The Beer-Lambert law describes how the light energy of a source passes through a transparent substance. It establishes a relationship between the intensity of the incoming light, that of the outgoing light, and the concentration of the substance the light passes through.
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How to measure it in class
FizziQ’s luxmeter measures the illuminance received by the smartphone’s light sensor: by placing it behind solutions of increasing concentrations, you can build a calibration curve.
Steps:
- Prepare a colored stock solution (copper sulfate, potassium permanganate or diluted mint syrup), then a series of five to six dilutions of known concentrations.
- Place the smartphone flat, light sensor facing up, and put an empty cuvette or stemmed glass on top of it. Illuminate from above with a fixed lamp and record the reference illuminance I₀ with FizziQ’s luxmeter.
- Without moving anything, fill the cuvette with the least concentrated solution and record the transmitted illuminance I. Empty, rinse, and repeat for each concentration, keeping strictly the same liquid height.
- For each trial, calculate the transmittance T = I/I₀ then the absorbance A = −log T, and plot A as a function of the concentration c.
- Verify that the points line up on a straight line through the origin at low concentrations, and identify from which concentration the points drop below the line.
- Use the calibration line to determine the concentration of an unknown solution prepared by the teacher.
Scientific activities on this topic
- Beer-Lambert law with a luxmeter - verify the proportionality between absorbance and concentration from illuminance measurements, then determine the concentration of an unknown solution (upper secondary and beyond).
- Colorimetry of autumn leaves
Learn more
The Beer-Lambert law owes its name to two distinct contributions. In 1760, Johann Heinrich Lambert stated that the attenuation of light is proportional to the thickness of matter traversed. In 1852, August Beer established that it is also proportional to the concentration of the absorbing species. The current law combines the two results. It is essential in many spectroscopy applications, particularly in analytical chemistry, biochemistry, and other fields where quantifying the concentration of absorbing compounds is necessary.
The Beer-Lambert law is generally expressed in the form of the following equation:
A = ε.c.l
where:
A represents the absorbance of the substance, which is a measure of the amount of light absorbed by the solution.
ε (epsilon) is the molar absorption coefficient, also called the molar extinction coefficient, which depends on the specific substance, the wavelength of the light and the temperature. It is characteristic of the substance.
c is the concentration of the substance in solution, generally expressed in moles per liter (mol/L).
l is the path length of the light through the solution, generally expressed in centimeters (cm).
The law is used in many fields, notably spectrophotometry, which is a technique used to measure the concentration of chemical substances using light. It is used to analyze blood, urine and tissue samples to detect diseases, infections and metabolic disorders. It is also used to analyze materials for optical properties, and for industrial applications such as the manufacture of glass products and pigments.
Conditions of validity
The law is not universal; it is a limiting law. It assumes a dilute solution, monochromatic light at a fixed wavelength, an absorbing species that does not react and does not associate, and a medium that does not scatter light. A turbid solution, a suspension or an emulsion put the law at fault, not because the absorption changes but because the light is deflected away from the detector axis, which the instrument wrongly counts as absorption.
Absorbance and transmittance
The two quantities must not be confused. The transmittance T is the fraction of light transmitted, between 0 and 1, and it multiplies when absorbing media are stacked. The absorbance A = −log T is dimensionless, it equals 0 for a perfectly transparent medium, and above all it adds. It is this additivity that makes A so useful: the absorbance of a mixture of two dyes that do not interact is the sum of their absorbances, and absorbance is directly proportional to concentration, whereas transmittance is not.
The choice of wavelength
The coefficient ε depends strongly on the wavelength. One therefore works at the absorption maximum of the species studied, identified on its spectrum: this is where the sensitivity is best and where the uncertainty on λ matters least, since the curve is flat there. A blue solution is analyzed in the orange-red, around 620 to 650 nm, that is, in its complementary color: what it absorbs, not what it reflects.
The deviation from linearity at high concentrations
This is the phenomenon students actually encounter in lab work: beyond a certain concentration, the points of the curve A = f(c) bend below the straight line. Several causes accumulate. The molecules become close enough to interact with each other and modify ε. The refractive index of the solution varies. Finally, the source is never perfectly monochromatic: the least absorbed wavelengths end up dominating the transmitted signal. In practice, one stays in the range where A is below about 1, that is, at least 10% of light transmitted.
Orders of magnitude
Molar absorption coefficients range from a few tens to a few hundred thousand L·mol⁻¹·cm⁻¹: about 10 for the hydrated Cu²⁺ ion around 800 nm, about 2,400 for the permanganate ion MnO₄⁻ at 525 nm, about 10⁵ for chlorophyll a around 430 nm. A spectrophotometer cuvette has a standardized width of 1.00 cm. An absorbance of 1 corresponds to 10% of light transmitted, an absorbance of 2 to 1%. The usual concentrations in lab work range from 10⁻⁵ to 10⁻³ mol/L.
Formula
The Beer-Lambert law relates absorbance to concentration, at a given wavelength:
A = ε × l × c
where:
- A: absorbance at wavelength λ (dimensionless)
- ε: molar absorption coefficient at this wavelength (L·mol⁻¹·cm⁻¹)
- l: optical path length through the solution (cm)
- c: concentration of the absorbing species (mol/L)
The absorbance is deduced from the measured light intensities:
A = −log(I / I₀) = −log T
where:
- I₀: incident light intensity, measured with the solvent alone (lux or W/m²)
- I: light intensity transmitted by the solution (lux or W/m²)
- T = I/I₀: transmittance, between 0 and 1 (dimensionless)
For a mixture of absorbing species that do not interact, the absorbances add:
A_total = A₁ + A₂ + … = (ε₁c₁ + ε₂c₂ + …) × l
Application examples
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A potassium permanganate solution at 1.0 × 10⁻⁴ mol/L, in a 1.00 cm cuvette at 525 nm where ε ≈ 2,400 L·mol⁻¹·cm⁻¹, has an absorbance A ≈ 0.24, or about 58% of light transmitted
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An absorbance A = 1.0 corresponds to T = 0.10: the solution lets through only 10% of the incident light
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The spectrophotometric determination of diiodine in a Betadine solution, part of the final-year French curriculum, relies on a calibration line A = f(c)
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A pulse oximeter compares the absorption of blood at two wavelengths, in the red and near infrared, to determine the oxygen saturation of hemoglobin
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Water quality monitoring stations measure nitrates by absorption in the ultraviolet, around 220 nm
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Doubling the thickness of solution traversed doubles the absorbance, but divides the transmittance by a factor of 10^A
FAQ
Q: Why are my points no longer aligned when the solution is very concentrated? A: The Beer-Lambert law is only valid for dilute solutions. At high concentrations, the molecules interact with each other, the refractive index changes and the source is not perfectly monochromatic: the curve bends below the straight line. The sample must be diluted to stay within the validity range, typically A below 1.
Q: Why do we use absorbance rather than transmittance? A: Because absorbance is proportional to concentration, which gives a usable straight line, and because it is additive. Transmittance, on the other hand, decreases exponentially with concentration and multiplies instead of adding.
Q: How do we choose the working wavelength? A: It is chosen at the absorption maximum of the species studied, read on its absorption spectrum. The sensitivity is maximal there and a small deviation in λ changes the result little. In practice, a solution is analyzed in its complementary color.
Q: Can the law be applied to a turbid solution or to milk? A: No. The law assumes that light is neither scattered nor reflected by suspended particles. In a turbid medium, light is deflected away from the detector and the instrument strongly overestimates the absorbance. The sample must be filtered or centrifuged beforehand.
Q: Does the smartphone luxmeter replace a spectrophotometer? A: No, it gives a simplified version of it. The sensor receives all visible light and not a single wavelength, and its spectral response is its own. Relative measurements, made under strictly identical conditions, allow you to verify the proportionality, but do not give a usable coefficient ε.
Related concepts
Absorbance - Colorimeter - Luxmeter - Illuminance