Beer’s-Lambert Law is a combination of Beer’s law and Lambert’s law. This law associates the intensity of absorbed light with the thickness of the absorbing medium and the concentration of the solution. This law was first enacted by Pierre Bouguer before 1729. According to the attribution to Johann Heinrich Lambert, the law included path length as an extinction-affecting variable. Finally, in 1852, beer extended the law to include the concentration of the solution, which was called Beer-Lambert’s law.
Beer’s law says that the absorption of light in a solution is directly proportional to the sample concentration in which the light passes.
A = log10(Io/I) ~ C
A = εC
where,
A = Absorption
C = Concentration of the solution
ε = Absorptivity or molar attenuation coefficient
This law is used to understand the chemical analysis and decay of optical systems.
Lambert’s law states that the absorption of light in a uniform solution is directly proportional to the length i.e. thickness of the transmitted sample.
A = log10(Io/I) ~ l
A = εl
where,
A = Absorption
l = Length of the sample through which light passes
ε = Absorptivity or molar attenuation coefficient
Beer-Lambert’s Law
According to Beer-Lambert’s law, energy absorption in a solution is directly proportional to the molar extinction coefficient and solute concentration of a given solution.
Scattering by particles in the solution and reflection at the interface causes some loss of light intensity, mainly due to absorption by the solution. The ratio of I to Io depends on the path length of the absorption medium l and the concentration of the absorption solution c. These factors are related to Lambert and Beer’s Law.
Derivation of Beer-Lambert Law
When an object is exposed to radiation, some of the incident radiation is absorbed, some are scattered, and some are transmitted. As a result of absorption, the intensity of the substance that transmits light, i.e, the Intensity of transmitted light. The absorption part of the incident light depends on the thickness of the absorption medium. Lambert derived a quantitative relationship between the decrease in the intensity of monochromatic light passing through a homogeneous medium of thickness dx and the light intensity I. This law is known as Lambert’s Law and can be given as. The decrease in light intensity due to the thickness of the absorbing medium at any point is directly proportional to the light intensity.
– dI / dx ∝ I —– (1)
Where dI is the slight decrease in light intensity when passing a small distance dx, and I is the intensity of monochromatic light just before entering the medium.
– dI / dx = aI —– (2)
Where –dI/dx is the rate of decrease in strength with thickness dx, and a is called the absorption coefficient.
By the integral of Eq. (2) after dislocation,
– ln I = ax + C —– (3)
Where C is the constant of integration. At x = 0, I = Io. Therefore, C = – ln Io. Substituting this into equation (3) yields:
ln I/ Io = – ax —- (4)
Equation (4) can also be written as,
I = Io e−ax —- (5)
Equation (5) can also be written as,
log I/ Io = − a/ 2.303 x —- (6)
or, log I/ Io = -a’ x —- (7)
Where a’ (= a / 2.303) is the extinction coefficient and -ln I / Io is the absorbance of the medium. Absorbance is represented by A.
Absorbance is a dimensionless quantity that depends on the optical path length, the concentration of the absorbing substance, and the wavelength of the light.
Lambert-Beer’s law shows that when light passes through a solution of a certain thickness, the rate at which incident light is absorbed depends not only on the intensity I of the light but also on the concentration c of the solution. Extended by Beer. This is known as Lambert-Beer’s Law.
-dI / dx ∝ c —– (8)
Combining the two laws – dI / dx ∝ b × I × c —– (9)
When expressed, b is called the molar absorption coefficient.
As with Lambert’s Law, equation (9) can be transformed as follows: log I / Io = −b / 2.303 × c × x —– (10)
log I / Io = – ε × c × x ——— (11)
Here, ε (= a / 2.303) is called the molar extinction coefficient and is expressed in L /mol/cm.
Molar extinction coefficient: It measures the probability of electronic transitions. The higher the molar extinction coefficient, the higher the likelihood of electronic transitions.
Molar extinction coefficient ε depends on the type of solute to be absorbed and the wavelength of the incident light used. Equation 11 is known as Beer-Lambert’s Law.
Necessary conditions for Beer-Lambert’s Law
To avoid deviations, certain conditions must be met for Lambert-Beer’s Law to be effective and the conditions are as follows:
- There should be no electromagnetic coupling and the attenuators must operate independently of each other.
- The sample to be inspected must be homogeneous. For obvious reasons, the solution should be uniform where the light interacts with the solution, as sample variability affects attenuation.
- Incident radiation must have parallel rays traveling the same length as the sample solution.
- The incident radiation must not be multicolored. Monochromatic light is desirable.
- Light changes the sample, and optical saturation can change the attenuation, so it must not affect atoms or molecules.
- The wave characteristics of the light used should be negligible so that the attenuation does not change.
Application of Beer-Lambert law
Here are some examples of how Beer-Lambert-‘s law can be applied.
The concentration of an unknown solution can be determined by comparing it with a solution of known concentration using a spectrophotometer. The principle is based on Beer-Lambert’s law. The purity of the substance can be confirmed by measuring the absorbance of the compound with a spectrophotometer.
Limitations of Beer-Lambert law
The limitations of Beer-lambert law are given as:
Beer-Lambert’s law applies only to monochromatic light. This law applies to low concentration ranges where intramolecular interactions are not considered. This law is also invalid when very high-intensity radiation is used.
Sample Problems
Problem 1: Find the relative amount of light that gets absorbed by the sample if the absorbance of the sample is 2 at a particular wavelength.
Solution:
According to the absorbance definition;
A = log10 (I0/I)
Rearrange the equation to determine the relative loss of intensity
10A = I0/I
10-A= I/I0
Substituting the value of A = 2
1-I/I0 = 1 – 10-2 = 1 – 1/100 = 0.99
Therefore, we can say that 99% of the light is absorbed and 1% of light is transmitted.
Problem 2: What is Beer-Lambert’s Law in Absorption Spectroscopy?
Solution:
In electromagnetic spectroscopy, we see many applications of Beer-Lambert’s law. This law gives a linear relationship between the absorbance and concentration of a sample of the solution, making it possible to determine the molarity of any number of solutions.
Problem 3: Why is Beer-Lambert’s Law used in science?
Solution:
Chemistry uses Beer-Lambert’s law to measure the concentration of chemical solutions, analyze oxidation, and measure polymer degradation. The law also describes the attenuation of radiation by the Earth’s atmosphere.
Problem 4: Why does Lambert-Beer’s law fail at high concentrations?
Solution:
Beer-Lambert’s law fails at high concentrations because the linearity of the law is limited to chemical and instrumental factors. At higher concentrations, the close proximity between solution molecules is so close that absorption discrepancies occur. The refractive index changes even if the density is high.
Problem 5: Find the absorbance of a solution if its concentration is 1 mole/liter, the molar absorption coefficient is 6000 M/cm and path length is 0.02 m.
Solution:
Given – c = 1 mole/liter , ε = 6000M/cm and L = 0.02 m
Using Beer Lambert law, we have
A = εLc
= 6000 (0.02) (1)
= 120.
Problem 6: Find the concentration of a solution if its absorbance is 5.25, the molar absorption coefficient is 2254 mM/cm and path length is 300 nm.
Solution:
Given – A = 5.25 , ε = 2254 mM/cm and L = 300 nm
By using Beer Lambert law,
A = εLc
5.25 = 2254 (300) c
c = 5.25/676200
c = 77 nM.
Problem 7: Find the path length of a solution if its absorbance is 0.45, the molar absorption coefficient is 3725 mM/cm and concentration is 50 μM.
Solution:
Given – A = 0.45, ε = 3725 mM/cm and c = 50 μM.
By using Beer Lambert law,
A = εLc
0.45 = (3725) (50) L
L = 0.45/186250
L = 2.42 μM.
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