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Therefore if this value of C may be substituted in the above equation as:

Thus Lambert’s Law was defined as: The intensity of parallel monochromatic radiation decreases exponentially as it passes through a medium of homogenous thickness. Simply put, the absorbance is proportional to the thickness ( path length b) of the solution.

Beer Derivation

In 1852, in a similar derivation to Lambert’s, Beer demonstrated the following relationship between absorbance and concentration of a solution:

where, kⁿ is a proportionality constant, c is the concentration and A is the absorbance.

Thus, Beer’s law can be defined as: the intensity of a beam of parallel monochromatic radiation decreases exponentially with the number of absorbing molecules. More simply, it stated that absorbance is proportional to the concentration (c) of the solution.

A combination of the above two laws i.e. the Lambert law and Beer’s law yields the modern derivation of the Beer-Lambert law.

Modern Derivation of the Beer-Lambert Law

The modern derivation of the Beer-Lambert law combines the two derivations in order to relate absorbance of a substance to both the concentration and the path length of the sample.

Note: A lot of websites have not explained this derivation well in terms of what is actually happening. I will attempt to simplify it by explaining each step carefully and give definitions which are important to know rather than jumping from one equation to another. I would recommend reading this at least two times in order to understand what is going on. Please feel free to comment below in case you have any doubts on this.

Consider a beam of parallel monochromatic light with intensity I₀ striking the sample at the surface as shown in the figure above, such that it is perpendicular to the surface. After passing through the path length b of the sample, which contains N molecules/cm³, the intensity of the light reduces to I_T.

Now, consider a cross-section of the block having an area S and an infinitesimal thickness dz placed at a distance z from the surface.

If the sample has N molecules/cm³, the number of molecules present in the infinitesimal block would be
= N × S × dz

If each molecule has a cross-sectional area σ, where photons of light get absorbed, then the fraction of the total area where light gets absorbed due to each molecule would be (fractional area for each molecule)
= σ / S.

The total fractional area for all molecules in the block where light gets absorbed would therefore be
= (number of molecules) × (fractional area for each molecule)
i.e.= (N × S × dz) x (σ / S)  = σ × N × dz

If I_z is the light entering the infinitesimal block considered, and the light absorbed due to the absorbing particles is dI, the light exiting the slab would be I_z – dI.
The fraction of light absorbed would therefore be = dI/I_z

Now, since the fractional area is the probability of light striking a molecule,
the fraction of light absorbed in the block = fractional area occupied by all the molecules in the slab.
i.e. dI / I_z =  – σ × N × dz
Note: The negative sign is placed to indicate absorption of light.

If we integrate this infinitesimal block for the whole sample from z=0 to z=b, where b is the path length of the entire solution we get

ln I_T – ln I₀ = – σ × N × b
OR
– ln ( I_T / I₀) = σ × N × b

One can correlate the number of molecules/cm³ of sample to the concentration of the sample in moles/liter using the relationship:
c = N × 1000/ (6.023 × 10²³)
OR
N = (6.023 × 10 ²³) × c / 1000 = 6.023 × 10²⁰ x c

where c is the concentration in moles/litre, N is the number of molecules/cm³, and (6.023 x 10²³) is the Avagadro’s number. N × 1000 would be therefore molecules / liter as 1 liter = 1000 cm³.

Substituting this term for N in the above equation we get:
– ln (I_T / I₀ ) =  σ × 6.023 × 10²⁰ × c × b

To simplify this equation further we convert to log using the term 2.303 × log (x) = ln (x).
therefore we get,
– log (I_T / I₀) =  (σ × 6.023 × 10²⁰ × c × b) / 2.303
OR
log (I₀ / I_T) = [ ( σ × 6.023 × 10²⁰)/2.303 ] × c × b

The term log (I₀ / I_T) is nothing but absorbance (A). While the term in the brackets [ ] is a constant and can be replaced by a constant ε = ( σ × 6.023 × 10²⁰)/2.303. Thus we get:
A =  ε × c × b
OR
A = εbc

Since we have used moles/liter as the units of concentration, ε stands for molar absorptivity (or molar extinction coefficient). If the units of concentration were g/liter then the ε would be replaced by “a” (absorptivity).

By definition, the molar absorptivity (ε) at a specified wavelength of a substance in solution is the absorbance at that wavelength of a 1 mole/liter solution of the substance in a cell having a path length of 1 cm. The units of molar absorptivity are liter mole^-1 cm^-1

Specific absorbance is the absorbance of a substance in solution of specific concentration in a cell of specific path length. The most commonly used term for specific absorbance is A^1%_1cm, which is the absorbance of a 1 g/100ml (1%) solution in a 1cm cell at a particular wavelength of light. The beer-lambert equation in such case becomes:
A = A^1%_1cmbc

where, c is the concentration measured in g/dl and b is the path length measured in 1cm.
Note: Generally the path length of most experiments performed is 1cm.

Books on Analytical Chemistry and Spectroscopy

Check out these good books for analytical chemistry and spectroscopy

References

1. Analytical Chemistry: An Introduction (Saunders Golden Sunburst Series) 7th Ed., by Douglas A. Skoog, Donald M. West, F. James Holler. 1999.
2. Practical Pharmaceutical Chemistry 4th Ed. Edited by A. H. Beckett and J. B. Stenlake. 1988.
3. Fundamentals of Analytical Chemistry 8th Ed., by Douglas A. Skoog, Donald M. West, F. James Holler, Stanley R. Crouch. 2003.