Beer's Law

Spectrophotometry/Beer’s Law Lecture
Chemistry 114

Overview:
Spectroscopy will be a tool that you will use as you continue in your chemistry, biology and physics courses. Already, you have used spectroscopy in this course and CHM 113. Earlier in the term, we used the spectrophotometer to monitor double stranded and single stranded DNA. We have also used it to measure concentration of chemicals. This lecture will discuss spectophotometry in more detail.

Background:
You will recall from CHM 113 (week 4) that light is described by both a particle theory (photons) and a wave theory. In CHM 113, we discussed the relationships between energy, wavelength, frequency, and speed of light waves. The wavelength, λ, (lambda) of any wave is the crest-to-crest distance between waves. The frequency, ν, (nu) is the number of complete oscillations that a wave makes each second. The speed of a wave is the distance one wave travels in a given time period. The relation between wavelength, frequency, and speed is

Speed = frequency * wavelength (1)

The speed of light is constant in a given medium. For example, the speed of light in a vacuum is 2.99792458 x 108 m/s and is symbolized by the letter ‘c’. The speed of light decreases in other media such as air or water. In water, the speed of light decreases enough to give illusions such as a spoon appearing bent when it is partially submerged in a glass of water. The decrease in speed is negligible in air so we will consider the speed of light to be 3.00 x 108 m/s in air. This leads to the equation

c = λν (2)

Where c = 3.00 x 108 m/s, λ = wavelength given in meter units, and ν = frequency given in Hertz units (Hz, oscillations/second).

Each light particle, or photon, carries a discrete amount of energy. This energy is proportional to the frequency of the light, and is therefore inversely proportional to the wavelength of light. This relation was discovered by Max Planck in 1900 when he determined that

Ephoton = hν (3)

where Ephoton = Energy of one photon (not a mole of photons!) given in Joule units and h = Planck’s Constant, 6.626 x 10-34 Joule*seconds.

Absorption and Emission of Light:
The particle nature of light is most often used to explain absorption and emission properties of atoms and molecules. Molecules absorb or emit energy only in discrete amounts or packets called quanta. Absorption or emission of light by an atom or a molecule can only occur if the energy of the photon is equal to the energy difference between two energy levels in the atom or molecule. For the purposes of CHM 114, we will concern ourselves only with absorption of light for the remainder of this discussion.

Molecules can exist only in discrete energy states. This means that they will only absorb photons of specific energies and therefore specific frequencies and wavelengths. This is shown graphically when the amount of light absorbed is plotted as a function of the frequency or wavelength. This graph is called the spectrum of a particular molecule or atom. Figure 1 shows how two molecules that are very similar structurally can have different spectra. We will use this difference in spectra to our advantage during our kinetics laboratory in week 8.

Figure 1: Two molecules NAD and NADH have very similar structures but have different spectra in the ultraviolet light range.
Transmittance and Absorbance:
It is difficult, if not impossible, to measure actual absorbance of light. Instead, we measure transmittance or the fraction of light that is able to pass through a solution of molecules. A spectrophotometer measures the intensity of light entering a sample and compares this to the intensity of light emerging from the sample (Figure 2).

Figure 2: Transmittance of light in a spectrophotometer. Transmittance = I0/IE.

There are three things that will effect the amount of light emerging from the sample. First, the concentration of molecules in the solution affects the transmittance. Each molecule can absorb light. As you increase the number of molecules in the solution, you also increase the photons absorbed. Therefore, as you increase the concentration of a sample, you decrease the transmittance. Second, the length of the sample path will affect the transmittance. By increasing the pathway that the light must travel through your sample, you are increasing the number of molecules that will interact with the light; in effect, you are increasing the apparent concentration. Third, the transmittance will be affected by specific properties of the molecules. Molecules absorb light at different efficiencies and at different energies. Therefore, the transmittance will be dependent upon the specific molecule in solution and the wavelength of light being passed through the sample.

In 1729 Pierre Bouguer discovered that the relationship between transmittance and concentration or sample path length is a logarithmic one. The reason for this can be illustrated if you consider the sample of solution in figure 2 above. The molecules nearest the source of light will experience IO the same as that which was measured by the spectrophotometer. However, molecules nearest the exit point of the sample will experience IO that is less than the original because molecules have absorbed light throughout the sample. In other words, as light travels through the sample, there is a drop in IO in each succeeding layer. The definition of transmittance tells us that that T = IE/IO. Because IO changes throughout the sample, we have a logarithmic, rather than linear, relationship between transmittance and concentration or sample path length.

The Lambert-Beer Law:
The fact that transmittance of light varies exponentially as it passes through an absorbing medium was rediscovered by Johann Heinrich Lambert in 1760. Later, in 1852, August Beer defined absorbance:

A = -log10(T) (4)

When no light is absorbed, IE = IO, T = 1.00 and A = 0. When 90% of the light is absorbed, T = 0.1 and A = 1. Beer then showed that absorbance was linearly related to concentration. These two men are given credit for the Lambert-Beer Law:

A = abc (5)

Where a = a constant which takes into account the specific properties of the molecules which are absorbing photons, b = sample path length, usually given in centimeter units, and c = concentration of solution. Note that Absorbance is unitless, so the units of the constant a must be the inverse of the units of path length and concentration. For example, if b is in centimeter units and c is in mg/mL units, then a must be in mL*mg-1*mL-1 units.

When b has M-1*cm-1 units, we call it the molar absorption coefficient (formerly called the extinction coefficient) and designate it as ε. This means the concentration of the solution must be expressed in Molarity units. Also, some chemists will use l instead of b to denote path length. In these cases, equation 5 becomes

A = εlc (6)

It is interesting to note that equations 5 & 6 are often referred to as Beer’s Law, with no reference to Lambert. Moreover, Pierre Bouguer, who first published the theory upon which this law is based does not get any credit in the name of the equation.

Practical Uses of Beer’s Law:
How does any of this help us in our study of chemistry? Most often we are using Beer’s Law to help us determine the concentration of a solution--if we know two of the three variables a, b, or c in equation 5, we can measure the absorbance and determine the unknown variable.

Determination of Concentration When a is Known:
For many chemicals, the constant a is known and is listed in tables. It is usually listed as its molar absorption coefficient, ε. In this case, you would simply use the literature value for ε, measure the path length of your sample, measure the absorption of your solution and solve for c in equation 6. For example, the cuvette is 1 cm, ε = 3600 M-1cm-1, and the absorption of solution measures 0.650, then you can solve for c:

A = εlc
0.650 = (3600 M-1cm-1)(1 cm)c c = 1.80 x 10-4 M

In practice, chemists rarely determine concentration in this manner. This method can result in answers that are incorrect due to differences in instrumentation and inherent errors made by the scientist. If the instrument that you are using is not calibrated the same as the instrument used to determine ε, the concentration calculated could be wrong. Also, it is possible for you to make consistent mistakes such as pipetting incorrectly or using a cuvette that has a scratch or is dirty. For these reasons, concentrations are most often determined using a standard curve.

Determination of Concentration Using a Standard Curve:
Standard curves are generated when a is not known and/or to minimize experimental error. In this case, the researcher determines the absorbance of several known concentrations of the solution. These known concentrations are referred to as standard solutions. The standards are then plotted as an absorbance vs concentration graph. This graph is called a standard curve. If you were careful in making the standards and measuring their absorbance, the graph should be a straight line (figure 3). The line should have a y intercept of zero (when the concentration is zero, there should be zero absorbance) and the slope of the line is equal to a in equation 5.

Figure 3: Standard curve.

After the standard curve has been plotted, the concentration of experimental solutions can be determined. The absorbance of the experimental solution is measured and compared to the standard curve. A quick, but less precise, method for doing this is to find the absorbance of the experimental solution on the y-axis of the standard curve, draw a line parallel to the x-axis until you reach the line on the graph, then draw a line parallel to the y-axis until you reach the x-axis. The point on the x-axis tells you the concentration of your solution (figure 4).

Figure 4: An experimental solution had an absorbance of 0.48. Using the graphic method, this indicates that the concentration was 0.38 M.

A more precise method for determining the concentration of an experimental solution requires you to determine the straight-line equation of your line. This can easily be done by computer. Once you have this equation, you simply put the measured absorbance of your experimental solution in as y, and solve for x. For example, the linear regression of the data in Figures 4 and 5 shows that the straight-line equation for these graphs is

y = 1.2x + 0 so, 0.48 = 1.2x x = 0.40 M

You will note that the answers from both methods is very similar, but you get a more precise measurement when you use the line equation.

Sources of Error in Beer’s Law Measurements:
Beer’s Law is only true for dilute solutions--the exact range of solutions must be determined experimentally. Beyond this range, measurements and calculations using Beer’s Law will be erroneous.

Other common sources of error include the use of dirty cuvettes, poorly mixed solutions, poor pipetting techniques, and incorrect light source or wavelength. Because you have control over these errors, you must make sure to minimize these problems in your laboratory exercises.