Beer–Lambert Law Calculator
Work out the absorbance of a solution from the Beer–Lambert law, A = ε·l·c — using the molar absorptivity, the cuvette path length and the concentration — and read off transmittance and percentage transmittance.
Absorbance (A)
0.5000
- Transmittance (T)
- 0.3162
- Transmittance
- 31.62 %
- Light absorbed
- 68.38 %
Beer–Lambert law: A = ε·l·c. Transmittance follows from A = −log₁₀(T), so T = 10^(−A).
How to use this calculator
Enter three values. First, the molar absorptivity ε (also called the molar attenuation or molar extinction coefficient) in units of L·mol⁻¹·cm⁻¹, sometimes written M⁻¹·cm⁻¹ — this is a property of the substance at the specific wavelength you are measuring, so look it up for your analyte and wavelength. Second, the path length l in centimetres: this is how far the light travels through the sample, and for a standard spectrophotometer cuvette it is 1 cm. Third, the concentration c of the absorbing species in moles per litre (mol/L, also written M). The calculator returns the absorbance A (a dimensionless number), together with the transmittance T (the fraction of light that passes through, between 0 and 1), the percentage transmittance %T, and the percentage of light absorbed. To go the other way — the usual laboratory task of finding a concentration from a measured absorbance — rearrange the same law to c = A / (ε·l): read A off your instrument, divide by ε and the path length. Keep concentration low enough that the law stays linear (typically A below about 1).
How the calculation works
The Beer–Lambert law (often just Beer’s law) states that the absorbance of a solution is directly proportional to the concentration of the absorbing species and to the distance the light travels through it: A = ε·l·c. Here ε is the molar absorptivity, a constant for a given substance at a given wavelength that measures how strongly it absorbs; l is the path length; and c is the molar concentration. Absorbance itself is defined from the amount of light that gets through. If I₀ is the intensity of light entering the sample and I is the intensity leaving it, the transmittance is T = I / I₀ and the absorbance is A = −log₁₀(T). Rearranging gives T = 10^(−A), which is how this calculator turns absorbance into transmittance. An absorbance of 0 means all the light passes through (T = 100 %); an absorbance of 1 means only 10 % gets through; an absorbance of 2 means only 1 %. Because A is logarithmic in transmittance but linear in concentration, doubling the concentration doubles the absorbance but does not simply halve the transmittance. This linearity in c is exactly what makes spectrophotometry quantitative: measure the absorbance of a sample of unknown concentration, and — with ε and the path length known — solve c = A / (ε·l). The law assumes dilute solutions, monochromatic light and no chemical or optical effects that change ε; at high absorbances (roughly A > 1–2) stray light and analyte interactions cause deviations from linearity.
Worked example
A dye has a molar absorptivity of ε = 5000 M⁻¹·cm⁻¹ at its peak wavelength. It is measured in a standard 1 cm cuvette at a concentration of 1×10⁻⁴ mol/L (0.0001 M). Applying the law, A = ε·l·c = 5000 × 1 × 0.0001 = 0.5. The transmittance is T = 10^(−0.5) = 0.3162, so 31.62 % of the light passes through and 68.38 % is absorbed. If the concentration were doubled to 2×10⁻⁴ M, the absorbance would double to A = 1.0 and the transmittance would fall to 10 % — the classic result that an absorbance of 1 corresponds to one-tenth of the light being transmitted. Working backwards instead: if you measured an absorbance of 0.5 for this dye in the 1 cm cuvette, the concentration would be c = A / (ε·l) = 0.5 / (5000 × 1) = 1×10⁻⁴ M, recovering the original value.
Frequently asked questions
What is the Beer–Lambert law?
The Beer–Lambert law (also called Beer’s law) states that the absorbance of a solution is proportional to the concentration of the absorbing species and to the path length the light travels through it: A = ε·l·c. In that equation A is the absorbance (dimensionless), ε is the molar absorptivity in L·mol⁻¹·cm⁻¹, l is the path length in cm, and c is the concentration in mol/L. It underpins quantitative spectrophotometry, letting you determine an unknown concentration from a measured absorbance.
How do I find concentration from absorbance?
Rearrange the law to c = A / (ε·l). Measure the absorbance A on the spectrophotometer at the analyte’s peak wavelength, then divide by the molar absorptivity ε and the path length l. For a 1 cm cuvette that is simply c = A / ε. In practice, laboratories often build a calibration curve from standards of known concentration and read the unknown off the best-fit line, which also corrects for small deviations from ideal behaviour.
What is the difference between absorbance and transmittance?
Transmittance T is the fraction of light that passes through the sample, T = I/I₀, and ranges from 0 to 1 (or 0–100 % as %T). Absorbance A is a logarithmic measure of how much light is absorbed, defined as A = −log₁₀(T). So T = 10^(−A): an absorbance of 0 means 100 % transmittance, A = 1 means 10 %, and A = 2 means 1 %. Absorbance is the more useful quantity for analysis because it is linear in concentration, whereas transmittance is not.
What are the units of molar absorptivity (ε)?
Molar absorptivity ε is usually given in L·mol⁻¹·cm⁻¹, which is the same as M⁻¹·cm⁻¹, because it must cancel the units of concentration (mol/L) and path length (cm) to leave absorbance dimensionless. It is a property of the substance at a specific wavelength: strongly absorbing dyes and conjugated molecules can have ε of 10⁴–10⁵, while weakly absorbing species may be only 10–100. Always use the ε value that corresponds to the wavelength you are measuring at.
Why does the Beer–Lambert law break down at high concentrations?
The law assumes dilute, non-interacting molecules and truly monochromatic light. At high concentrations (typically where absorbance exceeds about 1–2) several effects cause the absorbance-versus-concentration line to curve: molecules interact and shift ε, the refractive index changes, and stray light and the finite bandwidth of the instrument matter more when very little light gets through. For accurate work, dilute the sample so its absorbance falls in the linear range, roughly 0.1–1.
Does path length affect absorbance?
Yes — absorbance is directly proportional to path length, so doubling the path length doubles the absorbance for the same solution. This is why spectrophotometers use standardised cuvettes, almost always 1 cm, so that measurements are comparable and ε values (which are quoted per cm) apply directly. Longer path-length cells (for example 5 cm or 10 cm) are used to boost the signal from very dilute samples, and shorter ones for very concentrated samples.