Beer-Lambert law

The Beer-Lambert law describes the relationship between the concentration of a light-absorbing solute and the amount of light absorbed:

Equation 1: A = εcℓ

• A = the absorbance, a measure of how much light the solution absorbs. It has no units.
• ε = molar absorption coefficient, with units of M-1 cm-1.
• ℓ = length of the path of light through the sample, in centimeters. This is straightforward: twice as much light will be absorbed if the light has to travel through 2 centimeters than if the light has to travel through 1 centimeter.
• c = concentration of the solute (molar, M).

The equations says that the absorbance A equals the product of the molar absorption coefficient, the path length of the light in the sample, and the concentration of the solute.

Figure 1. A beam of light is passed through a clear blank solution and a colored sample solution. More light is absorbed by the sample than the blank. The radiant power of the light that has passed through the blank (Po) is higher than the radiant power of the light that has passed through the sample (P). Electronic circuits can measure the radiant power of the light. The circuit can have various outputs, such as a change in voltage, resistance or frequency.

The absorbance is defined by how much light is transmitted through the sample (P) vs how much light is transmitted through a blank (Po). Actually, by the log to the base of 10 of this ratio. See Equation 2.

Equation 2. A = log (light from blank/light from sample) = log (Po/P)

• A = absorbance
• Light from blank = amount of light that shines through that solution when there is no absorbing substance (the blank) = Po
• Light from sample = amount of light that shines through the sample containing a light-absorbing substance = P

Equation 2 says that the absorbance A is equal to the log of the ratio of “light from blank” over “light from sample”. Note that as the “light from sample” number gets smaller, the absorbance gets bigger, which is what you would expect. If you have a way to determine the amount of light leaving the sample, you can calculate the absorbance.

The amount of light can be measured as a change in voltage produced by a light sensing electronic circuit. The circuit can have any of several light-sensitive components, such as a photoresistor, a phototransistor, or a photodiode. The amount of light can also be measured as a change in frequency of a signal produced by a light sensing integrated circuit such as the TSL230R chip. Practically, “light from blank” is taken as the 100% transmission standard since this accounts for absorbance due to the sides of the sample container, etc. If the signal is measured a as voltage, Equation 2 can be re-written as follows:

Equation 3. A = log (Vo/V)

• A = absorbance
• Vo = voltage produce by light shining through the blank.
• V = voltage produced by light shining through the sample.

Combining Equations 2 and 3 yields a relationship that will be used to make a graph to find the concentration in an unknown sample.

Equation 4. log Vo/V =αc

• Vo = voltage output with blank
• V = voltage output with sample
• α = a constant, which depends on ε, path length, and other factors.
• c = concentration of the solute that is absorbing the light.

Equation 4. says that a graph of the log of Vo/V vs the concentration of the solute should yield a straight line with a slope of α and a y-intercept of 0. To find the concentration of an unknown, we will make a series of standard solutions with known concentrations, use these to make a graph based on equation 4, and then read the concentration of the unknown from the graph. In reality the line is not linear over the higher range of the standards, but even so the line can be used to find the concentration of the “unknown” accurately.

The table below has some example data.

Table 1. Example data from a Bradford assay (protein concentration) experiment.

Three voltages are read for each sample. Vo is the voltage obtained with a blank solution. The value of log(Vo/V) is the absorbance and is proportional to the concentration.

Analyzing the data.

Figure 2. Example data for Bradford assay. Note that the response becomes non-linear at higher concentrations.

Use the graph to estimate the concentration of protein in the unknown sample.

1. Draw a horizontal line on the graph showing the value of log Vo/V for the sample.

2. Locate the concentration on the x-axis that corresponds to this absorbance.

3. Correct for the dilution factor. a. For example, if the unknown sample has a concentration of 400 micrograms per ml after it was diluted 100 X, the original solution must have been 100 times more concentrated.

4. Record the value obtained for the protein concentration of the unknown sample or samples.

Variations

Author(s)

David Whyte

Terms and definitions

Solute - chemical that is dissolved in a solution.

Molar absorption coefficient, ε - a measurement of how strongly a chemical species absorbs light at a given wavelength. ε is the Greek letter epsilon.

M - molar concentration

Blank - a sample with no solute. It is used to correct for light that is absorbed or scattered by the solvent, the sample tube, etc.

Standard solutions - solutions with known concentrations of the solute. Used to make a standard curve.

Standard curve - a graph showing the concentration of standard solutions on the x-axis and the absorbance on the y-axis. Used to determine the concentration of the unknown.