**Ultraviolet-Visible
(UV-Vis) Spectroscopy – Derivation of Beer-Lambert Law**

__Objectives__

**At the end of the
session students will be able to**

• State
Lambert and Beer’s laws

• Derive
the fundamental equation of quantitative spectroscopy

• Explain
the terms absorbance and transmittance

• Distinguish
between absorption coefficient, specific absorption coefficient and molar
absorption coefficient

__Lambert’s
Law__

• Lambert’s
law states that when monochromatic light passes through a transparent medium of
uniform thickness, the rate of decrease in the intensity of light is directly
proportional to the intensity of light

• The
candela (abbreviation, cd) is the standard unit of luminous intensity in the
International System of Units (SI).

• It
is formally defined as the magnitude of an electromagnetic field, in a
specified direction, that has a power level of 1/683 watt (1.46 x 10 ^{-3}
W) per steradian at a frequency of 540 terahertz (540 THz or 5.40 x 10 ^{14}
Hz).

• .
A frequency of 540 THz corresponds to a wavelength of about 556 nanometers
(nm), which is in the middle of the visible-light spectrum.

• A
steradian is the standard unit solid angle in three dimensions; a sphere
encloses 4 pi (approximately 12.57) steradians.

Where

Ã˜ I
is the intensity of radiation

Ã˜ b is pathlength (length of the medium
through which light travelled)

Ã˜ dI and db are the differences in intensity and pathlength respectively

__Beer’s Law__

• August
Beer

• Beer’s law states that when monochromatic light is passed through a solution of uniform concentration, the rate of decrease in intensity of light is proportional to the intensity of light

__Beer-
lambert’s Law__

• Combining
both the equations,

• A= log〖I0/It=abc〗 Where a is specific absorption coefficient, the value
of which is dependent on the way concentration is expressed and on the unit of
path length.

• This
is the fundamental equation of spectroscopy

__Terms used
in spectrophotometry__

• Transmittance
is the ratio of intensity of transmitted light to that of incident light

• Absorbance:
logarithmic ratio of intensity of incident light to that of transmitted light

• Synonyms:
Optical density, Extinction

• Absorption
coefficient: Absorbance per unit path length

• Classical
definition: the reciprocal of path
length in cm that is required to reduce the intensity of incident light to 1/10^{th}
of its value

• Synonyms:
Absorptivity, extinction coefficient

• Specific
absorption coefficient : Absorbance per unit concentration per unit path length

• a=A/bc

• A1%_{1cm} : A specific absorption coefficient when concentration is expressed as % solution and
path length in cm

• Molar absorption Coefficient (É›): Specific absorption coefficient when concentration is expressed as moles per liter solution and path length in cm

__Molar absorption coefficient and Molecular weight__

• Molecular
weight in g = 1 Mole

• Assume
Concentration of the solution is c % solution

• c
g is c/M.wt moles

• Concentration of the solution is 10c/M.wt
moles/litre

• Îµ=A/bc ( M.wt)/10

__Summary__

• Lambert’s
law explains the effect of path length
on the intensity of light

• Beer’s
law explains the effect of concentration of the solution on the intensity of
light

• Both
laws assume monochromatic light

• Absorbance
is logarithmic ratio of intensity of incident light to that of transmitted
light

• Transmittance
is the ratio of intensity of transmitted light to that of incident light

• The
combined law gives the fundamental equation of quantitative spectroscopy

• The
fundamental equation of spectroscopy is

A= log〖I0/It=abc〗 Where a is specific absorption coefficient, the value of which is dependent on the way concentration is expressed and on the unit of path length.

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