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Beer-Lambert's Law states that “Equal fractions of incident light are absorbed by successive layers of equal thickness and equal concentration of absorbing medium.” We know that Lambert's Law is $$ 2.303\log { \frac { { I }_{ 0 } }{ { I }_{ t } } } = K'l $$ And Beer's Law is $$ 2.303\log { \frac { { I }_{ 0 } }{ { I }_{ t } } } = K''c $$ On combining both, we get Beer-Lambert's Law which is, $$ 2.303\log { \frac { { I }_{ 0 } }{ { I }_{ t } } } = K'l K''c $$ $$ 2.303\log { \frac { { I }_{ 0 } }{ { I }_{ t } } } = Kcl $$ $$ \log { \frac { { I }_{ 0 } }{ { I }_{ t } } } = \frac { K }{ 2.303 } cl $$ $$ \log { \frac { { I }_{ 0 } }{ { I }_{ t } } } = \varepsilon cl $$ where \(\epsilon\) is constant and \(\epsilon = \frac { K }{ 2.303 }\) is known as molar extinction coefficient Or molar absorptivity and it is defined as “the reciprocal of the thickness which produces transmitted ray \(\frac { 1 }{ 10 }\) of its incident rays if the concentration of the absorbing species in the solution is 1mole.”
The unit is \({ dm }^{ 3 } { mole }^{ -1 } { m }^{ -1 }\).
But Optical density is defined as logarithm of ratio of \({ I }_{ 0 }\) to \({ I }_{ t }\). Therefore, $$ \boxed { \log { \frac { { I }_{ 0 } }{ { I }_{ t } } } = Optical \ density \ (O.D.) = \ Absorbance \ (A) } $$ OD has no unit and its value lies between 0 to 1. Also Transmittance T is defined as the fraction of the light transmitted. $$ T = \frac { { I }_{ t } }{ { I }_{ 0 } } $$ The value of T is always small to convert in to appreciable it is always expressed in percentage.
Thus Beer-Lambert's Law becomes, $$ \boxed { O.D = A = \log { \frac { { I }_{ 0 } }{ { I }_{ t } } } = -\log { T } = \varepsilon cl } $$
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