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The Beer-Lambert law

Shine light through a colored solution and the more concentrated it is, the more light it swallows. The Beer-Lambert law makes that exact: absorbance is a straight line in concentration, which turns a spectrometer into a concentration meter.

§1

Absorbance is linear in concentration.

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The Beer-Lambert law, A = εbc, relates a solution's absorbance (A) to its concentration (c), the path length of light through it (b), and a constant ε specific to the substance and wavelength.

The key feature is linearity: absorbance is directly proportional to concentration. Double the concentration and the absorbance doubles. (Transmittance — the fraction of light passing through — is not linear in concentration; absorbance is.)

Because of that linearity, a calibration curve — absorbance measured for several known concentrations — is a straight line. Measure an unknown's absorbance, read across to the line, and get its concentration.

UNIT 3 TOPIC 3.13 • BEER-LAMBERT LAW CUVETTE LAB Light passes through a solution in a cuvette — more absorbance means less light gets through. LIGHT THROUGH A CUVETTE DILUTE SOLUTION (LOW CONCENTRATION) MORE LIGHT TRANSMITTED DETECTOR SHORT PATH b = 1.00 cm LOW ABSORBANCE (HIGH TRANSMITTANCE) CONCENTRATED SOLUTION (HIGH CONCENTRATION) LESS LIGHT TRANSMITTED DETECTOR SHORT PATH b = 1.00 cm HIGH ABSORBANCE (LOW TRANSMITTANCE) BEER–LAMBERT LAW A = εbc A = absorbance (unitless) ε = molar absorptivity (L·mol⁻¹·cm⁻¹) b = path length (cm) c = concentration (mol·L⁻¹) INCREASING CONCENTRATION OR PATH LENGTH INCREASE CONCENTRATION (c) DECREASES TRANSMITTED LIGHT (I) INCREASES ABSORBANCE (A) low A high A OR INCREASE PATH LENGTH (b) DECREASES TRANSMITTED LIGHT (I) INCREASES ABSORBANCE (A) low A high A CALIBRATION: A vs c A 0 c (mol·L⁻¹) A c calibration standards unknown sample Read A → get c off the line. KEY TAKEAWAY A is directly proportional to c and b · more c or longer b → more absorbance CED SAP-8.C AP Chemistry · Unit 3 · Properties of Substances & Mixtures
Fig. 3.13.1 The Beer-Lambert law, A = εbc. Absorbance rises in direct proportion to concentration (and path length), so a more concentrated solution transmits less light. A calibration curve of absorbance versus known concentrations lets you read off an unknown's concentration.
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Using a calibration curve.

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Absorbance in, concentration out — along a straight line.

  1. Build the calibration line. Measure absorbance for several standards of known concentration; plot A versus c. It should be a straight line through the origin.
  2. Measure the unknown's absorbance. Read the absorbance of the unknown at the same wavelength and path length.
  3. Read across to the line. Find the concentration on the line that gives the unknown's absorbance.
  4. Use absorbance, not transmittance. Only absorbance is linear in concentration; do not draw a straight-line calibration from transmittance.
§3

The pieces you'll meet.

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One law, a few terms.

A = εbc
Beer-Lambert law
Absorbance equals ε times path length times concentration.
A
Absorbance
Directly proportional to concentration; the linear quantity.
c
Concentration
What you usually solve for from absorbance.
b
Path length
How far the light travels through the solution.
ε
Molar absorptivity
A constant for the substance at a given wavelength.
calibration
Calibration curve
A straight line of absorbance vs known concentration.
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Worked example: find an unknown concentration.

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Question. A calibration curve gives absorbance A = 0.20 × c (with c in mol/L). An unknown sample reads an absorbance of 0.50. What is its concentration?

Linearity. Absorbance is proportional to concentration, so the calibration line A = 0.20c applies.

Solve. 0.50 = 0.20 × c, so c = 0.50 / 0.20 = 2.5 mol/L.

Check. Doubling the concentration would double the absorbance, consistent with the straight-line law. Reading concentration from absorbance is exactly what a calibration curve is for.

§5

Mistakes that cost real points.

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Pitfall · 01

"Transmittance is linear in concentration, just like absorbance."

Only absorbance is directly proportional to concentration. Transmittance (the fraction of light that passes through) falls off nonlinearly as concentration rises. Building a straight-line calibration from transmittance gives wrong concentrations.

Fix. Plot and use absorbance, not transmittance, for a linear calibration in concentration.

Pitfall · 02

"You can find an unknown's concentration from a single absorbance reading with no calibration."

A single absorbance number means nothing without a calibration curve (or a known ε and path length) to convert it. The calibration line, built from known standards, is what turns absorbance into concentration.

Fix. Always calibrate: measure absorbances of known concentrations first, then read the unknown against that line.

Pitfall · 03

"Doubling concentration does not necessarily double absorbance."

By the Beer-Lambert law, absorbance is directly proportional to concentration (at fixed path length and wavelength), so doubling the concentration does double the absorbance, within the law's linear range.

Fix. Use the proportionality A ∝ c: within the linear range, absorbance scales directly with concentration.

§6

Skill Check.

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