Beer's law and Lambert's law describe the absorption of light in materials. Beer's law states that absorbance is directly proportional to concentration, while Lambert's law states absorbance is directly proportional to path length. Beer and Lambert combined their laws into the Beer-Lambert law, which states absorbance is equal to the molar absorptivity (a constant for a given substance and wavelength) multiplied by the path length and concentration. The Beer-Lambert law is commonly used for quantitative analysis but has limitations at very high concentrations due to interactions between molecules.

1. BEER-LAMBERT LOW
Topics:
• Beer’s low
• Lambert’s low
• Beer-Lambert low
• Derivation of beer lambert low
• Limitations
Asif Pappu
Dept. of Applied Chemistry & Chemical Engineering
Noakhali Science & Technology University
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2. BEER’S LOW
Absorbance(A) Vs. Concentration (c)
(For a dilute solution)
If a light is pass through a solution then the absorbance of light is proportional to
the concentration of the solution. Which is
Iin
Iin
Iout [2]
Iout [1]
cA 
cA  2

3. LAMBERT’S LOW
Absorbance(A) Vs. Path length (b)
The path distance that travels in the sample is proportional to the absorbance.
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Iin
Iin
Iout [2]
Iout [1]
bA
Path length- the distance that light
travels in the sample

4. BEER-LAMBERT LOW
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cA  bA 
A abc=
bcA
Here a is molar absorptivity
Units of molar absorptivity:
))()(( cofunitbofunitaofunitAofunit 
))(( cofunitbofunit
Aofunit
aofunit 
1111
))((
1
))((

 cmMMcm
McmMcm
unitless
aofunit
“A” is unitless but “a” has units M-1cm-1
Beer’s low Lambert’s low

5. DERIVATION OF BEER-LAMBERT’S LOW
Here dx is the thickness of medium and I is the intensity of light then from
lamberts low-
-dI/dx ∞ I ----------------(1) or,
-dI/dx = aI---------------(2) 5

6. 6
Where - 𝑑𝐼/𝑑𝑥 is the rate of decrease of intensity with thickness dx , a is called the
absorption co-efficient. Integration of equation (2) after rearrangement given
- ln I = ax+C --- --- --- --- --- --- (3)
Where C is a constant of integration.
At x=0, I=Io. So, C = - ln Io.
Introducing this in equation (3) we get,
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)
Log I/Io = -a’x-------------------(7)
Where a` (= a /2.303 ) is called extinction co-efficient and -ln I/ Io is termed absorbance of
the medium. Absorbance is represented by A.

7. 7
Lambert’s law was extended by beer who showed that when light passes through a solution of a
given thickness the fraction of incident light absorbed is dependent not only on the intensity I of
light but also on the concentration c of the solution. This is known as the Beer’s law.
- 𝑑𝐼/ 𝑑𝑥 ∝ 𝑐 --- --- --- --- --- --- (8)
The two laws may be combined to write
- 𝑑𝐼/ 𝑑𝑥 ∝ 𝐼 × 𝑐 Or, - 𝑑𝐼/ 𝑑𝑥 = 𝑏 × 𝐼 × 𝑐 --- --- --- --- --- (9)
When the concentration, c, is expressed in mol /L, b is called the molar absorption co-efficient.
As in the case of Lambert’s law equation (9) may be transformed into,
log I/ Io = − 𝑏/ 2.303 × 𝑐 × 𝑥 --- --- --- --- --- (10)
log I/ Io = - ∈× 𝑐 × 𝑥 --- --- --- --- --- (11)
Where ∈ (= 𝑏/ 2.303 ) is called the molar extinction co-efficient which is expressed in L/mol/cm.
The molar extinction co-efficient ∈ is dependent on the nature of the absorbing solute as well as
on the wave length of the incident light used. The expression (equation 11) is commonly known
as Beer-Lambert’s law

8. LIMITATIONS
• Deviations in absorptivity coefficients at high concentrations (>0.01M) due to
electrostatic interactions between molecules in close proximity.
• Scattering of light due to particulates in the sample.
• Fluoresecence or phosphorescence of the sample.
• Changes in refractive index at high analyte concentration.
• Shifts in chemical equilibria as a function of concentration.
• Non-monochromatic radiation, deviations can be minimized by using a relatively flat
part of the absorption spectrum such as the maximum of an absorption band.
• Stray light.
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REFERENCE:
PRINCIPLES OF PHYSICAL CHEMISTRY.
-DR. MUHAMMAD MAHBUBUL HUQUE &
-DR. MOHAMMAD YOUSUF ALI MOLLAH.