Beer Lambert’s Law-Deviations and IR Vibrations

According to this law, when a beam of monochromatic radiation is passed through a solution of absorbing species, the intensity of a beam of monochromatic light decreases exponentially with an increase in the concentration of absorbing species.

-dI/dc α I

Lambert’s law states that the rate of decrease of intensity of monochromatic light with the thickness of the medium is directly proportional to the intensity of incident light.

-dI/dt α I

Derivation of Beer and Lambert’s Law

According to beer’s law,

-dI/dc α I

The decrease in the intensity of light (I) with concentration(c) is proportional to the intensity of incident light(I)

-dI/dc = K.I { removing & introducing the dc constant of proportionality “K”}
-dI/I = K.dc { rearranging terms}

On Integrating the equation,

-∫dI /I= K.∫dc
-ln I = K.c + b  ——- equation 1 { b constant of integration}

When concentration =0, there is no absorbance,
Hence I=I₀ Substituting in equation 1

-ln I₀ = K*0+b
-ln I₀ = b Substituting the value of b in equation 1
-ln I = K.c-ln I₀
ln I₀ – ln I = Kc {since log A-log B = log A /B}
ln I₀/I = Kc
I₀ /I = ekc { removing natural log }
I/ I₀ = e-kc {making inverse on both sides}
I = I₀.e-kc ——— equation 2                        [equation for beer’s law]

According to lambert’s law,

dI/dt α I

This eqn can be simplified by replacing ‘c’ with ‘t’ in
I = I₀.e-kt  ——— equation 3

Eq . 2 & Eq . 3 can be combined to get
I = I₀. e-kct

Converting natural log to base 10 & K= k x 0.4343

I /I₀ = ₁₀-kct { rearranging terms }
I₀ /I= ₁₀kct { inverse on both sides}

Taking log on both sides,
log I₀/I = Kct ——— equation 4

Here, Transmittance T = I/I₀ , Absorbance A = log 1/T
A = log 1/ T
A = log 1/I/I₀ { Since T = I/I₀ }
A =log I₀/ I ——— equation 5

Using eqn 4 & 5,
since A =log I₀/I and log I₀/I = Kct ,
A= Kct
Instead of K, we can use ε
A = εct                           { Mathematical eqn for beer lambert’s law}

For any particular wavelength absorbance can be calculated by
A = εbc

A= Absorbance
ε = Molar absorptivity (L/(mol cm))
b = Path length (cm)
c = Concetration (Mol/L)

Deviation from Beer-Lambert’s Law

A system is said to obey Beer’s law, when a plot a graph of concentration and absorbance gives a straight line. When a non-linear curve is obtained then the system is said to undergo deviation from beer law.

There are two types of deviation

• Positive deviation: It results in when a small change in concentration produces a greater change in absorbance.
• Negative deviation: It results in when a large change in concentration produces a smaller change in absorbance.

These deviations from the Beer law can be classified into three categories:

REAL DEVIATIONS: – These are fundamental deviations due to the limitations of the law itself.
CHEMICAL DEVIATIONS– These are deviations observed due to specific chemical species of the sample which is being analyzed.
INSTRUMENT DEVIATIONS – These are deviations which occur due to how the absorbance measurements are made.

Beer’s law describes the absorption behaviour of dilute solutions only so it is a limiting law. At high concentration (exceeding 0.01M) solute molecules can cause different charge distribution on their neighbouring species in the solution. Since at high concentration it results in a shift in the absorption wavelength of the analyte. Some large ions or molecules show deviations even at very low concentrations.

For e.g., methylene blue absorptivity at 436 nm fails to observe beer’s law even at concentrations as low as 0.01M. High analyte concentrations alter the refractive index (η) of the solution, it affects the absorbance obtained.

Chemical deviations occur due to some chemical phenomenon. Association, dissociation and interaction with the solvent to give a different product.
Example, phenol red undergoes a resonance transformation when moving from the acidic form (yellow) to the basic form (red). Due to this resonance, the electron distribution of the bonds of molecule changes with the pH of the solvent in which it is dissolved.

Acid and Base forms of phenol red along with their UV spectra at different pH demonstrate chemical deviations of Beer law in UV- Visible spectroscopy.

a. Polychromatic radiation: Beer t law is strictly followed when a monochromatic source of radiation exists. It is common to use a polychromatic source of radiation with a continuous distribution of wavelengths along with a monochromator is used to create a monochromatic beam from this source.

b. Stray radiation:
Scattered radiation is the radiation from the instrument that is outside the nominal wavelength band selected. This radiation is due to reflection and scattering by the surfaces of lenses, mirrors, gratings, filters and windows. The wavelength of the stray radiation is different from the wavelength band selected.

The radiation exiting from a monochromator is often contaminated with minute quantities of scattered or stray radiation. If the analyte absorbs at the wavelength of the stray radiation, a deviation from Beer law.

c. Mismatched cell:
If the cells holding the analyte and the blank solutions are having different path-lengths, or unequal optical characteristics it leads to deviation from Beer’s law.

d. Due to improper slit width
If the width of the slit is not proper, deviations occur due to undesirable radiation to fall on the detector. These undesirable radiations are absorbed by the impurities present in the solution of the sample. It leads to a change in the absorbance of the sample.

IR Vibrations

Principle of IR Spectroscopy

  • When the energy in the form of IR is applied and if the applied IR frequency = Natural frequency of vibration, the absorption of IR takes place and a peak is observed.
  • Molecules are excited to the higher energy state from the ground state when they absorb IR radiation.
  • When a compound is exposed to IR radiation, it selectively absorbs the radiations resulting in vibration of the molecules of the compound, giving rise to closely packed absorption bands, called as IR absorption spectrum.
  • The bands correspond to the characteristic functional groups and the bonds present in a chemical substance. Thus, an IR spectrum of a compound is considered as the fingerprint for its chemical identification.

IR spectra is considered as vibrational-rotational spectra. All the bonds in a molecule are not capable of absorbing red energy but only those bonds which are accompanied by a change in dipole moment will absorb in the IR region.

It means,

  1. When the frequency of the IR radiation is equal to the natural frequency of vibration, the molecule absorbs IR radiation. Absorption of IR radiation causes excitation of the molecule from a lower to the higher vibrational level. Each vibrational level is associated with a number of closely placed rotational level. Therefore the IR spectroscopy is also called as “vibrational-rotational spectroscopy”.
  2. All the bonds in a molecule are not capable of absorbing IR energy but those bonds which are accompanied by a change in dipole moment will absorb in the IR region and such transitions are called IR active transitions. The transitions which are not accompanied by a change in the dipole moment of the molecule are not directly observed and are considered as IR inactive

Modes of Vibrations in Poly Atomic Molecules

Two main types of absorption bands occur in IR spectra:
i.  Fundamental: Vibrations which appear as a band in the spectra.

The different types of fundamental vibrations are:

A. Stretching vibrations: Stretching vibration Involves a continuous change in the inter atomic distance along the axis of the bond b/w two atoms. It requires more energy so appear at shorter wavelength.

(i) Symmetrical:

Beer Lambert's Law-Deviations and IR Vibrations 1

Inter atomic distance b/w tow atoms increases/decreases in the same direction. i.e both atoms move towards and move outwards from the central atom in the same direction.

(ii) Asymmetrical

Beer Lambert's Law-Deviations and IR Vibrations 3

Inter atomic distance b/w two atoms is alternate/opposite. i.e. both atoms move towards and move outwards from the central atom in the opposite direction.

B. Bending vibrations: Bending vibrations are characterized by a change in the angle b/w two bonds. It requires less energy so appear at a longer wavelength.

(i) In-plane bending

 If all the atoms are on the same plane

(a) Scissoring

Beer Lambert's Law-Deviations and IR Vibrations 5

When the two atoms move away or close towards each other results the scissoring type of molecular vibration. This is a type of symmetric bending vibration in a plane.

(b) Rocking

Beer Lambert's Law-Deviations and IR Vibrations 7When a change in angle b/w a group of atoms, that results in the rocking type of vibration. This is a type of asymmetric type of bending vibration in a plane.

(ii) Out-plane bending

 If two atoms are on the same plane while the 1 atom is on the opposite plane

(a) Wagging

Beer Lambert's Law-Deviations and IR Vibrations 9

Change in angle b/w the plane of a group of the atoms. Two atoms move to one side of the plane. They move up and down the plane.

(b) Twisting

Beer Lambert's Law-Deviations and IR Vibrations 11

Change in angle b/w the plane of two groups of atoms. One atom moves above the plane and another atom.

Credit Reference Link for above images:

ii.Non-Fundamental: Vibrations which appears as a result of fundamental vibrations. They are followed as below.

Combination bands are observed when more than two or more fundamental vibrations are excited simultaneously. These combination modes arise from the anharmonicities of the oscillators which leads to an interaction of the vibrational states in polyatomic molecules.

(multiples of a given frequency) results from excitation from the ground state to higher energy states. Overtones occur when a vibrational mode is excited from v=0 to v=2, which is called the first overtone, or v=0 to v=3, the second overtone.

  • 0 –> 1 (fundamental)
  • 0 –> 2 (first overtone)
  • 0 –> 3 (second overtone)
  • 0 –> 4 (third overtone)
  • 0 –> 5 (fourth overtone)


Fermi Resonance:
Interactions which occur between fundamental and overtone or combinational bands are known as Fermi resonance. This phenomenon can be observed whenever two fundamental or fundamental and overtone bands have nearly the same energy.  Here molecule transfers its energy from fundamental to overtone and back again and so that each level becomes partially fundamental or partially overtone in character. As a result, two strong bands are observed in the spectrum, instead of the expected strong and weak bands.

E.g.: CO2

◦ It normally shows the fundamental band at 1337 cm-1 and overtone at 1334.6 cm-1 .
◦ But due to the effect of Fermi resonance the first band shift towards higher frequency and give rise to two bands at 1285.5 cm-1 and 1388.3 cm-1

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