The radiant power of a beam of light is the number of photons per second per unit area perpendicular to the beam. When light passes through a solution that contains a light-absorbing species, photons are absorbed, and the radiant power of the beam decreases. If the radiant power of a beam of monochromatic light incident to the solution is Po and the radiant power of the beam exiting the solution is P, then
where c is the molarity of the absorbing species in the solution, b is the length of solution through which the light passed, and e is a constant called the molar absorptivity. The magnitude of e depends on the wavelength, l , of the incident light and the nature of the absorbing species. If b is expressed in units of cenimeters, then e has the units (M-cm)-1. The absorbance of the solution, A, is defined as , and the percent transmittance, %T, of the solution is expressed as .
Equation 2 is the Beer-Lambert law or Beer's law and is valid only when the incident light is monochromatic.
The light-absorbing solution is held in a glass or quartz container called a cell or cuvette. A significant amount of radiant power is lost through reflection of the light at the air/cell and cell/solution interfaces and through absorption of the light by the cell. Consequently, Po and P cannot be easily measured. However, the absorbance and percent transmittance of the absorbing solution can be approximated by comparing the radiant power of the light exiting the absorbing solution, Psolution , to the radiant power of the light exiting a blank solution, Pblank .
The blank is a reference solution that contains all of the components found in the absorbing solution except the absorbing species. When the cuvettes that contain the blank solution and the absorbing solution are identical, the approximation expressed in eq 3 corrects for the loss of radiant power due to the reflection and absorption of the light by the cuvette.
If the values for e and b are known and the absorbance or percent transmittance of a solution which contains an unknown amount of an absorbing species is measured, then the molarity of the absorbing species, cunk, may be obtained with the aid of Beer's law, eq 4.
Generally, the absorbances are measured at a wavelength l corresponding to a maximum in the absorption spectrum because at lmax there is the greatest change in absorbance with a change in the molarity of the absorbing species and hence, the sensitivity of the spectrophotometric method to small changes in the molarity of the absorbing species is maximized. Also, at lmax the absorbance changes little with small changes in wavelength, and thus Beer's law is approximated when the incident radiation, found in most spectrophotometers, is a narrow band of polychromatic light. To evaluate e a set of standard solutions with known concentrations of the absorbing species are prepared and the absorbances of these solutions are measured at lmax. Since the standard and unknown solutions contain the same absorbing species and since lmax is the same in each measurement , then e is a constant and the plot of absorbances versus molarities of the standard solutions should be a straight line with an intercept of zero .
Such a plot is called a calibration curve and the slope of the straight line is equal to eb. A linear regression program may be used to construct a calibration curve from the experimental data and to determine the slope and intercept of the curve. Often the linear equation that is fitted to the experimental data by the program has a nonzero intercept. If the values for the slope and intercept are substituted into eq 5 , then cunk can be calculated
or cunk may be read from the calibration curve . The linearity of the calibration curve also confirms that the chemical system under study does comply with Beer's law. Deviations from Beer's law can occur, and therefore it is prudent to verify that Beer's law is obeyed particularly if the value of e is to be determined.1,2
The JAVA applet below simulates an analytical experiment in which the spectrophotometric method is employed to determine the molarity of the hexaaquochromium(III) ion, Cr(H2O)63+, in an aqueous solution. The absorption spectrum of an aqueous solution of Cr(H2O)63+ can be accessed by clicking the button. The Ace Spectrophotometer is a single-beam spectrophotometer . The scrollbar in the upper right-hand corner of the spectrophotometer is used to set the wavelength of the incident light, and that wavelength is displayed in the box to the left of the scrollbar. The cuvette with the blank solution is inserted into the spectrophotometer by clicking the button. After the cuvette has been place in the cuvette holder and the cuvette cover has been closed, adjust the scrollbar labeled 100 %T until the display reads 100 %T. Replace the blank with an identical cuvette that contains a standard Cr(H2O)63+ solution by selecting the molarity of the standard solution from the pull-down menu labeled Standard Solutions of Cr(III). The blank may also be replaced with an identical cuvette that contains a Cr(H2O)63+ solution of unknown molarity by clicking the button. After the cuvette has been positioned in the holder and the cover closed, the %T of the solution will appear in the display. The blank should always be placed in the spectrophotometer and the 100 %T set or reset before the %T of a standard or unknown solution is measured. If you wish to have the output from the spectrophotometer displayed as absorbance rather than %T, chick the button. The absorbance of the blank solution should be adjusted to 0.000.
Use the applet in the design and performance of experiments
to answer the following questions.
- At which wavelength of light should the absorbance or percent transmittance measurements be made?
- What are the absorbances of the 0.0100, 0.0200, 0.0300, 0.0400 M standard solutions of Cr(H2O)63+ at the selected wavelength?
- Do the Cr(H2O)63+ solutions obey Beer's law? What evidence do you have to support your answer?
- If b has a value of 1.16 cm, what is the value of the molar absorptivity, e, for the Cr(H2O)63+ ion at the selected wavelength ?
- What is the molarity of your unknown Cr(H2O)63+ solution ?
If you have preformed all of the experiments, answered all of the questions, and completed the report to be submitted for credit, then you may check the
Answers to the Spectrophotometry Questions
1) Harris, Daniel, C. Quantitative Chemical Analysis, 5th ed.; W. H. Freeman and Company: New York, 1998; pp 511-575.
2) Skoog, Douglas A.; West, Donald M.; Holler, F. James Analytical Chemistry, An Introduction, 6th ed.; Saunders College
Publishers: Philadelphia, 1994; pp 383-442.