The Beer-Lambert law (Beer-Bouguer) is one that relates the absorption of electromagnetic radiation of one or more chemical species, with its concentration and the distance that light travels in particle-photon interactions. This law brings together two laws in one.
Bouguer's law (although the recognition has fallen more on Heinrich Lambert), establishes that a sample will absorb more radiation when the dimensions of the absorbent medium or material are greater; specifically, its thickness, which is the distance l that the light travels when entering and leaving.
The upper image shows the absorption of monochromatic radiation; that is, made up of a single wavelength, λ. The absorbent medium is inside an optical cell, the thickness of which is l, and contains chemical species with a concentration c.
The light beam has an initial and final intensity, designated by the symbols I0 and I, respectively. Note that after interacting with the absorbent medium, I is less than I0, which shows that there was absorption of radiation. The older they are c Y l, smaller will be I with respect to I0; that is, there will be more absorption and less transmittance.
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The image above perfectly encompasses this law. Radiation absorption in a sample increases or decreases exponentially as a function of c or l. In order to fully understand the law in a simple way, it is necessary to skirt its mathematical aspects.
As just mentioned, I0 and I are the intensities of the monochromatic light beam before and after the light, respectively. Some texts prefer to use the symbols P0 and P, which refer to the energy of the radiation and not to its intensity. Here, the explanation will be continued using the intensities.
To linearize the equation of this law, the logarithm must be applied, generally the base 10:
Log (I0/ I) = εlc
The term (I0/ I) indicates how much the intensity of the radiation product of absorption decreases. Lambert's law considers only l (εl), while Beer's law ignores l, but places c instead (εc). The upper equation is the union of both laws, and therefore is the general mathematical expression for the Beer-Lambert law.
The absorbance is defined by the term Log (I0/ I). Thus, the equation is expressed as follows:
A = εlc
Where ε is the extinction coefficient or molar absorptivity, which is a constant at a given wavelength.
Note that if the thickness of the absorbent medium is kept constant, like ε, the absorbance A will depend only on the concentration c, of absorbent species. Also, it is a linear equation, y = mx, where Y is A, and x it is c.
As absorbance increases, transmittance decreases; that is, how much radiation manages to be transmitted after absorption. They are therefore inverse. Yes I0/ I indicates the degree of absorption, I / I0 equals transmittance. Knowing this:
I / I0 = T
(I0/ I) = 1 / T
Log (I0/ I) = Log (1 / T)
But, Log (I0/ I) is also equal to absorbance. So the relationship between A and T is:
A = Log (1 / T)
And applying the properties of logarithms and knowing that Log1 is equal to 0:
A = -LogT
Transmittances are usually expressed in percentages:
% T = I / I0∙ 100
As previously stated, the equations correspond to a linear function; therefore, it is expected that when graphing them they will give a line.
Note that to the left of the image above there is the line obtained by graphing A against c, and to the right the line corresponding to the graph of LogT against c. One has a positive slope, and the other negative; the higher the absorbance, the lower the transmittance.
Thanks to this linearity, the concentration of the absorbent chemical species (chromophores) can be determined if it is known how much radiation they absorb (A), or how much radiation is transmitted (LogT). When this linearity is not observed, it is said that it is facing a deviation, positive or negative, of the Beer-Lambert law.
In general terms, some of the most important applications of this law are mentioned below:
-If a chemical species shows color, it is an exemplary candidate to be analyzed by colorimetric techniques. These are based on the Beer-Lambert law, and allow to determine the concentration of the analytes as a function of the absorbances obtained with a spectrophotometer..
-It allows the construction of calibration curves, with which, taking into account the matrix effect of the sample, the concentration of the species of interest is determined.
-It is widely used to analyze proteins, as several amino acids have significant absorptions in the ultraviolet region of the electromagnetic spectrum..
-Chemical reactions or molecular phenomena that involve a change in color can be analyzed using absorbance values, at one or more wavelengths..
-Using multivariate analysis, complex mixtures of chromophores can be analyzed. In this way, the concentration of all the analytes can be determined, and also, the mixtures can be classified and differentiated from one another; for example, rule out whether two identical minerals come from the same continent or specific country.
What is the absorbance of a solution exhibiting a transmittance of 30% at a wavelength of 640 nm??
To solve it, just go to the definitions of absorbance and transmittance.
% T = 30
T = (30/100) = 0.3
And knowing that A = -LogT, the calculation is straightforward:
A = -Log 0.3 = 0.5228
Note that it lacks units.
If the solution from the previous exercise consists of a species W whose concentration is 2.30 ∙ 10-4 M, and assuming the cell is 2 cm thick: what should its concentration be to obtain a transmittance of 8%??
It could be solved directly with this equation:
-LogT = εlc
But, the value of ε is unknown. Therefore, it must be calculated with the previous data, and it is assumed that it remains constant over a wide range of concentrations:
ε = -LogT / lc
= (-Log 0.3) / (2 cm x 2.3 ∙ 10-4 M)
= 1136.52 M-1∙ cm-1
And now, you can proceed to the calculation with% T = 8:
c = -LogT / εl
= (-Log 0.08) / (1136.52 M-1∙ cm-1 x 2cm)
= 4.82 ∙ 10-4 M
Then, it is enough for the W species to double its concentration (4.82 / 2.3) to reduce its transmittance percentage from 30% to 8%..
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