Beer-Lambert Law (Photometry)

Multiply the molar extinction coefficient ε, the concentration c and the path length d: $A = \varepsilon{}\cdot{}c\cdot{}d$. The absorbance A is dimensionless. Insert ε in $L\cdot{}mol^{-1}\cdot{}cm^{-1}$, c in $\frac{mol}{L}$ and d in cm so the units cancel. Example: a protein with $\varepsilon{} = 40 000 L\cdot{}mol^{-1}\cdot{}cm^{-1}$ in a 1 cm cuvette at $c =\frac{ 10^{-5} mol}{L}$ gives $A = 40 000\cdot{}10^{-5}\cdot{}1 = 0.40$. From the absorbance the transmission follows via $T = 10^{(-A)}$, here T = 10⁻⁰·⁴ ≈ 40 percent. Make sure the unit of ε and the path length match, usually both in centimetres.

Rearrange the law for the concentration: $c =\frac{ A}{(\varepsilon{}\cdot{}d)}$. You measure the absorbance A of the sample in the photometer, know the molar extinction coefficient ε of the substance and the path length d of the cuvette. From these three quantities the concentration follows. In practice one often records a calibration curve by measuring the absorbance of several known concentrations; its slope is $\varepsilon{}\cdot{}d$. From the absorbance of the unknown sample the concentration is then read off via the line. Watch for consistent units and stay in the linear range. This method is a standard procedure of quantitative analysis, for example to determine the concentration of dyes or proteins.

At high absorbance only very little light reaches the detector, because almost everything is absorbed. At $A = 2$ only 1 percent of the light gets through, at $A = 3$ only 0.1 percent. This small residual signal is dominated by stray light, noise and instrument imprecision, so small measurement errors cause large relative errors in the concentration. In addition the linear relationship loses its validity at high concentrations, because particles influence each other and the law holds only for dilute solutions. Therefore one measures most accurately in the absorbance range between about 0.1 and 1. If the sample is too concentrated, you dilute it instead of measuring an unreliable high absorbance.

The molar extinction coefficient ε is a substance-specific quantity and states how strongly a substance absorbs light of a certain wavelength. It has the unit $L\cdot{}mol^{-1}\cdot{}cm^{-1}$ and describes the absorbance per concentration and path length. A large value means that even small concentrations absorb strongly, which allows sensitive measurements. ε depends on the substance, the wavelength of the light and partly on the solvent; therefore one usually measures at the absorption maximum, where ε is largest. Dyes and proteins with extended electron systems often have very high extinction coefficients. ε is the proportionality factor that links concentration and measured absorbance in the Lambert-Beer law.

The law holds only under several conditions: dilute solutions, monochromatic light and no scattering or chemical interactions. At high concentrations the absorbance deviates from linearity, because the particles influence each other and the extinction coefficient is no longer constant. If the light is not single-coloured, different wavelengths with different ε mix and distort the result. Turbid or scattering samples, chemical reactions, association or dissociation of the dissolved particles also lead to deviations. A common mistake is to apply the law at too high concentrations. For clear, dilute solutions in a photometer, by contrast, it gives very reliable, linear relationships.