Convert absorbance, transmittance and percent transmittance; solve any variable of A = εbc (absorbance, molar absorptivity, path length or concentration) and work the typical exam problem: a sample with two, three or more species measured at several wavelengths, where the concentrations are found from the molar absorptivities of each species.
A = −log₁₀ T = ε b c. In a mixture of species that do not react with each other the absorbances add up: A(λ) = b Σ εᵢ(λ) cᵢ, one linear equation per wavelength.
When light passes through a colored solution, part of it is absorbed, and the more substance there is, the more light is absorbed. The Beer–Lambert law turns that observation into a linear relationship that lets you measure concentrations with a spectrophotometer. This guide explains absorbance and transmittance, the law, its limits and how a mixture of two species is analyzed.
If I0 is the intensity of the light entering the sample and I the intensity leaving it, the transmittance is T = I/I0 (or %T = 100·T) and the absorbance is
A = −log10 T
For example, %T = 50 corresponds to A ≈ 0.301, and A = 1 means only 10 % of the light gets through. Absorbance has no units.
A = ε · b · c
b is the path length (the width of the cuvette, usually 1 cm), c the concentration and ε the molar absorptivity (in L·mol⁻¹·cm⁻¹), specific to each substance at each wavelength. Since A is proportional to c, measuring A gives c, provided ε is known or a calibration curve has been prepared with standards.
A sample gives A = 0.45 in a 1 cm cuvette, at a wavelength where the species has ε = 2000 L·mol⁻¹·cm⁻¹ (given in the exercise).
The absorbances of species that do not react with each other add up. With two species X and Y, measurements at two wavelengths are needed, giving a system of two equations. With b = 1 cm and these absorptivities (given in the exercise):
1000·cX + 200·cY = 0.50 100·cX + 800·cY = 0.42
Solving the first for cX and substituting in the second: 780·cY = 0.37, so cY ≈ 4.74 × 10⁻⁴ M and cX ≈ 4.05 × 10⁻⁴ M. The calculator solves this system for any number of species; if there are more wavelengths than species, it uses least squares. Both examples give the same results in it.
That is why real analysis prefers a calibration curve with standards of known concentration, measured on the same instrument, to using an ε from the literature.
D. C. Harris, Quantitative Chemical Analysis (W. H. Freeman); D. A. Skoog, F. J. Holler and S. R. Crouch, Principles of Instrumental Analysis (Cengage). Simuveliq is not affiliated with these authors or publishers.
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