce{V_1} = \ce{M_2}\ce{V_2}\).
• For example, the 0.01 M (\( \ce{M_2}\)) solution is prepared by pipetting 2 mL (\( \ce{V_1}\)) aliquot of the 0.05 M (\( \ce{M_1}\)) stock solution into the 10 mL (\( \ce{V_2}\)) volumetric flask.
• After the proper amount of stock solution has been transferred to the volumetric flask, add deionized water with a squeeze bottle to bring the total volume to the horizontal mark on the volumetric flask.
• Using parafilm cover the flask, and mix the solution well by inverting the flask about 15 times.
• Transfer the solution from the volumetric flask to a labeled test tube for storage and cover the top of the test tube with a piece of Parafilm.
• Rinse the volumetric flask several times with a small amount of deionized water, discarding the waste into your waste container. Prepare the remaining dilutions by changing \( \ce{M_2}\) and resolving for \( \ce{V_1}\).

Dilution Prep for \( \ce{Co^{2+}} \)

• Following the same procedure, prepare 0.0300, 0.0600, 0.0900, and 0.1200 M solutions of \( \ce{Co^{2+}} \) from the 0.1500 M stock solution.
• You will need to calculate the required volume of stock solution to pipette into the 10 mL volumetric flask. 

 

Analyzing the Dilutions 

• Starting with the least concentrated solution, measure the absorbance of each \( \ce{Cr^{3+}} \) and \( \ce{Co^{2+}} \) dilution at the two wavelengths that you determined earlier.
• Measure the absorbance for each dilution at both wavelengths before changing the solution.
• Then, empty the sample cuvette into your waste container, rinse it with a small amount of the new solution, pour the waste into your waste beaker, fill the cuvette with the new solution, and measure its absorbance at the two wavelengths.
• Record the absorbances at each of the two wavelengths and the concentration.
• After lab, you will use a spreadsheet to prepare 2 Beer's Law plots of absorbance versus molar concentration for \( \ce{Cr^{3+}} \) and \( \ce{Co^{2+}} \) respectively. For each species, perform a least-squares fit of the data for each plot and obtain the product of the molar absorptivity and the path length for each species and wavelength from the slopes. These correspond to the various \( k \)s in equations \(\PageIndex{6}\) and \(\PageIndex{7}\). Include the best-fit lines on the graphs.

Analysis of an Unknown \( \ce{Cr^{3+}} \) and \( \ce{Co^{2+}} \) Mixture

• Obtain an unknown \( \ce{Cr^{3+}} \):\( \ce{Co^{2+}} \) sample from your TA and record its number.
• Measure the absorption spectrum of your unknown, and determine its absorbance at the two analysis wavelengths, following the earlier procedures. 
• Clean up your station, disposing of the chemical waste in the appropriate waste beaker. All glass waste should go in the glass disposal.

Calculations

• After lab, you will use a spreadsheet to prepare 2 Beer's Law plots of absorbance versus molar concentration for the two ions. For each species, perform a least-squares fit of the data for each plot and obtain the product of the molar absorptivity and the path length for each species and wavelength from the slopes. These correspond to the various ks in equations 7.6 and 7.7. Include the best-fit lines on the graphs.
• From the two absorbances at the analysis wavelengths and the \( k \)s, calculate the concentration in moles per liter of each of the components in the unknown by setting up a pair of simultaneous equations analogous to equations \(\PageIndex{6}\) and \(\PageIndex{7}\) and solving for the two unknowns. Graph the absorption spectrum of your unknown, restricting the wavelength range from 350 to 750 nm. On the same graph, show the expected spectra of the individual components for the concentrations that you determined and the sum of these expected spectra. The expected spectra are obtained from your measured spectra for 0.05 M \( \ce{Cr^{3+}} \) and 0.15 M \( \ce{Co^{2+}} \) through Beer's Law by scaling for the concentration since the path length is constant. For example, if the \( \ce{Cr^{3+}} \) concentration in your unknown is \( x \) M, then the expected absorbance from \( \ce{Cr^{3+}} \) at any wavelength is \( x/0.05 \) times the absorbance of the 0.05 M solution at this wavelength for a constant path length. The expected absorbance of the unknown mixture is then the sum of the absorbances of the individual components at each wavelength. Your absorption spectrum for the unknown mixture should closely agree with the calculated sum of the individual spectra since the absorbances are additive in the studied system.
• Give the results of the analysis for your unknown mixture (and the unknown number) and discuss the accuracy of your results by propa­gating the errors from the \( k \)s. The graphs and tables should be sequentially numbered so that these can be referred to as, say, Figure 1 or Table 1 in your written text. Discuss any possible systematic errors in the experiment and what you could have done to eliminate these. 

 

Questions

1. The percent transmittance of a sample is 24.7% in a cell with a 5.00 cm path length. What is the percent transmittance of the same sample in a cell with a 1.00 cm path length?
2. The absorbance of an \( 8.54 \times 10^{-5} \) M \( \ce{AuBr_4^{-}} \) solution was determined to be 0.410 at 382 nm in a 1.00 cm cell. Assuming that the absorbance is due entirely to \( \ce{AuBr_4^{-}} \), what is the molar absorptivity of \( \ce{AuBr_4^{-}} \) at 382 nm?

3. A mixture containing \( \ce{MnO_4^{-}} \) and \( \ce{Cr_2O_7^{-}} \) was analyzed spectrophotometrically at 440 and 545 nm. The observed absorbances were 0.385 and 0.653, respectively, at each wavelength for a 1.00 cm cell, and due entirely to \( \ce{MnO_4^{-}} \) and \( \ce{Cr_2O_7^{-}} \). The molar absorptivities for \( \ce{MnO_4^{-}} \) are 93.8 \( {\textrm{M}}^{-1} {\textrm{cm}}^{-1} \) at 440 nm and 2350 \( {\textrm{M}}^{-1} {\textrm{cm}}^{-1} \) at 545 nm, while those for \( \ce{Cr_2O_7^{-}} \) are 370 \( {\textrm{M}}^{-1} {\textrm{cm}}^{-1} \) at 440 nm and 11 \( {\textrm{M}}^{-1} {\textrm{cm}}^{-1} \) at 545 nm. Determine the molar concentrations of \( \ce{MnO_4^{-}} \) and \( \ce{Cr_2O_7^{-}} \) in the mixture.

4. The molar absorptivity at 500 nm of a compound in aqueous solution is 2500±10 \( \ce{MnO_4^{-}} \) and \( \ce{Cr_2O_7^{-}} \). An aqueous solution of the compound in a 3.00±0.01 cm cell shows an absorbance of 0.410±0.005 at 500 nm. What are the molarity and its uncertainty of the compound in the solution?