**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.

-dI/db∝ I

Where

Ø I

is the intensity of radiation

Ø b is pathlength (length of the medium

through which light travelled)

Ø 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 (ɛ): 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

• ε=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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