Unit 3 · Topic 3.13 Beta

Beer-Lambert Law

A spectrophotometer measures how much light of one wavelength a solution absorbs.

Practice 2: Question and MethodPractice 5: Mathematical Routines

Question set for this topic

Part 1 · Hook

Why this matters

Hospitals measure blood glucose, water labs measure nitrate, and food scientists check the dye in sports drinks by shining one color of light through a sample. The more of the colored substance there is, the more light it absorbs, in exact proportion.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. What is the molarity of 0.0200 mol of solute in 0.500 L of solution?

  1. 0.0400 M
  2. 0.0100 M
  3. 25.0 M
Show the answer

0.0200 mol ÷ 0.500 L = 0.0400 M.

  • Correct: 0.0400 M:
  • 0.0100 M:
  • 25.0 M:

2. A solution looks blue. Which light does it absorb most?

  1. Orange-red
  2. Blue
  3. None
Show the answer

We see the light that is not absorbed.

  • Correct: Orange-red:
  • Blue:
  • None:

Part 4 · See it

See it first

A spectrophotometer: light from a lamp passes through a wavelength selector, then through a cuvette of path length b holding the sample, and a detector measures how much gets through. The instrument reports absorbance, A = εbc.
Light of one wavelength passes through the sample; the instrument reports absorbance. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. Each absorbing particle in the beam can absorb a photon of matching energymore particles in the path absorb more light
  2. Doubling the concentration or the path length doubles the particles in the beamabsorbance is proportional to both: A = εbc
  3. ε is largest at λmaxmeasurements there are most sensitive to concentration
  4. Standards of known concentration fall on a straight line of A against can unknown's absorbance on that line gives its concentration

Part 6 · Key ideas

Key ideas

  • A = εbc: A unitless, ε in L/(mol·cm), b in cm, c in mol/L.
  • Measure at λmax, the wavelength of maximum absorbance.
  • Calibration curve: A against c is a straight line through the origin with slope εb.
  • Errors that add absorbance (fingerprints, no blank) make the concentration too high.

Part 7 · Misconception

A common mistake

The wrong idea: A solution that absorbs more light must have a larger molar absorptivity.

What actually happens: ε is fixed for a substance at one wavelength. More absorbance at the same wavelength and path length means a higher concentration.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Graph

Calibration curve for a blue food dye

A student measures five standard solutions of a blue dye at 630 nm in 1.00 cm cuvettes after zeroing the instrument with water. A diluted sports drink gives an absorbance of 0.565.

00.20.40.60.811.20246810Dye concentration (× 10⁻⁵ M)Absorbance
Data table
Dye concentration (× 10⁻⁵ M)Standards
00
20.229
40.45
60.681
80.902
101.131

1. Use the calibration curve to find the dye concentration in the diluted drink, in units of 10⁻⁵ M (for example, 3.00 means 3.00 × 10⁻⁵ M).

Type a number in × 10⁻⁵ M.

Show the answer

The line passes through the origin with slope 1.131 / 10.0 = 0.1131 per 10⁻⁵ M. c = 0.565 / 0.1131 = 4.996, so 5.00 × 10⁻⁵ M.

  • Answer: 5.00 × 10⁻⁵ M

2. The slope of the line equals εb. With b = 1.00 cm, what is the molar absorptivity of the dye?

Type a number in L/(mol·cm).

Show the answer

Slope = ΔA/Δc = 1.131 / (1.00 × 10⁻⁴ M) = 11,310 M⁻¹. ε = slope / b = 11,310 M⁻¹ / 1.00 cm = 1.13 × 10⁴ L/(mol·cm).

  • Answer: 1.13e+4 L/(mol·cm)

3. Before measuring the drink, the student touches the clear faces of its cuvette, leaving fingerprints. How does this affect the concentration she reports?

  1. Too high, because the fingerprints absorb or scatter light and raise the absorbance.
  2. Too low, because the fingerprints let more light reach the detector.
  3. No effect, because the fingerprints are outside the cuvette and not in the solution.
  4. Too low, because fingerprints raise the path length b of the cuvette.
Show the answer

Any light lost at the cuvette walls is counted as absorbed by the sample. A higher absorbance read on the calibration line gives a higher concentration.

  • Correct: Too high, because the fingerprints absorb or scatter light and raise the absorbance.: Right: extra absorbance, higher concentration.
  • Too low, because the fingerprints let more light reach the detector.: Fingerprints block or scatter light; they do not let more through.
  • No effect, because the fingerprints are outside the cuvette and not in the solution.: The beam passes through the walls, so anything on them affects the reading.
  • Too low, because fingerprints raise the path length b of the cuvette.: Path length is the inside width, which does not change.

4. The drink was diluted by mixing 10.0 mL of it with water to a total of 50.0 mL. What was the dye concentration in the undiluted drink?

  1. 1.00 × 10⁻⁵ M
  2. 5.00 × 10⁻⁵ M
  3. 6.25 × 10⁻⁵ M
  4. 2.50 × 10⁻⁴ M
Show the answer

The diluted solution is 5.00 × 10⁻⁵ M. The dilution factor is 50.0 / 10.0 = 5.00, so the original is 5.00 × 5.00 × 10⁻⁵ M = 2.50 × 10⁻⁴ M (M₁V₁ = M₂V₂).

  • 1.00 × 10⁻⁵ M: This divides by the dilution factor instead of multiplying.
  • 5.00 × 10⁻⁵ M: This is the diluted concentration, not the original.
  • 6.25 × 10⁻⁵ M: This used the 40.0 mL of water added as V₁ (50.0/40.0); the volume of drink taken was 10.0 mL.
  • Correct: 2.50 × 10⁻⁴ M: Right: multiply back by the dilution factor.

Data table

Measuring copper(II) ions

Copper(II) solutions absorb strongly at 810 nm. A student measures three samples with a spectrophotometer.

Absorbance at 810 nm
SampleConcentration (M)Path length (cm)Absorbance
10.05001.000.592
20.10001.00?
30.05002.00?

5. What absorbance should sample 2 give?

  1. 0.296
  2. 0.592
  3. 1.18
  4. 2.37
Show the answer

A is proportional to c. Doubling c from 0.0500 to 0.1000 M doubles A: 2 × 0.592 = 1.184, so 1.18.

  • 0.296: This halves instead of doubling.
  • 0.592: This ignores the change in concentration.
  • Correct: 1.18: Right: twice the concentration, twice the absorbance.
  • 2.37: This quadruples; A is proportional to c, not c².

6. What absorbance should sample 3 give?

  1. 0.0592
  2. 0.296
  3. 0.592
  4. 1.18
Show the answer

A is proportional to the path length b. Same concentration as sample 1 but twice the path: 2 × 0.592 = 1.18.

  • 0.0592: This divides by 10 for no reason.
  • 0.296: A longer path increases A; it does not halve it.
  • 0.592: A longer path passes more absorbing ions, so A rises.
  • Correct: 1.18: Right: twice the path, twice the absorbance.

7. In A = εbc, what does b stand for?

  1. The path length of light through the sample
  2. The molar absorptivity of the substance
  3. The concentration in mol/L
  4. The wavelength of the light used
Show the answer

b is the path length, the distance light travels through the solution, usually 1.00 cm.

  • Correct: The path length of light through the sample: Right: usually 1.00 cm.
  • The molar absorptivity of the substance: That is ε.
  • The concentration in mol/L: That is c.
  • The wavelength of the light used: The wavelength is chosen (λmax) but is not a variable in the equation.

Part 9 · Summary

Summary

A spectrophotometer measures how much light of one wavelength a solution absorbs. The Beer-Lambert law, A = εbc, makes absorbance proportional to concentration and path length. Measurements are made at λmax, and a calibration curve of standards gives an unknown's concentration; anything that adds absorbance, such as fingerprints or a missed blank, makes it read too high.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections