According to Beer–Lambert law, the absorbance is proportional to the number of molecules that absorb radiation at a specified wavelength:

A _λ = log ( I _o / I _t )= ε _λ c l (1)

where A _λ = absorbance, I _o and I _t are the incident and transmitted intensities, ε _λ = absorptivity (formerly called the extinction coefficient ) at wavelength λ , and l is the sample path length. The absorptivity is proportional to the transition probability of the absorbing molecules, whichdepends on photon energy, i.e., on the wavelength. In addition to the wavelength of light, the absorptivity depends on the identity of the absorbing substance as well as of the solvent. If the concentration is measured in mol.L ⁻¹ , the absorptivity is called the molar absorptivity .

The absorption of light by two or more compounds at the same wavelength is additive. That is, the absorbance at any wavelength of a mixture is equal to the sum of the absorbance values of each component in the mixture at that wavelength. In other words, Beer-Lambert law is valid simultaneously for all chemically non interacting absorbers in a solution. Thus total absorbance a particular wavelength, λ is given as:

A _λ (total) = (A ₁ ) _λ + (A ₂ ) _λ + (A ₃ ) _λ + …… = (ε ₁ ) _λ c ₁ l+(ε ₂ ) _λ c ₂ l+(ε ,sub>3 ) _λ c ₃ l+ . . . (2)

Where A i is the absorbance and ε _I is the molar absorptivity of the i -th species at a given wavelength, λ , and c _i is the concentration of i -th species in the mixture. For a two component mixture (say, the species present are X & Y) when each component absorbs appreciably at λ _max of one another and for a unit sample path length ( l = 1), one can write:

A _λ (total) = (A _X ) _λ + (A _Y ) _λ = (ε _X ) _λ .c _X + (ε _Y ) _λ .c ,sub>Y (3)

This can easily be verified in following two ways by taking two UV-visible light absorbing solutions of known concentrations.

Method I : One can first measure the absorbance valuesof individual components and then mix them in a given concentration ratio and measure the absorbance values for the mixture. Suppose,

concentration of pure X = C _X0 and corresponding absorbance = A _X0 at wavelength, λ ; and concentration of pure Y = C _Y0 and corresponding absorbance = A _Y0 at the same wavelength, λ .

When two solutions are mixed, original concentrations of pure substances change due to dilution effect. Say, in the mixture the concentration of X = C _X and the concentration of Y = C _Y and the measured absorbance value of the mixture= A mix at the wavelength, λ.

Then, the sum of absorbance values of individual components in the mixture,

A _X+Y = (A _X0 /C _X0 ).C _X + (A _Y0 /C _Y0 ).C _Y (4)

If the principle of additivity holds good, then

A _mix = A _X+Y . (5)

Relationship given in eq (5) should be valid for any given wavelength.

Method II: This method requires knowledge of absorptivity (ε) values of individual components at different wavelengths. If ε values are known, it is evident from eq (3) that one can solve for two unknowns c _X and c _Y by constructing two equations. Two equations are constructed by measuring absorbance values for standard and sample solutions at two different wavelengths as given below:

A ^mix λ1 = (ε _X ) _λ1 .c _X + (ε _Y ) _λ1 .c _Y (6)

A ^mix λ2 = (ε _X ) _λ2 .c _X + (ε _{Y ) λ2 .c Y (7)}

where A ^mix λ1 and A ^mix λ2 are measured absorbance values of the mixture (sample) at two different wavelengths λ ₁ and λ ,sub>2 , respectively. One can solve the above simultaneous equations for theconcentrations of species X and Y when the molar absorptivities of both species at both wavelengths are known. ε values for two components can be determined from absorbance measurements done on a set of different concentration solutions (standards) of individual components followed by the plot of absorbance (at a given wavelength)vs. concentration values (Beer-Lambert law plot). ε values are calculated from the slopes of the absorbance-concentration plots. The absorbance values of the mixture at each of the two wavelengths are obtained from measurements. Inserting the ε and absorbance values in above equations 6 and 7, one can solve for c _X and c ,sub>Y . In order to verify the validity of the principle of additivity of absorbance values for multicomponent systems, these concentration values determined from absorbance measurements should be compared with the concentration values calculated from the actual concentrations taken to prepare the mixture. (While comparing one should take into consideration the measurement errors in absorbance values, in addition to the calibration errors in the values of ε _i , which result in errors in the concentrations determined from the above two equations.)

In this experiment, a mixture of coumarin 343 and coumarin 6 is analyzed. Since the compounds are similar, one might expect a considerable overlap of their absorption spectra. One should note that the wavelength at which two components have the same extinction coefficient is called the isosbestic point . One or more isosbestic points may be found for a pair of components.