Logarithmic Relation Between Transmittance and Concenctration

 

 

Why is there a logarithmic relation between transmittance and concentration?

Beer’s law (), states that the absorbance of a sample is directly proportional to the concentration of the absorbing species. The fraction of light passing through the sample (the transmittance) is related logarithmic, not linearity, to the sample concentration. Why should be this?

Imagine light of radiant power P passing through an infinitesimally thin layer of solution whose thickness is dX. The decrease power, dP, is proportional to the incident power P, to the concentration of absorbing species c, and to the thickness of section dX:

Where b is constant of proportionality. The Equation above can be arranged and integrated quite simple:

The limits of integration are P=P₀ at X=0 and P=P at X=b. Evaluating the integrals gives us:

Finally, converting ln to log, using relation Ln z = (ln 10)(log z), gives

                              A Constant

OR

Which is Beer’s law!

 

 

The logarithmic relation of P₀/P and concentration arises from the fact that in each infinitesimal portion of total volume, the decrease in power is proportional to the power incident upon that section. As light travels through the sample, the drop in power in each is succeeding layer decrease, because the magnitude of incident power that reaches each layer is decreasing.