Beer-Lambert Law Equation, Derivation & Rearrangements

Core idea

Overview

The Beer-Lambert Law defines the linear relationship between the absorbance of a substance and its concentration in a solution. It posits that as light passes through a medium, the intensity of light absorbed depends on the chemical properties of the solute, the distance the light travels, and the molar density of the sample.

When to use: Use this equation when performing spectrophotometry to determine the concentration of a known solute in a solution. It assumes monochromatic light is used, the solution is dilute (typically below 0.01 M), and there are no chemical fluctuations or light scattering within the sample.

Why it matters: It is the foundational principle for modern chemical analysis, enabling everything from monitoring pollutants in water to quantifying DNA or proteins in biological research. Its simplicity allows for rapid, non-destructive testing in pharmaceutical and industrial quality control.

Symbols

Variables

A = Absorbance, $ϵ$ = Molar Absorptivity, l = Path Length, c = Concentration

A

Absorbance

Variable

ϵ

Molar Absorptivity

L/mol cm

l

Path Length

cm

c

Concentration

mol/L

Walkthrough

Derivation

Formula: Beer-Lambert Law

Relates absorbance to concentration for light passing through a homogeneous solution at a fixed wavelength.

Absorbing medium is homogeneous.
Incident light is monochromatic.

1

State the Equation:

Absorbance A is proportional to molar absorptivity $ε$ , concentration c, and path length l.

A = ε c l

Result

A = ε c l

Source: OCR A-Level Chemistry A — Analytical Techniques

Free formulas

Rearrangements

Solve for $A$

Make A the subject

A = ϵ l c

A is already the subject of the formula.

Difficulty: 1/5

Solve for $c$

Beer-Lambert Law: Make c the subject

c = \frac{A}{ϵ l}

Rearrange the Beer-Lambert Law to solve for concentration, $c$ . This involves isolating $c$ by dividing both sides of the equation by the product of molar absorptivity and path length.

Difficulty: 2/5

Solve for $ϵ$

Make epsilon the subject

ϵ = \frac{A}{l c}

Rearrange the Beer-Lambert Law to solve for molar absorptivity ( $ϵ$ ).

Difficulty: 2/5

Solve for $l$

Beer-Lambert Law

l = \frac{A}{ϵc}

Rearrange the Beer-Lambert Law, $A = ϵ l c$ , to isolate the path length, $l$ .

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph displays a straight line passing through the origin where the slope represents the product of epsilon and l. For a chemistry student, low concentration values result in minimal absorbance while high concentration values indicate the substance is absorbing significantly more light. The most important feature of this curve is the linear relationship which means that doubling the concentration results in a proportional doubling of the absorbance.

Graph type: linear

Why it behaves this way

Intuition

Imagine a beam of light as a stream of particles (photons) attempting to pass through a crowded room; the more people (absorbing molecules)

A

A quantitative measure of the fraction of incident light absorbed by a sample. It is defined as log10(I_0 / I), where I_0 is the incident light intensity and I is the transmitted

The higher the absorbance, the less light passes through the sample; it's a direct indicator of how 'dark' or 'opaque' the solution is to the specific wavelength of light.

ϵ

Molar absorptivity (or molar extinction coefficient), an intrinsic property of a substance at a specific wavelength, representing how strongly it absorbs light per unit

This value reflects the inherent 'light-blocking efficiency' of a single molecule of the substance at a given wavelength; different molecules have different efficiencies.

l

The optical path length, which is the distance the light beam travels through the sample.

The longer the path light travels through the solution, the more opportunities it has to interact with and be absorbed by the solute molecules.

c

The molar concentration of the absorbing substance in the solution.

A higher concentration means more absorbing molecules are present in a given volume, increasing the probability of light absorption as the beam passes through.

Free study cues

Insight

Canonical usage

The units of molar absorptivity, path length, and concentration are chosen such that their product yields a dimensionless value for absorbance.

Common confusion

A common mistake is forgetting that absorbance is dimensionless, or incorrectly assigning units to molar absorptivity ( $ϵ$ ) that do not cancel out with path length and concentration.

Dimension note

Absorbance (A) is a dimensionless quantity, representing the logarithm of the ratio of incident to transmitted light intensity.

Unit systems

$A$ dimensionless · Absorbance is a logarithmic ratio of light intensities and thus has no units.

$ϵ$ L mol^-1 cm^-1 · Molar absorptivity's units are derived to ensure the product \epsilon l c is dimensionless when l is in cm and c is in mol/L.

$l$ cm · Path length is conventionally measured in centimeters in most spectroscopic applications.

$c$ mol/L · Concentration is typically expressed in moles per liter (Molar, M).

Ballpark figures

Quantity:

One free problem

Practice Problem

A chemical dye with a molar absorptivity of 5000 M⁻¹cm⁻¹ is analyzed in a spectrophotometer. If the concentration of the solution is 0.0002 M and the path length of the cuvette is 1.0 cm, what is the measured absorbance?

Molar Absorptivity5000 L/mol cm

Path Length1 cm

Concentration0.0002 mol/L

Solve for: $A$

Hint: Multiply the molar absorptivity, path length, and concentration together (e × l × c).

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

When measuring concentration of a colored solution, Beer-Lambert Law is used to calculate Absorbance from Molar Absorptivity, Path Length, and Concentration. The result matters because it helps connect measured amounts to reaction yield, concentration, energy change, rate, or equilibrium.

Study smarter

Tips

Ensure the spectrophotometer is zeroed with a blank solution.
Work within the linear range of the instrument, typically an absorbance between 0.1 and 1.0.
Match the wavelength to the substance's maximum absorbance peak for highest sensitivity.

Avoid these traps

Common Mistakes

Forgetting path length l.
Confusing absorbance with transmittance.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Relates absorbance to concentration for light passing through a homogeneous solution at a fixed wavelength.

Use this equation when performing spectrophotometry to determine the concentration of a known solute in a solution. It assumes monochromatic light is used, the solution is dilute (typically below 0.01 M), and there are no chemical fluctuations or light scattering within the sample.

It is the foundational principle for modern chemical analysis, enabling everything from monitoring pollutants in water to quantifying DNA or proteins in biological research. Its simplicity allows for rapid, non-destructive testing in pharmaceutical and industrial quality control.

Forgetting path length l. Confusing absorbance with transmittance.

When measuring concentration of a colored solution, Beer-Lambert Law is used to calculate Absorbance from Molar Absorptivity, Path Length, and Concentration. The result matters because it helps connect measured amounts to reaction yield, concentration, energy change, rate, or equilibrium.

Ensure the spectrophotometer is zeroed with a blank solution. Work within the linear range of the instrument, typically an absorbance between 0.1 and 1.0. Match the wavelength to the substance's maximum absorbance peak for highest sensitivity.

References

Sources

Atkins' Physical Chemistry
Wikipedia: Beer-Lambert law
IUPAC Gold Book: Beer-Lambert law
Atkins' Physical Chemistry, 11th ed.
Principles of Instrumental Analysis, Skoog, Holler, Crouch, 7th ed.
Skoog, D. A., Holler, F. J., & Crouch, S. R. (2017). Principles of Instrumental Analysis (7th ed.). Cengage Learning.
Atkins, P., & de Paula, J. (2014). Atkins' Physical Chemistry (10th ed.). Oxford University Press.
IUPAC Gold Book (Compendium of Chemical Terminology).

Beer-Lambert Law

Overview

Variables

Derivation

State the Equation:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Nernst Equation

pH definition

Rate law

Frequently Asked Questions

Sources