Beer's Law

Absorbance and Fluorescence

When light hits a substance, it is either absorbed, transmitted, or reflected. Typically, a substance interacts with a range of wavelengths of light, each one interacting with the molecules or atoms differently. A substance may absorb a specific range of wavelengths, reflect another range of wavelengths, and transmit the other wavelengths of light.

When a molecule absorbs light, the energy is used in four different ways: (1) translation, which causes the molecule to change its molecular velocity; (2) vibration, which causes the distance between the molecules to change rapidly; (3) rotation, which causes the atoms to rotate around the bonds in a molecule; and (4) electron excitement, which causes electrons to transition to higher energy levels.

Energy Levels

In 1913, Niels Bohr proposed a model for the hydrogen atom in which electrons travel around the nucleus in fixed, circular orbits, called stationary states. The energy associated with each orbit, or stationary state, exists only at fixed, discrete energies. Only when an electron moves to another orbit is energy absorbed or emitted. The electron is never in-between states. This change only occurs if the absorbed or emitted energy equals the difference between the two energy states.

In Bohr’s model, the quantum number n represents the electron’s energy. When an electron occupies the lowest possible energy state, it is said to occupy the ground state, which is n = 1. When an electron absorbs a photon, whose energy equals the difference between the first and second states, the electron becomes excited and transitions from the ground state to the excited state, where n = 2. If the energy of the photon equals the difference between the first and third states, the electron transitions to the third state, or n = 3, and so on.

Electrons can spontaneously return to the ground state or any other lower, excited state. When this happens, the excess energy that was gained from excitation is released in the form of an emitted photon. The energy of the photon is equal to the difference between the two energy states and corresponds to different wavelengths of light.

Absorption and Emission Spectra

While most substances absorb or emit the maximum amount of light at one wavelength, they also tend to absorb or emit light at a range of wavelengths. This range of wavelengths is called a spectrum. The energy of the absorbed light is quantified and visualized using an absorption spectrum, while the energy of the emitted light is quantified and visualized using an emission spectrum.

Absorption and emission spectra are measured using a spectrophotometer, which is a device that transmits light through a sample and then measures both the wavelength and intensity of light that passes through it. Inside the spectrophotometer is either a diffraction grating or a prism, which separates incoming light into its component wavelengths. The different wavelengths are then transmitted through the sample, and the intensity is recorded on a linear charge-coupled device (CCD) detector. The CCD is an integrated circuit etched onto a silicon surface that forms light-sensitive elements called pixels. The CCD collects and sorts the diffracted light and reads it back at an absorption wavelength.

When measuring the absorbance of a sample, the solute is usually dissolved in a solvent and placed in a container known as a cuvette. Then, the sample is placed inside the spectrophotometer, and the intensity of the transmitted light is measured along with the wavelengths of light to obtain an absorbance spectra. As expected, the intensity of the transmitted light is lower than when there is no sample present inside the spectrophotometer.

This is because the transmitted light is absorbed by the sample, the cuvette, and the solvent. Before measuring samples, the spectrophotometer must be calibrated with a ‘blank’. A blank is a cuvette that only contains the solvent that is used to dissolve the solute. The spectrophotometer is calibrated so that the total absorbance due to the cuvette and solvent is subtracted from the sample’s measured absorbance. This allows us to record the absorbance that is only attributed to the species of interest.

The absorbance is often measured at one wavelength, the maximum absorbance wavelength. However, the absorption can also be measured at a range of wavelengths to acquire the absorption spectrum. For this, the sample is exposed to a range of wavelengths of incident light, and the absorption is recorded at each wavelength. If the sample emits light, the emission spectrum is measured similarly, except the incident wavelength is fixed at the wavelength of maximum absorbance. The instrument then measures the intensity of emitted light over a range of wavelengths.

Beer-Lambert Law

The absorbance of a sample at the wavelength of maximum absorbance provides information about the sample, namely its concentration. The Beer-Lambert Law is an equation that relates transmittance to sample concentration. The transmittance, or intensity of transmitted light, is the fraction of original light that passes through the sample, I, divided by the intensity of the incident light, I₀.

The Beer-Lambert Law states that the optical absorbance, A, of a species in solution is related to the negative log of the transmittance.

An alternative version of the Beer-Lambert Law states that the optical absorbance, A, of a species in solution is linearly proportional to the concentration, c, of that species when the wavelength, λ, and pathlength, l, are held constant.

The molar attenuation coefficient, ε, is a measure of how strongly a species absorbs light at a given wavelength. The greater the molar attenuation coefficient, the greater the absorbance. The pathlength, l, is the distance that the light travels through the sample, which is the width of the cuvette. Standard cuvettes have a pathlength of 1 cm.

This linear relationship between absorbance and concentration is a powerful tool that is used to determine the concentration of an unknown sample based on its absorbance. To do this, a standard curve is generated using a gradient of known concentrations of the solute. The absorbance at the peak absorbance wavelength, λ_max, is measured for each concentration.

By plotting concentration versus absorbance, a linear relationship is observed that corresponds to the Beer-Lambert equation. The slope of this line is equal to the product of the pathlength and the molar attenuation coefficient. Using this calculated linear function, if the absorbance of the unknown sample is known, the concentration can easily be determined.

If the sample being analyzed is a reaction at equilibrium, Beer’s Law can be used to determine the equilibrium concentration of a product or reactant if the absorbance is measured at λ_max specific to that product or reactant. Once the concentration is known, you can determine the equilibrium concentrations of the remaining reactants and products and then solve for the equilibrium constant K_eq.

References

1. Kotz, J.C., Treichel Jr, P.M., Townsend, J.R. (2012). Chemistry and Chemical Reactivity. Belmont, CA: Brooks/Cole, Cengage Learning.
2. Silderberg, M.S. (2009). Chemistry: The Molecular Nature of Matter and Change. Boston, MA: McGraw Hill, Boston.
3. Harris, D.C. (2015). Quantitative Chemical Analysis. New York, NY: W.H. Freeman and Company.