When does the Beer–Lambert law break down?

The Beer–Lambert law breaks down at high concentrations, when there are strong solute–solute or solute–solvent interactions, when the solution scatters light (turbidity, bubbles, particulates), when the incident light is not monochromatic, or when the chemical species changes with concentration or pH.

What units should I use for molar absorptivity in the Beer–Lambert law?

For the common form A = εℓc, molar absorptivity ε is usually expressed in L·mol⁻¹·cm⁻¹, concentration c in mol·L⁻¹ (M), and path length ℓ in cm. With these units, absorbance A is dimensionless.

Can I use the Beer–Lambert law with mass concentration instead of molarity?

Yes. In that case, the proportionality constant is often called an absorptivity with units such as L·g⁻¹·cm⁻¹, and concentration is expressed in g·L⁻¹. The relationship remains linear as long as the system obeys Beer–Lambert behavior.

Beer–Lambert Law Calculator: Absorbance, Concentration & Path Length

Q: What is the Beer–Lambert law?

The Beer–Lambert law (or Beer’s law) relates the absorbance A of a solution to its concentration c, the optical path length ℓ, and the molar absorptivity ε: A = εℓc. It is widely used in spectrophotometry to determine concentrations from measured absorbance.

What is the Beer–Lambert law?

The Beer–Lambert law (often shortened to Beer’s law) describes how light is absorbed as it passes through a homogeneous medium. In its most common form for solutions:

Beer–Lambert law (molar form):

\[ A = \varepsilon \,\ell\, c \]

\(A\) – absorbance (dimensionless)
\(\varepsilon\) – molar absorptivity (L·mol⁻¹·cm⁻¹)
\(\ell\) – optical path length (cm)
\(c\) – concentration (mol·L⁻¹)

Absorbance is defined from the ratio of incident intensity \(I_0\) to transmitted intensity \(I\):

\[ A = -\log_{10}\!\left(\frac{I}{I_0}\right) = \log_{10}\!\left(\frac{I_0}{I}\right) \]

Combining these relationships gives a powerful tool: if you know any three of \(\{A, \varepsilon, \ell, c\}\), you can solve for the fourth. This is the basis of quantitative UV–Vis spectrophotometry.

Rearranging the Beer–Lambert law

From \(A = \varepsilon \ell c\), we can solve for any variable:

Absorbance: \( A = \varepsilon \ell c \)
Concentration: \( c = \dfrac{A}{\varepsilon \ell} \)
Molar absorptivity: \( \varepsilon = \dfrac{A}{\ell c} \)
Path length: \( \ell = \dfrac{A}{\varepsilon c} \)

Typical units and conventions

Path length \(\ell\): usually 1.00 cm cuvettes in UV–Vis spectroscopy.
Concentration \(c\): mol·L⁻¹ (M), but mM or µM are common in biochemistry.
Molar absorptivity \(\varepsilon\): L·mol⁻¹·cm⁻¹ at a specific wavelength and solvent.
Absorbance \(A\): unitless; values between 0.1 and 1.0 are generally most reliable.

Worked examples

Example 1 – Find concentration from absorbance

Given:

\(\varepsilon = 15{,}000\ \text{L·mol}^{-1}\text{·cm}^{-1}\)
\(\ell = 1.00\ \text{cm}\)
\(A = 0.600\)

Find: concentration \(c\).

Use \( c = \dfrac{A}{\varepsilon \ell} \):

\[ c = \frac{0.600}{(15{,}000)\,(1.00)} = 4.0 \times 10^{-5}\ \text{mol·L}^{-1} = 40\ \mu\text{M} \]

Example 2 – Predict absorbance for a standard

Given:

\(\varepsilon = 5{,}200\ \text{L·mol}^{-1}\text{·cm}^{-1}\)
\(\ell = 1.00\ \text{cm}\)
\(c = 2.5 \times 10^{-4}\ \text{mol·L}^{-1}\)

Find: absorbance \(A\).

\[ A = \varepsilon \ell c = (5{,}200)(1.00)(2.5 \times 10^{-4}) = 1.30 \]

An absorbance of 1.30 is relatively high; you might dilute the sample to bring \(A\) closer to 0.1–1.0 for better accuracy.

When does the Beer–Lambert law hold?

The law is an approximation that works best when:

The solution is dilute (typically \(A \lesssim 1\) and \(c\) not too high).
The absorbing species is chemically stable and does not associate, dissociate, or change with concentration or pH.
The medium is homogeneous and non‑scattering (no turbidity, bubbles, or particulates).
The incident light is monochromatic and the bandwidth is narrow compared to spectral features.
The path length is well defined and the cuvette is clean and properly aligned.

Common sources of deviation

High concentration – solute–solute interactions and refractive index changes.
Stray light or instrument limitations – especially at very high absorbance.
Polychromatic light – using a wide bandwidth where \(\varepsilon\) varies with wavelength.
Chemical equilibria – acid–base indicators, complex formation, aggregation.

Practical tips for lab use

Always run a blank (solvent + reagents, no analyte) and zero the instrument.
Use the same cuvette for standards and samples when possible.
Prepare a calibration curve (A vs c) to verify linearity instead of relying on a single ε value.
Record the wavelength, solvent, temperature, and path length in your lab notebook.

FAQ

What is the difference between Beer’s law and Lambert’s law?

Historically, Lambert’s law describes the exponential decrease of light intensity with path length in an absorbing medium, while Beer’s law describes the dependence on concentration. The combined relationship is called the Beer–Lambert law.

Can I use absorbance units directly as “AU” or “OD”?

Yes. Absorbance is dimensionless, but in practice people write “absorbance units (AU)” or “optical density (OD)” as shorthand. The calculator treats absorbance as a pure number.

How do I get ε for my compound?

You can obtain molar absorptivity from literature, from a certificate of analysis, or by measuring a series of standards and fitting a straight line to a plot of A vs c. The slope of the line equals \(\varepsilon \ell\); with a 1 cm cuvette, the slope is ε.