You have encountered the above expression while studying Beer – Lambert’s law ,right? And did you wonder what this means? How do you arrive at this equation? Isn’t A = εbc , the expression for this law? Then why do we need the above expression? Let us try to answer all of the above questions in this post by studying the above mentioned equation!
This is another expression for Beer-Lambert’s law in terms of intensity of incident light( Ɪo) and intensity of transmitted light ( Ɪt).
Why is this expression useful?
While conducting experiments in the laboratory, we often study the intensities of light. We decide the incident intensity ( Ɪo) and then measure the transmitted ( Ɪt) intensity using a detector. α in the above expression is a constant. Thus, we can measure the concentration(C) of the solution under study.
How do we derive this equation?
We know from Beer’s law that – the intensity of light (Ɪ) is directly proportional to concentration C.
Using differentiation approach one more time , we can write,
-dꞮ ⇒ decrease in intensity of incident light (negative sign indicates decrease in intensity)
Ɪ ⇒ intensity of incident light.
dc ⇒ decrease in concentration.
The above expression tells us that the decrease in the intensity of light is directly proportional to the decrease in concentration.
Introducing constant of proportionality (α) in the above expression, we get –
Integrating the above equation ,
Solving for the integrals –
Thus, we get an equation –
Case 1 – Now initially , when C=0 , Ɪ = Ɪ0
Equation (1) will be ,
-ln Ɪ0 = α(0) + b
∴ -ln Ɪ0 = b
Substituting the value of ‘b’ in equation (1) we get ,
In post 158 , we have already mentioned that ,
Logarithm means opposite of an exponential.
i.e log b x = y means by = x
Thus, the above expression can be written as –
And this is how we arrived at our equation !!
In the next post we will start our discussion on spectrophotometer. Till then,
Be a perpetual student of life and keep learning….
Good day !