Beer-Lambert Law Calculator
Solve the Beer-Lambert law for absorbance, transmittance, concentration, path length or absorptivity.
All five quantities solve in reverse, with ten volume units and seven path-length units.
Updated October 19, 2026
Introduction
A spectrophotometer does not measure concentration. It measures how much light survived a trip through your sample, and the Beer-Lambert law is what turns that number into a concentration. This page holds the five quantities of that law together, and solves whichever one you leave empty.
The trick to reading the law is that the sample is described by three quantities and the light by two. The sample has a molar absorption coefficient — how strongly one mole per litre of the substance absorbs at the wavelength you chose — a concentration and a path length , which is simply how far the beam travels inside the liquid. The light itself is described by the absorbance and the transmittance , which are two ways of saying the same thing.
That makes the page useful anywhere an optical reading has to become a number you can report: quantifying a dye or a pigment, checking a DNA or RNA sample by its ultraviolet reading, running a colorimetric assay for protein, phosphate or nitrate, following a reaction kinetically, or marking a teaching lab where the point is to see the law work. The two menus that carry the unit of the coefficient are worth a second look the first time you use the page — that is where a factor of a thousand hides most easily.
Quick start
Type the absorption coefficient in the units your reference used
Textbooks, data sheets and instrument software quote this constant differently. The page gives you a molar menu and a length menu, and the row reads as one reciprocal unit — means , per mole per litre per centimetre. Both menus are converted before they are used, so a coefficient of 8400 per molar per centimetre can be entered as or as and gives the same answer. What it cannot do is guess: the value has to belong to the substance and the wavelength you actually measured, in the solvent you actually used.
Fill in what you already know and leave the unknown empty
Enter the concentration if you prepared the solution yourself, and the path length of the cell you used — 1.00 cm for a standard cuvette. Then leave the box you are looking for alone: the page solves whichever quantity is left blank, in either direction. If your reading came first, start from the absorbance or the transmittance instead and add the coefficient and the cell afterwards.
Work through a dye measurement
A solution of a dye has an absorption coefficient of at its peak wavelength, and you measure it in a 1.00 cm cell. The prepared concentration is , which the page accepts as 43.3 µmol/L. The law then gives:
and the transmittance follows from the same measurement, reading 43.2793% on the page.
Sanity-check the reading against the decade scale
Absorbance and transmittance are logarithmic, so it helps to know a few landmarks by heart. These pairs are exactly what the page returns when you enter the transmittance:
| Transmittance | Absorbance | What it means |
|---|---|---|
| 100% | 0 | Nothing is absorbed — a blank or an empty reference cell. |
| 50% | 0.30103 | Half the light gets through. |
| 10% | 1 | The comfortable middle of the scale, and the easiest decade to remember. |
| 1% | 2 | One part in a hundred; the top of the range most instruments still handle well. |
| 0.1% | 3 | One part in a thousand — treat anything at or beyond this as instrument-limited rather than a measurement. |
Decide whether the number is worth reporting
A finite answer is not automatically a good measurement. The page flags two results as implausible rather than hiding them: an absorbance above 5, which almost no instrument measures linearly, and an absorption coefficient above 1,000,000, which no chromophore reaches. A red number means the arithmetic is fine but the input deserves a second look — most often the sample needs diluting, and the dilution has to be multiplied back in.
Three conversions that do the most work
- — and the same statement again in micromoles per cubic millimetre. The concentration menus are all one equality wearing different units.
- — a coefficient quoted per millimetre is numerically smaller than the same coefficient per centimetre.
- Absorbance adds while transmittance multiplies, which is why concentrations, not transmittances, are what you average.
How the five quantities are connected
One equation carries the sample and one carries the light, and they meet at the absorbance. That shared box is why the page can solve in either direction, and why editing a derived value is a perfectly reasonable thing to do.
The law itself
With the coefficient in , the path length in centimetres and the concentration in , the units cancel and the absorbance comes out as a plain number, which is what an instrument reports. Whenever an instrument reading is the thing you trust, the page inverts the same statement:
Solving for the coefficient, worked through
A sample reads in a 1.00 cm cell, and the concentration is known independently to be . Leaving the coefficient empty returns:
The same measurement also reports , and the coefficient is exactly if your reference prefers the millimolar scale.
From a reading percentage to an absorbance
These two are the whole of the conversion. An absorbance of 0 means nothing was absorbed and every photon arrived; an absorbance of 1 means one photon in ten arrived. Because the relation is logarithmic, absorbance is the quantity that is proportional to concentration — transmittance is not, which is why a graph of transmittance against concentration curves while a graph of absorbance against concentration is (within the useful range) a straight line.
What the two unit menus are doing
The coefficient is a quantity per molar per unit of length, so its unit is a reciprocal: , sometimes written . Both menus are converted to that scale before the multiplication happens:
In practice you only have to remember one direction: millimolar is 1000 times smaller than molar, so a coefficient expressed per millimolar is 1000 times smaller — a factor that turns a sensible answer into one that is three orders of magnitude out if it is entered in the wrong menu. The concentration behaves the same way, with the volume menu adding its own factor on top.
Getting a reading you can trust
The arithmetic is the easy part. Nearly every disappointing result comes from the measurement rather than the equation, and four habits cover most of it.
Blank on the solvent, not on water
Zero the instrument with the pure solvent — or with the complete reagent blank, if the assay adds colour reagents — in the same cell, at the same wavelength, at the same temperature as the sample. The reading you then get is the absorbance of your analyte alone. A blank taken on a different liquid carries that liquid's absorption into every concentration you calculate.
Make the cell match the path length you typed
A 1.00 cm cuvette is the default convention, and the coefficient in your reference is almost certainly quoted for it. Short-path and flow-through cells exist to keep strong absorbers on scale, but then the path length in the box has to change with them. Two cells that are not matched to each other also add a small offset that looks exactly like a concentration.
Stay in the linear part of the scale
Below about 0.1 the reading is mostly detector noise, and above roughly 2 stray light and detector non-linearity start to bend the response, which is where a plot of absorbance against concentration visibly flattens. The page warns you once an absorbance passes 5. Between those two ends the law is at its best, and diluting a strong sample is almost always the quickest fix.
Dilute by a measured factor and multiply it back
If a sample is too concentrated, dilute a measured volume to a measured final volume, read the diluted solution, and multiply the concentration the page reports by the dilution factor. Keep the solvent and the matrix the same, because the coefficient assumes the same environment. For the volumetric step itself, the solution dilution calculator works out the volumes from .
When a series of standards is the better route
If you have several standards, a calibration line built from them absorbs the errors that a single literature coefficient cannot: the exact wavelength setting, the cell you are using, the reagent batch, and the day. That slope is an effective coefficient for your conditions, and it is usually more trustworthy than a handbook number for the same substance. The calibration curve calculator fits the line and reads an unknown concentration from it.
If all you have is a mass concentration — milligrams per millilitre from a preparation sheet — convert it to molarity first; the molar mass calculator gives the grams per mole that the conversion needs.
Limitations and assumptions
- •One absorbing substance at one wavelength. If two species absorb where you are measuring, their absorbances add and a single coefficient cannot tell them apart. The model also has no wavelength input: it uses the coefficient you supply, so a coefficient measured at the wrong wavelength silently biases everything.
- •Absorption only, no scattering. A cloudy, turbid or particulate sample scatters light as well as absorbing it, and the instrument cannot distinguish the two. The apparent absorbance is then too high, and the calculated concentration with it.
- •Beer's law is a dilute-solution limit. At high concentration the molecules interact, the substance may associate or change form with pH, and the response flattens. The law is not a law of nature; it is an excellent approximation over the range where the substance obeys it.
- •Ranges and gentle warnings are not verdicts. The page refuses a transmittance outside 0–100%, keeps an implausible absorbance or coefficient on screen in red, and leaves a result blank when the mathematics is genuinely undefined — for instance dividing by a zero coefficient, or asking for the absorbance of a sample that transmits no light at all. A blank is information, not a bug.
- •Educational and planning use. This is a calculation aid, not a validated analytical method. For results that will be reported, regulated or used to release a batch, follow your laboratory's documented procedure, its own standards and its quality control.
Frequently asked questions
Why did a box go blank instead of showing a number?
Because the arithmetic has no answer for that combination of inputs, and the page would rather say so than print something plausible. Two cases come up most often. The first is a zero coefficient while you are solving for concentration:
The second is a transmittance of exactly 0%. That is a legal measurement — an opaque sample — but its absorbance grows without limit, so there is no finite number to display. Both cases carry their own explanation next to the affected box, and other fields on the page keep working while you fix the input.
Can I use this to analyse a mixture of two substances?
Not directly. One coefficient describes one absorbing species, and a mixture's total absorbance is simply the sum of the contributions:
Two unknowns need two equations, so a mixture is measured at two wavelengths where the two substances have different coefficients, and the pair of readings is solved together — the simultaneous-equations step that instrument software or a spreadsheet handles. What this page can do is supply the individual terms: enter each substance's coefficient, path length and concentration separately to see what each contributes to the total, then compare the sum with the reading you actually got.
Which wavelength should the coefficient have been measured at?
The same one you measured at. The coefficient is not a property of the molecule in general; it is a property of that molecule at one wavelength in one solvent, and it changes steeply near an absorption band. Two numbers for the same dye can differ several-fold between its peak and a wavelength a few nanometres off it.
In practice, use the wavelength of maximum absorption for sensitivity, and use a coefficient taken at exactly that setting. Measuring on the shoulder of a band is sometimes done on purpose to stay within the linear range of a strong absorber, but only with a coefficient measured on that shoulder too. If your coefficient and your reading come from different sources, that mismatch — not the arithmetic — is the most likely reason the concentration looks wrong.
Related Calculators
Cell Dilution Calculator
Solve any one of four dilution variables (C1, V1, C2, V2) with LRU bidirectional editing and volume unit support.
Chemical Equation Balancer
Balance a chemical reaction and get the smallest whole-number coefficients. Supports bracketed groups such as Fe(OH)3, user-supplied coefficients, and unicode subscripts.
Net Ionic Equation Calculator
Reduce a molecular equation to its net ionic form. Balances the reaction, splits soluble compounds into ions, and names the spectator ions that cancel out.
Cubic Cell Calculator
Convert an atomic radius into the edge of a cubic unit cell, or measure a cell edge back to the atomic radius. Simple, body centered and face centered packing are each solved in both directions.
Atom Calculator
Calculate atomic properties including protons, neutrons, electrons, atomic number and mass number
Electron Configuration Calculator
Get the electron configuration for any element from the periodic table. View atomic number, atomic mass, and valence electrons.