Chemkit

Beer-Lambert Law Calculator — Solve A = ε·b·c

The Beer-Lambert law relates how much light a solution absorbs to how much stuff is in it. In its working form: A = ε · b · c, where A is the (dimensionless) absorbance the spectrometer reports, ε is a wavelength- and compound-specific molar absorptivity (M⁻¹·cm⁻¹), b is the optical path length through the sample (cm), and c is the molar concentration (mol/L). The law lets you solve for any of the four variables given the other three — which is why every quantitative spectrophotometric assay, from enzyme kinetics to protein quantification to dye chemistry, reduces to a Beer-Lambert calculation.

Beer-Lambert Calculator

Enter any three of A, ε, b, c and pick which one to solve for. ε is in the SI-default M⁻¹·cm⁻¹; the calculator handles mm/m path lengths and mM/µM/nM concentrations internally.

Solve for

Equations used

Base relation and each rearrangement:

Internal units: ε in M⁻¹·cm⁻¹, b in cm, c in mol/L, A dimensionless. User inputs convert to those at the boundary; the answer converts back to the chosen display unit.

The equation

For a beam of light travelling distance b through a solution of concentration c, the fraction transmitted falls off exponentially with the amount of absorbing material in the beam. Taking the base-10 logarithm gives the absorbance:

A = log₁₀(I₀ / I) = ε · b · c

Standard unit convention: ε in M⁻¹·cm⁻¹, b in cm, c in mol/L. Those choices make A dimensionless, which is what every spectrophotometer displays. Path length is almost always 1 cm — that is the standard cuvette — but short-path (0.5, 1, 2 mm) cells are used for highly absorbing or precious samples.

Where it breaks

Beer-Lambert is a dilute-solution law. Real samples deviate from linearity for four separate reasons:

  • High absorbance (A > ~1-1.5). Detector stray light and shot noise dominate when only a few percent of the incident light reaches the detector. Standard analytical practice keeps A between 0.1 and 1.0 by diluting the sample or shortening b.
  • High concentration (> ~0.01 M for many dyes). Solute-solute interactions — dimers, aggregates, changes in the local dielectric — alter ε itself. The apparent extinction coefficient is no longer constant, and the plot of A vs. c curves downward.
  • Chemical equilibria that shift with concentration. Acid-base indicator forms, metal-complex speciation, and self-association of aromatic molecules all change ε as a function of c. The law still holds for each species individually, but the total absorbance is a weighted sum.
  • Scattering samples. Turbid suspensions and aggregated proteins scatter light out of the beam; the detector reads “missing” photons as absorbance, so the measured A overstates true absorption. Always centrifuge or filter before quantifying, and subtract a scattering blank if needed.

Worked example — NADH at 340 nm

Every NAD(H)-linked enzyme assay (lactate dehydrogenase, alcohol dehydrogenase, glucose-6-phosphate dehydrogenase, and dozens of others) depends on the same measurement: reduced NADH absorbs at 340 nm, oxidized NAD⁺ does not, and the rate of change of A₃₄₀ reports the enzyme rate. The canonical extinction coefficient is ε ≈ 6.22 × 10³ M⁻¹·cm⁻¹ (Horecker & Kornberg 1948).

Worked example — 100 µM NADH at 340 nm
ε (NADH, 340 nm)
6220 M⁻¹·cm⁻¹ (Horecker & Kornberg 1948)
b
1.00 cm (standard cuvette)
c
100 µM = 1.00 × 10⁻⁴ M
  1. Formula
  2. Substitute
  3. Note

    Reversed: measuring A₃₄₀ = 0.622 for an unknown NADH sample in a 1 cm cell gives c = A / (ε · b) = 0.622 / 6220 ≈ 1.00 × 10⁻⁴ M. This is the arithmetic behind every enzyme assay reported in µmol·min⁻¹·mg⁻¹ — the observed dA₃₄₀/dt divided by ε yields dc/dt, which is the reaction rate.

  4. Result

Common ε values (verify against your source)

A few well-cited molar absorptivities used across teaching labs, enzyme kinetics, and molecular biology. Always confirm against the primary literature or your reagent’s spec sheet before using — ε can drift 10-30 % between buffers, temperatures, and instrument calibrations.

Compoundλ (nm)ε or shorthandSource
NADH340ε ≈ 6220 M⁻¹·cm⁻¹Horecker & Kornberg, J. Biol. Chem. 175 (1948) 385
NADPH340ε ≈ 6220 M⁻¹·cm⁻¹ (same chromophore as NADH)Horecker & Kornberg 1948
BSA (protein)280A₂₈₀ ≈ 0.66 for 1 mg/mL, 1 cm cellThermo Fisher Pierce BSA spec
dsDNA2601 A₂₆₀ ≈ 50 µg/mLSambrook, Molecular Cloning 4e
ssRNA2601 A₂₆₀ ≈ 40 µg/mLSambrook, Molecular Cloning 4e

The NAD(H) coefficient in particular has been quoted anywhere from 6220 to 6300 in the literature over the decades; Horecker & Kornberg’s 6.22 × 10³ M⁻¹·cm⁻¹ remains the value most commonly used by enzymologists and by Sigma-Aldrich product data.

Common pitfalls

PitfallWhat to do
Reading A > 1.5 and trusting the numberDilute the sample (or switch to a shorter-path cell) so A falls in 0.1-1.0.
Using an ε value from the wrong buffer or temperatureConfirm against the paper you got ε from; solvent and temperature shift ε by 10-30 %.
Forgetting to subtract a blankSubtract a buffer-only reading before applying A = ε·b·c — scattering and cuvette absorption both bias the raw number.
Mixing mM⁻¹·cm⁻¹ with mol/LConvert ε to M⁻¹·cm⁻¹ before using this calculator (multiply mM⁻¹·cm⁻¹ values by 1000).
Reporting protein concentration from A₂₈₀ alone for an unknown proteinUse Bradford or BCA for unknowns; A₂₈₀ only works when you know ε₂₈₀ from the sequence (or a published shortcut like BSA’s 0.66/(mg/mL)).

Practice problems

Attempt each on paper, then expand the worked solution to check your arithmetic and sig-figs.

Problem 1 Easy A 1.00 cm cuvette contains 100 µM NADH measured at 340 nm (ε ≈ 6220 M⁻¹·cm⁻¹). What absorbance do you expect?

Answer: A ≈ 0.622

Solve for absorbance A
ε (NADH, 340 nm)
6220 M⁻¹·cm⁻¹
b
1.00 cm
c
100 µM = 1.00 × 10⁻⁴ M
  1. Formula
  2. Substitute
  3. Note

    The NADH extinction coefficient at 340 nm was measured by Horecker & Kornberg in 1948 and is still the reference value used in essentially every NAD(H)-linked enzyme assay.

  4. Result
Problem 2 Easy You measure A = 0.415 for an unknown NADH sample in the same 1.00 cm cell at 340 nm. What is c?

Answer: c ≈ 66.7 µM

Solve for concentration c
A
0.415
ε (NADH, 340 nm)
6220 M⁻¹·cm⁻¹
b
1.00 cm
  1. Formula
  2. Substitute
  3. Note

    Back-calculating c from a measured A with a known ε is the single most common student use of the law. Keep A in the 0.1-1.0 range for reliable readings.

  4. Result
Problem 3 Intermediate A purified protein reads A₂₈₀ = 0.660 in a 1.00 cm cuvette. If it has the ε₂₈₀ of BSA (43 824 M⁻¹·cm⁻¹, MW 66 463 g/mol), what is the concentration in mg/mL?

Answer: ≈ 1.00 mg/mL

Concentration from A₂₈₀, then convert to mg/mL
A₂₈₀
0.660
ε (BSA, 280 nm)
43 824 M⁻¹·cm⁻¹ (ExPASy ProtParam)
b
1.00 cm
MW (BSA)
66 463 g/mol
  1. Formula
  2. Substitute
  3. Note

    Multiply the molar concentration by MW to get mg/mL: 1.506 × 10⁻⁵ mol/L × 66 463 g/mol ≈ 1.00 g/L = 1.00 mg/mL. The A₂₈₀ ≈ 0.66 shortcut for 1 mg/mL BSA in the Thermo Pierce spec sheet is exactly this calculation.

  4. Result
Problem 4 Intermediate A dye has ε = 15 000 M⁻¹·cm⁻¹ at its λ_max. You want to measure a 0.5 mM stock, but the cuvette gives A = 7.5 — off-scale. What path length b should you use to bring A into the 0.1-1.0 range (target A ≈ 0.5)?

Answer: b ≈ 0.067 cm (a ~0.7 mm short-path cell)

Solve for path length b
A (target)
0.5
ε
15 000 M⁻¹·cm⁻¹
c
0.5 mM = 5.0 × 10⁻⁴ M
  1. Formula
  2. Substitute
  3. Note

    Instead of diluting a precious sample, drop into a short-path cuvette (0.5 or 1 mm) — a common trick for turbid or highly absorbing solutions.

  4. Result

Frequently asked questions

What is the Beer-Lambert law?
The Beer-Lambert law relates the absorbance A of a solution to its concentration: A = ε·b·c, where ε is the molar absorptivity (a wavelength- and compound-specific constant, in M⁻¹·cm⁻¹), b is the optical path length the light travels through the sample (in cm; standard cuvettes are 1 cm), and c is the molar concentration (in mol/L). Absorbance itself is dimensionless because ε·b·c is dimensionless with those unit choices.
When does the Beer-Lambert law break down?
Linearity holds well for A ≲ 1 in dilute, homogeneous, non-scattering solutions. Common failure modes: (1) high concentration (>~0.01 M for many dyes), where solute-solute interactions change ε; (2) stray light and detector non-linearity at high A (typically A > 1.5-2); (3) chemical equilibria that shift with concentration — indicator forms, metal-complex speciation, protein aggregation — where ε itself becomes a function of c; and (4) turbid or scattering samples, where light lost to scatter is counted as absorption.
What units should ε be in?
The SI-default molar absorptivity is in M⁻¹·cm⁻¹ (equivalent to L·mol⁻¹·cm⁻¹). This is the convention used in Harris QCA and by the IUPAC Gold Book, and it makes A dimensionless when b is in cm and c is in mol/L. You may see ε reported in mM⁻¹·cm⁻¹ for very absorbing compounds (cytochromes, dyes) — multiply by 1000 to convert to M⁻¹·cm⁻¹.
How do I use A₂₈₀ to estimate protein concentration?
For a purified protein whose ε₂₈₀ you know from its amino-acid composition (calculators like ExPASy ProtParam use the Pace 1995 tryptophan/tyrosine sum), c = A₂₈₀ / (ε₂₈₀ · b). For bovine serum albumin (BSA), a handy shortcut is that A₂₈₀ ≈ 0.66 corresponds to 1 mg/mL in a 1 cm cuvette (Thermo Fisher Pierce spec sheet). For unknown proteins the Bradford or BCA assay is more reliable than raw A₂₈₀.
Why is a reading of A > 1 unreliable?
A = 1 means 90 % of the incident light is absorbed; A = 2 means 99 %. As A climbs above ~1, the detector measures the tiny remaining transmitted intensity against instrument stray light and shot noise, and the linear relationship between A and c breaks down. Standard analytical practice is to dilute the sample (or shorten the path) so the reading falls in the 0.1-1.0 range where the measurement is most precise.

Sources

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