3.13 Beer-Lambert Law

Beer-Lambert Law

A spectrophotometer bench where the beam takes the colour of the wavelength you select. Tune λ, path length and concentration to see A = εbc build the calibration line whose slope is εb.

A = εbcChoosing λmaxCalibration Curve5 Real Solutions
Topic 3.13

Beer–Lambert Law

Explain the amount of light absorbed by a solution of molecules or ions in relationship to the concentration, path length, and molar absorptivity.

The Beer–Lambert law relates the light absorbed by a solution to three variables:

A = εbc

  • ε (molar absorptivity) — how intensely a chemical species absorbs light at a specific wavelength. It is a property of the substance and the wavelength, not of the sample.
  • b (path length) — the distance the light travels through the solution, usually 1.00 cm for a standard cuvette.
  • c (concentration) — proportional, along with b, to the number of light-absorbing particles in the light path.

In most experiments path length and wavelength are held constant, so absorbance is directly proportional to concentration. That turns the law into a practical measuring tool: prepare standards of known concentration, plot A versus c, and read an unknown off the resulting straight calibration line. The slope is εb.

EK 3.13.A.2 also specifies the experimental design point: the spectrophotometer is set to the wavelength of maximum absorbance (λ_max) for the species being analyzed, to ensure maximum sensitivity of measurement. At λ_max a small change in concentration produces the largest change in absorbance.

Sources of error — this topic's science practice is about identifying them: a fingerprinted or scratched cuvette, failing to blank the instrument with pure solvent, bubbles in the light path, a colored impurity that absorbs at the same wavelength, or working outside the linear range at very high absorbance.

Key points

  • A = εbc, and with b and λ fixed, A ∝ c.
  • ε is a property of the substance at a given wavelength; it does not change with concentration.
  • Choose λ_max for maximum sensitivity — not the color you see.
  • Absorbance is unitless; a calibration curve is a straight line through the origin.

Equations

  • on the exam sheetPrinted on the AP equation sheet as A = εbc.
    • absorbance (unitless)
    • molar absorptivity (M⁻¹cm⁻¹)
    • path length (cm)
    • concentration (M)

Common mistakes

  • Absorbance, not transmittance. They are not proportional to each other; only A is linear in c.
  • A calibration curve must pass through the origin. Zero concentration absorbs nothing.
  • ε changes with wavelength but not with concentration.
  • Do not extrapolate far past your standards. The linear relationship fails at high absorbance.

Worked example

A student prepares standards of a blue dye and measures absorbance at λ_max in a 1.00 cm cuvette: 0.0200 M → A = 0.410; 0.0400 M → A = 0.815; 0.0600 M → A = 1.220. An unknown gives A = 0.590. Find the molar absorptivity and the unknown concentration.

Molar absorptivity. The data are linear, so use the slope:
slope = (1.220 − 0.410)/(0.0600 − 0.0200) = 0.810/0.0400 = 20.25 M⁻¹
Since slope = εb and b = 1.00 cm,
ε = 20.3 M⁻¹cm⁻¹

Unknown concentration.
c = A/(εb) = 0.590 ÷ (20.25 × 1.00) = 0.0291 M

Check: 0.590 lies between the 0.410 and 0.815 standards, so the concentration should lie between 0.0200 and 0.0400 M ✓ — and the answer is safely inside the calibrated range.

Full notes for topic 3.13 →