What is the Beer-Lambert Law formula?

The Beer-Lambert Law is A = epsilon x l x c, where A is absorbance (dimensionless), epsilon is molar absorptivity in L mol-1 cm-1, l is path length in cm, and c is molar concentration in mol/L. Absorbance is directly proportional to both concentration and path length.

How do you calculate concentration from absorbance?

Rearrange Beer-Lambert law: c = A divided by (epsilon x l). For example A=0.45, epsilon=15000, l=1 cm gives c=0.45/15000=0.00003 mol/L (30 micromolar). This is the most common use of Beers law in analytical chemistry.

How do you convert absorbance to transmittance?

Transmittance T = 10 to the power of -A. Percent transmittance %T = 100 x 10^(-A). Absorbance of 1.0 = 10% transmittance. Absorbance of 0.301 = 50% transmittance. Reverse: A = -log10(T).

What is molar absorptivity?

Molar absorptivity (epsilon) is the intrinsic ability of a chemical species to absorb light at a given wavelength. Units: L mol-1 cm-1. It is constant for a specific molecule at a specific wavelength, temperature, and solvent. High values above 10000 indicate strong absorbers.

What is path length in Beer-Lambert law?

Path length (l) is the distance light travels through the sample in centimeters. Standard laboratory cuvettes have 1 cm path length. Shorter 0.1 cm cuvettes are used for concentrated samples. Longer 5 or 10 cm cuvettes are used for dilute samples.

When does Beer-Lambert law fail?

Beer-Lambert law deviates at: (1) high concentrations above 0.01 mol/L due to molecular interactions; (2) polychromatic light; (3) turbid or scattering samples; (4) absorbance above 2.0 where stray light dominates; (5) concentration-dependent chemical changes such as dimerization. Keep absorbance between 0.1 and 1.0 for best accuracy.

What is the difference between absorbance and percent transmittance?

Absorbance has a linear relationship with concentration via Beer-Lambert law. Percent transmittance is nonlinear with concentration. Use absorbance for calibration curves because it gives straight-line plots. %T is useful for expressing filter cutoffs and instrument specs.

Can Beer-Lambert law be used for mixtures?

Yes. For non-interacting absorbers, total absorbance equals the sum of individual absorbances: A_total = (epsilon1 x l x c1) + (epsilon2 x l x c2). This additivity allows simultaneous determination of multiple components at different wavelengths.

What units are used in Beer-Lambert law?

Absorbance A: dimensionless. Molar absorptivity epsilon: L mol-1 cm-1 (or M-1 cm-1). Path length l: cm. Concentration c: mol/L. These units are self-consistent: (L/mol/cm) x cm x (mol/L) = dimensionless.

What does absorbance zero mean?

Absorbance of zero means the sample absorbs no light at the measured wavelength, corresponding to 100% transmittance. In practice this is the blank reference reading when pure solvent is measured against itself.

How does temperature affect Beer-Lambert measurements?

Temperature changes molar absorptivity (shifts molecular energy levels) and concentration (thermal expansion, chemical equilibria). For high accuracy, maintain samples at constant temperature matching the epsilon reference conditions, typically 20 to 25 degrees C.

How is Beer-Lambert law used in UV-Vis spectrophotometry?

Procedure: (1) Measure at wavelength of maximum absorbance lambda-max; (2) Blank with pure solvent; (3) Measure sample absorbance; (4) Calculate c = A divided by (epsilon x l). For highest accuracy, build a calibration curve with 5-7 known standards to verify linearity.

What is the history of Beer-Lambert law?

Bouguer (1729) first described atmospheric light attenuation. Lambert (1760) quantified the path length relationship. Beer (1852) added the concentration dependence. The combined A=elc law is sometimes called the Beer-Lambert-Bouguer law, though Beer-Lambert is the standard term in analytical chemistry.

Beer-Lambert Law Calculator

Beer-Lambert Law: A = ε × l × c
Select which variable to solve for, then fill in the other three values.

Solve For

Molar Absorptivity (ε) Units: L mol⁻¹ cm⁻¹ Enter molar absorptivity.

Path Length (l) cm (standard cuvette = 1 cm) Enter path length.

Concentration (c) Units: mol/L (M) Enter concentration.

Result

--

⚠️ Disclaimer: Results assume Beer-Lambert law conditions: dilute solution below ~0.01 mol/L, monochromatic light, no scattering. For absorbance above 2.0 or concentrated solutions, verify linearity with a calibration curve.

Was this calculator helpful?

A ↔ T ↔ %T Live Converter
Enter any one value and all three update instantly. A = −log₁₀(T) | T = 10⁻ᴀ | %T = T × 100

Absorbance (A) Dimensionless

Transmittance (T) Fraction 0 to 1

Percent Transmittance (%T) 0% to 100%

Absorbance (A)

Dimensionless

Transmittance (T)

Fraction 0-1

% Transmittance

Percent

Was this converter helpful?

Sources & Methodology

All formulas follow IUPAC definitions. The Beer-Lambert law A = εlc is the universally accepted standard in analytical chemistry and UV-Vis spectrophotometry.

Source	Coverage	Reference
IUPAC Gold Book	Official A = εlc definition, unit conventions	goldbook.iupac.org
NIST Spectrophotometry	Molar absorptivity reference values, uncertainty standards	nist.gov
Chemistry LibreTexts	Derivation, limitations, UV-Vis application guide	chem.libretexts.org

// Beer-Lambert Law — IUPAC Gold Book
A = ε × l × c
c = A / (ε × l)
l = A / (ε × c)
ε = A / (l × c)
// Transmittance conversions
T = 10⁻ᴀ A = −log₁₀(T) %T = T × 100

Beer-Lambert Law: Complete Spectrophotometry Guide

The Beer-Lambert Law is the foundation of quantitative UV-Vis spectrophotometry, used daily in chemistry, biochemistry, pharmaceuticals, environmental monitoring, and food science labs worldwide. Expressed as A = ε × l × c, it links four physically measurable quantities in a linear relationship that enables precise concentration determination of absorbing species in solution.

What is the Beer-Lambert Law and How Does It Work?

The Beer-Lambert Law states that the absorbance of a solution is directly proportional to both the molar concentration of the absorbing species and the path length of light through the sample. Each variable in A = εlc has a precise physical meaning: Absorbance (A) is a dimensionless logarithmic ratio of incident to transmitted light intensity defined as log₁₀(I₀/I). Molar absorptivity (ε) is the intrinsic light-absorbing property of the molecule at a specific wavelength in L mol⁻¹ cm⁻¹. Path length (l) is the distance light travels through the sample in centimeters. Concentration (c) is the amount of absorbing species per unit volume in mol/L.

The physical basis is photon-molecule interaction probability: each absorbing molecule has a fixed probability of capturing a photon at a given wavelength (quantified by ε). Doubling the concentration doubles the number of absorbing molecules in the light path, doubling the probability of absorption and thus doubling the absorbance. The same logic applies to path length. This linearity is why Beer-Lambert law is so analytically powerful.

Beer-Lambert Formula A = εlc: Every Variable Explained

Absorbance A is defined as log₁₀(I₀/I), where I₀ is incident and I is transmitted light intensity. At A = 0, 100% of light passes through. At A = 1, only 10% passes (90% absorbed). At A = 2, only 1% passes. This logarithmic scale creates a linear relationship with concentration, making quantitative analysis straightforward. Molar absorptivity ε varies enormously: from below 10 L mol⁻¹ cm⁻¹ for weak forbidden transitions to above 100,000 for highly conjugated organic dye molecules. Standard 1 cm cuvettes are used so that path length cancels cleanly in calculations. Concentration is expressed in mol/L to maintain dimensional consistency with standard ε values.

Key Fact: Optimal absorbance range for Beer-Lambert accuracy is 0.1 to 1.0 (79% to 10% transmittance). Above A = 2.0, stray light and detector noise cause negative deviations. Below A = 0.1, measurement uncertainty becomes large. Always dilute samples with A above 1.5 before measuring.

Common Molar Absorptivity Reference Values

Compound	Wavelength (nm)	ε (L mol⁻¹ cm⁻¹)	Application
NADH / NADPH	340	6,220	Enzyme kinetics assays
Oxyhemoglobin	542	14,300	Clinical blood oxygen
Bovine Serum Albumin (BSA)	280	44,300	Protein concentration
DNA (per base pair)	260	6,600	Nucleic acid quantification
p-Nitrophenol (pH 10)	400	18,300	Enzyme activity assays
Potassium permanganate	525	2,420	Water quality, titration
Caffeine	273	9,800	Pharmaceutical analysis
Bilirubin	453	60,700	Clinical jaundice diagnosis
Copper sulfate (CuSO₄)	750	12	Biuret protein assay
Ethanol	210	~1,000	Purity testing

Source: NIST Spectrophotometry Database and analytical chemistry literature. Values are approximate and vary with solvent, pH, and temperature.

When Beer-Lambert Law Fails: Limitations and Deviations

Beer-Lambert law applies reliably only under specific conditions. High concentrations above ~0.01 mol/L cause molecular interactions that alter the effective molar absorptivity and produce positive or negative deviations. Polychromatic light sources introduce errors because the logarithmic relationship only holds strictly at a single wavelength. Turbid or scattering samples from suspended particles add apparent absorbance unrelated to concentration. Very high absorbance above 2.0 produces negative deviations because stray light and detector noise become significant fractions of the transmitted signal. Chemical equilibria that shift with concentration such as protonation, dimerization, or complexation also violate the assumptions. For all these cases, verify linearity experimentally with a calibration curve using known standards before measuring unknowns.

Frequently Asked Questions

The Beer-Lambert Law is A = ε × l × c, where A is absorbance (dimensionless), ε is molar absorptivity in L mol⁻¹ cm⁻¹, l is path length in cm, and c is molar concentration in mol/L. Absorbance is directly proportional to both concentration and path length. It is defined by IUPAC and is the universally accepted standard in analytical chemistry and spectrophotometry.

Rearrange Beer-Lambert law to: c = A / (ε × l). Example: A = 0.45, ε = 15000 L/mol/cm, l = 1 cm gives c = 0.45 / (15000 × 1) = 0.00003 mol/L = 30 μM. This is the most common use of Beer's law in analytical chemistry for quantifying unknown concentrations. Use the Solve for Concentration option in Tab 1 above.

Transmittance T = 10⁻ᴀ. Percent transmittance %T = 100 × 10⁻ᴀ. A = 1.0 gives T = 0.10 (10% transmittance). A = 0.301 gives T = 0.50 (50% transmittance). A = 2.0 gives T = 0.01 (1% transmittance). To convert transmittance back to absorbance: A = −log₁₀(T). Use the A ↔ T ↔ %T tab above for instant conversion.

Molar absorptivity (ε), also called the molar extinction coefficient, is the intrinsic ability of a chemical species to absorb light at a given wavelength. Units are L mol⁻¹ cm⁻¹. Values range from below 10 for weak absorbers to above 100,000 for highly conjugated organic dyes. The value depends on the molecule's electronic structure, the wavelength, solvent, pH, and temperature. Always use ε determined under conditions matching your experiment.

Path length (l) is the distance light travels through the sample solution, measured in centimeters. Standard laboratory cuvettes have exactly 1 cm path length. Shorter path cuvettes (0.1 cm) are used for concentrated samples that would otherwise give A above 2.0. Longer path cuvettes (5 or 10 cm) are used for dilute solutions to increase sensitivity. Always measure l in cm when using standard ε units (L/mol/cm).

Beer-Lambert law deviates at: (1) high concentrations above ~0.01 mol/L where molecular interactions alter ε; (2) polychromatic light, since the law strictly requires monochromatic light; (3) turbid or scattering samples where particles scatter light and artificially increase apparent absorbance; (4) absorbance above 2.0 where stray light dominates; (5) chemical equilibria such as dimerization or protonation that change with concentration. Keep absorbance between 0.1 and 1.0 for reliable linear behavior.

The optimal absorbance range is 0.1 to 1.0, corresponding to 79% to 10% transmittance. Below 0.1, signal-to-noise ratio is poor and small errors become relatively large. Above 2.0, detector saturation and stray light cause significant negative deviations from linearity. The range 0.2 to 0.8 gives the best balance of signal strength and linear response. If your sample reads above 1.5, dilute it 10-fold and remeasure, then multiply the calculated concentration by 10.

Common reference values: NADH at 340 nm: ε = 6,220 L/mol/cm. BSA at 280 nm: ε = 44,300 L/mol/cm. DNA at 260 nm: ~6,600 per base pair. Potassium permanganate at 525 nm: 2,420 L/mol/cm. Bilirubin at 453 nm: 60,700 L/mol/cm. Caffeine at 273 nm: 9,800 L/mol/cm. Values above 10,000 are considered high absorbers. See the full reference table in the content section above.

Absorbance (A) has a linear relationship with concentration through Beer-Lambert law, making it ideal for quantitative analysis and straight-line calibration curves. Percent transmittance (%T) is a direct linear measurement of how much light passes through but has a non-linear relationship with concentration. Convert %T to absorbance before doing Beer-Lambert calculations: A = −log₁₀(%T / 100). Spectrophotometers typically display both, but absorbance is always used for quantification.

Yes. For mixtures of non-interacting absorbing species, total absorbance equals the sum of individual absorbances (additivity principle): Aₜₒₛₐₗ = (ε₁ × l × c₁) + (ε₂ × l × c₂) + ... This allows simultaneous determination of multiple components by measuring absorbance at multiple wavelengths and solving the system of equations. This principle is used in clinical blood analyzers, food quality testing, and environmental monitoring.

Standard units: Absorbance (A) is dimensionless (no units). Molar absorptivity (ε): L mol⁻¹ cm⁻¹ (equivalently M⁻¹ cm⁻¹). Path length (l): cm. Concentration (c): mol/L (M). Dimensional consistency: (L/mol/cm) × (cm) × (mol/L) = dimensionless. Using these standard units throughout ensures calculations are consistent and directly comparable to published molar absorptivity values.

An absorbance of zero means no light is absorbed at the measured wavelength, corresponding to 100% transmittance. The sample is completely transparent at that wavelength. In practice, the blank (pure solvent measured against itself) reads A = 0 after baseline correction. This establishes the reference point. True A = 0 indicates no electronic transitions in the molecule match the photon energy at that specific wavelength.

Temperature affects Beer-Lambert measurements in two main ways: (1) It changes molar absorptivity ε because molecular energy levels shift slightly with temperature, altering absorption probability; (2) It changes concentration through thermal expansion of the solution and through chemical equilibria (e.g. pH-dependent equilibria in indicators). For high-accuracy work, maintain samples at constant temperature matching the ε reference conditions, typically 20 to 25 degrees C.

Standard procedure: (1) Select wavelength at the sample's absorption maximum (λ-max) where ε is highest and sensitivity is greatest; (2) Measure blank (pure solvent) for baseline correction; (3) Measure sample absorbance; (4) Calculate concentration: c = A / (ε × l). For rigorous quantification, build a calibration curve using 5 to 7 known concentration standards to verify linearity and determine the exact working range before measuring unknowns.

The Beer-Lambert law combines discoveries by three scientists across two centuries. Pierre Bouguer (1729) first described light attenuation in the atmosphere. Johann Heinrich Lambert (1760) quantified the proportionality between absorbance and path length. August Beer (1852) independently discovered the concentration dependence for colored solutions. The combined A = εlc equation is sometimes called the Beer-Lambert-Bouguer law, though Beer-Lambert is the standard term in analytical chemistry and spectrophotometry worldwide.