Beer-Lambert Law Calculator - Calculate Absorbance

What is the Beer-Lambert law?

The Beer-Lambert law is a formula that describes how much light a solution absorbs. It links absorbance to three things: how concentrated the solution is, how strongly the dissolved substance absorbs light, and how far the light travels through the sample. It is also called Beer's law. A Beer-Lambert law calculator applies this formula to find absorbance, transmittance, and the percentage of light absorbed from a few simple inputs.

The law is a core tool in spectroscopy, the study of how matter absorbs and releases light. It is used daily in chemistry, biology, and medical labs to measure the concentration of substances in a solution.

Beer-Lambert law formula

The Beer-Lambert law states:

$A = \varepsilon \times c \times l$

• A is absorbance, the amount of light a solution absorbs at a given wavelength. It has no units.
• ε (epsilon) is the molar absorptivity, also called the molar extinction coefficient. It measures how strongly a substance absorbs light at a specific wavelength. Units: L/(mol·cm).
• c is the concentration of the substance, in mol/L (molarity).
• l is the path length, the distance light travels through the sample, in cm.

Absorbance is related to transmittance, the fraction of light that passes through the sample:

$A = -\log_{10}(T)$

$T = 10^{-A}$

Here T is transmittance, a number between 0 and 1. Multiplying T by 100 gives the percentage of light transmitted. The percentage of light absorbed is:

$\text{Percent absorbed} = (1 - T) \times 100\%$

For example, an absorbance of 1.0 means T = 0.1, so 90% of the light was absorbed. An absorbance of 2.0 means T = 0.01, so 99% was absorbed.

How to calculate absorbance

To find absorbance with the calculator, enter three values:

1. Path length (l) — the width of the sample container, usually a cuvette. Most standard cuvettes are 1 cm wide, though some specialized ones are 0.1 cm or 10 cm.
2. Molar absorptivity (ε) — a property of the specific substance at a specific wavelength. It is usually found in published reference tables or calculated from a molecule's structure. Proteins, for instance, absorb light around 280 nm because of two amino acids, tryptophan and tyrosine, and the exact value depends on how many of these a protein contains.
3. Concentration (c) — the amount of substance dissolved, in mol/L. Concentrations given in other units, such as mg/mL, must first be converted using the molecule's molar mass.

The calculator multiplies the three numbers together to get absorbance, then derives transmittance and percent absorbed from that result.

Worked example

A researcher purifies a protein and needs to check its concentration before running an experiment. The protein's molar absorptivity at 280 nm is known from its sequence: ε = 44,000 L/(mol·cm). The researcher measures the sample in a standard 1 cm cuvette and reads an absorbance of A = 0.523.

Rearranging the formula to solve for concentration:

$c = \frac{A}{\varepsilon \times l} = \frac{0.523}{44{,}000 \times 1} = 1.19 \times 10^{-5} \text{ mol/L}$

That is 11.9 micromolar (μM). This falls in the range instruments read most accurately, so the researcher can trust the number without diluting the sample further.

Interpreting absorbance values

• A = 0: no light absorbed at this wavelength.
• A = 0.3 to 0.9: the range where most spectrophotometers give the most accurate, linear readings.
• A = 1.0: 90% of light absorbed, 10% transmitted.
• A = 2.0: 99% absorbed, 1% transmitted.
• A above 2.5: outside the reliable range of most instruments. The sample should be diluted and remeasured.

Readings below about 0.3 are affected by instrument noise. Readings above about 1.5 are affected by small amounts of stray light inside the instrument, which introduces error. Diluting a sample and remeasuring is the standard fix when a reading falls outside the accurate range.

Where the Beer-Lambert law is used

Analytical chemistry. Labs measure the absorbance of a sample and use the formula to back-calculate an unknown concentration. This is common for testing nitrate, phosphate, and metal ion levels in water, and for checking the concentration of a chemical batch during manufacturing.

Biochemistry. Absorbance at 280 nm estimates protein concentration. Absorbance at 260 nm estimates DNA or RNA concentration; by convention, an absorbance of 1.0 corresponds to about 50 μg/mL of double-stranded DNA or 40 μg/mL of RNA. The ratio of the two readings (A260/A280) gives a rough measure of sample purity.

Pharmaceuticals. Drug makers use absorbance readings to test how quickly a tablet releases its active ingredient, and to track how a drug degrades over time in storage.

Medicine. Many automated blood-test machines measure absorbance from color-changing chemical reactions to report glucose, liver enzyme, and bilirubin levels.

Food and drink. Beer color is measured by absorbance at 430 nm under the European Brewery Convention (EBC) method. Similar methods check wine and soft drink color.

Limitations of the Beer-Lambert law

The law assumes the light is a single, precise wavelength and that the sample does not scatter light. It breaks down in a few common situations:

• High concentration. Above roughly 0.01 mol/L, dissolved molecules start to interact with each other, and absorbance stops rising in a straight line with concentration.
• Cloudy or particle-filled samples. Suspensions and turbid liquids scatter light in addition to absorbing it, which inflates the absorbance reading. The Beer-Lambert law does not account for scattering; a related model called Kubelka-Munk theory is used for opaque or scattering samples such as powders.
• Chemical changes during measurement. Some substances break down under light or shift with pH while being measured, which changes the reading partway through.
• Temperature changes. Molar absorptivity depends on temperature, so readings taken at inconsistent temperatures are not directly comparable.

History

The law developed over more than a century through the work of three scientists. Pierre Bouguer described in 1729 how each equal layer of a material absorbs an equal fraction of the light passing through it. Johann Heinrich Lambert restated this relationship mathematically in his 1760 book Photometria, showing that absorbance rises in direct proportion to path length. August Beer extended the law in 1852 to show that absorbance also rises in direct proportion to concentration. The combined law is sometimes called the Bouguer-Beer-Lambert law in recognition of all three.

Frequently asked questions

What is the Beer-Lambert law used for? It calculates how much light a solution absorbs, and it works in reverse to find an unknown concentration from a measured absorbance. This makes it the basis of UV-Vis spectrophotometry, a common lab technique for measuring the amount of a substance in a solution.

What units does the Beer-Lambert law use? Path length in centimeters, molar absorptivity in L/(mol·cm), and concentration in mol/L. Absorbance itself has no units, though it is sometimes labeled "AU" for absorbance units. Mixing other units into the formula without converting them first gives a wrong answer.

Why isn't my absorbance reading proportional to concentration? The most common causes are a concentration above about 0.01 mol/L, a cloudy or particle-filled sample, or a reading far outside the 0.3–0.9 range where instruments are most accurate. Diluting the sample and remeasuring usually confirms which cause applies.

What is the difference between absorbance and transmittance? Transmittance is the fraction of light that passes through a sample. Absorbance is the negative base-10 logarithm of transmittance: A = −log₁₀(T). As absorbance rises, transmittance falls.

What is the difference between absorbance and optical density? They are the same quantity. "Optical density" is the term more often used in biology, while "absorbance" is more common in chemistry.

What absorbance range is most accurate? Roughly 0.3 to 0.9. Below that range, instrument noise dominates the signal. Above about 1.5, stray light inside the instrument introduces error, and above about 2.5 most instruments cannot give a reliable reading at all.

References

1. Beer, A. (1852). "Bestimmung der Absorption des rothen Lichts in farbigen Flüssigkeiten." Annalen der Physik und Chemie, 86: 78–88.
2. Swinehart, D. F. (1962). "The Beer-Lambert Law." Journal of Chemical Education, 39(7): 333–335.
3. Mayerhöfer, T. G., Pahlow, S., & Popp, J. (2020). "The Bouguer-Beer-Lambert Law: Shining Light on the Obscure." ChemPhysChem, 21(18): 2029–2046.