Derivation of Beer-Lambert Law

The Beer-Lambert law states that the absorption of energy in a solution is directly proportional to the molar absorptivity and the solute concentration of the given solution. Beer-Lambert law is the combination of 2 laws given by Johann Lambert and August Beer, hence it came to be known as Beer-Lambert law.

Lambert’s law states that the Absorption of light in a homogenous solution is directly proportional to the length of the sample through which it passes.

A = log₁₀(_I0/I) ~ l

where,

A = Absorption

l = Length of the sample through which light passes

Beer’s law said that the absorption of light in a solution is directly proportional to the sample concentration in which the light passes.

A = log₁₀(I₀/I) ~ C

Where, 

A = Absorption

C= Concentration of the solution

This law is used in chemical analysis and the understanding of attenuation in optics. 

Beer-Lambert Law Formula

Lambert said that the loss in the intensity of light is directly proportional to the path length. Later, in 1852, Beer extended the law and found out that the transmittance remains constant if the product of concentration and the length of the sample through which the light passes remains constant. 

The modern-day definition of Beer-Lambert law combines the two definitions and says that the attenuation of the transmitted light is directly proportional to the length of the sample solution and its concentration.

The mathematical derivation of the Beer-Lambert law is as follows.

Let I and I0 be the transmitted and the incident ray of light, respectively.

The incident light I0 falls upon the sample and the transmitted light I is radiated out.

Here, 

A = I₀/I

T = I/I₀

Where,

A = Absorbance 

T = Transmission

We get,

A = log₁₀(1/T)

A = log₁₀(I₀/I)

So according to Beer-Lambert Law:

A = log₁₀(I₀/I) ~ l

A = log₁₀(I₀/I) ~ C

Mathematically, Beer-Lambert Law is given as follows.

A = εlC

Where,

A = Absorbance

l = Length of the sample solution

C = Concentration of the solution

Derivation of Lambert's Law

Lambert’s law states that the absorbance of the solution is directly proportional to the length of the sample of the solution through which the light passes.

So mathematically,

A = log₁₀(I₀/I) ~ l

A = εl

Where,

A = Absorbance

l= Length of the sample solution

ε = Absorptivity or molar attenuation coefficient

Derivation of Ber's Law

Beer’s law states that the absorbance of the solution is directly proportional to the concentration of the solution.

So mathematically,

A = log₁₀(I₀/I) ~ C

A = εC

Where, 

A = Absorbance

C = Concentration of the solution

ε = Absorptivity or molar attenuation coefficient

Importance of Concentration of the Solution

Let us discuss how the concentration of the solution affects its absorbance. The amount of incident light absorbed will depend on how many molecules it interacts with in the solution. In the presence of a concentrated solution, there will be a number of molecules to interact with the light. However, in the case of a dilute solution, the number of molecules to react with the incident light will be far less, hence, the attenuation of the incident light will be less. This is why absorbance is directly proportional to the concentration of the solution.

Importance of the Length of the Sample

The length of the sample interacting with the incident light amounts to the number of molecules coming in contact with the light. So, if the path length of the incident light is increased or decreased the number of molecules interacting with it will also increase and decrease respectively. This is the reason for the directly proportional relationship between the absorbance and length of the sample.

Conditions Required for Beer-Lambert’s Law

To avoid deviations, certain conditions are to be fulfilled for the Beer-Lambert law to be valid. These conditions are as follows.

• There should be no electromagnetic coupling present and the attenuators must act independently of each other.

• The sample to be tested must be homogenous. The solution should be homogenous where the light interacts with the solution for obvious reasons because any variation in the sample will affect the attenuation.

• The incident radiation should have parallel rays travelling the same length along with the sample solution.

• The incident radiation should not be polychromatic. It should be monochromatic radiation, preferably.

• The light should not influence the atoms or molecules as it will change the sample, and the attenuation may vary due to optical saturation.

• The wave properties of the light used should be negligible, so it does not vary the attenuation.

Beer-Lambert’s Law 

Beer-Lambert law relates to the linear relationship between the concentration and absorbance of adsorbed species. Moreover, it is the relationship between the properties of a particular substance and the attenuation of light by that particular substance. According to Beer-Lambert's law, the absorption of a certain amount of light by a substance dissolved in a completely permeable solvent is directly proportional to the concentration of the substance and the length of the optical path obtained through the solution. 

The derivation of Beer-Lambert’s law helps to define the ratio of the intensity of visible UV light to the right amount of substance present. The derivation of Beer-Lambert's law has many uses in modern science. Used in modern laboratories for drug testing, organic chemistry, and quantitative testing. 

Beer-Lambert  Formula 

I = Io x e^−μ (X) 

Where,  

I is strength 

Io is the initial intensity 

X is the depth in meters 

𝜇 is the absorption coefficient 

Beer-Lambert Derivation 

Beer-Lambert's law can be derived from an approximation of the absorption coefficient of a molecule by approximating the molecule through an opaque disc. This cross-sectional area represents the effective domain seen by photons of frequency w. 

Also, if the frequency of the light is far from the resonance, the area will be about 0. In addition, the area is maximized when w is close to resonance. 

Take an infinitesimal dz for a particular sample. Io is the intensity that goes into the sample at z = 0. Also, Iz is the strength with which z enters the infinitesimal. 

dI is the intensity of absorption in the plate and I is the intensity of light leaving and exiting the sample. According to this, the total opaque area of ​​the plate through the absorber is * N * A * dz. 

 The ratio of absorbed photons is 

 * N * A * \[\frac {dz} {A}\]  , so 

\[\frac {dI} {IZ}\]  = -* N * dz

 The integral of this equation is possible from z = 0 to z = b. Result: 

 ln (I) -ln (Io) = - * N * b 

 Also -ln (I / Io) is =  * N * b.  

Further N  \[\frac {molecule} { cm^3}\]   *  \[\frac {1 mol} {6.023 \times 1023 molecule}\]   * \[\frac {1000 cm^3} {litre}\] = c \[\frac {mol} {litre}\]  and  2.303 * log (x) = ln (x) and 

-log (I / Io) ) Is =  *  \[\frac {6.023 \times 1020} {2.303}\]   * c * b  and -log (I / Io) is = A =  * b * c

 Here 

 =  *   \[\frac {6.023 \times 1020} {2.303}\]   =  * 2.61 × 1020

Example Related to Beer-Lambert Law 

1. Find the relative amount of light that gets absorbed by the sample if the absorbance of the sample is 1 at a particular wavelength.

Ans: According to 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 = 1

1-I/I0 = 1 – 10-1 = 1 – 1/10 = 0.9

Therefore, we can say that 90% of the light is absorbed and 10% of light is transmitted. 

Application of Beer-Lambert Law 

• Analytical chemistry uses spectrophotometry to handle the separation, measurement, and identification of substances. No extensive pretreatment of the material is required to obtain results. For example, you can use a spectrophotometer to measure the number of bilirubin in a blood sample.  

• This law can be used to describe solar radiation in the atmosphere.

• It is used for qualitative and quantitative analysis of biological and dosimetric materials that may contain organic or inorganic elements. By analyzing the absorption spectra of various chemicals in the cell structure, their concentrations can be measured.