Beer-Lambert Law
Build and use an absorbance calibration with a justified intercept, range, blank and error direction.
AP Chemistry Topic 3.13 · CED printed p. 72 / PDF p. 79. This teaching, practice and feedback were added for the AP edition; original author material remains separately traceable.
More absorbing particles along the light path means more absorption
Beer–Lambert law is , where A is absorbance, ε is molar absorptivity at the chosen wavelength, b is path length, and c is concentration of the absorbing species. If b is in cm and c in mol L−1, ε has units L mol−1 cm−1, so A is dimensionless. Path length and concentration increase the number of absorbing particles encountered by the light. At fixed ε, doubling either doubles A; doubling both quadruples A.
Absorbance is related to transmitted intensity: . It is not the fraction of photons absorbed, so A = 0.50 does not mean “50% absorbed.” The corresponding transmittance is 10−0.50 ≈ 0.316. Transmitted intensity falls nonlinearly as concentration rises even though absorbance is linear in c under the model. This additional relationship clarifies what the instrument reports.
Control wavelength and path length to measure concentration
Use a wavelength near the analyte’s absorbance maximum to get large ε and strong sensitivity: the slope of A versus c is εb. For a fixed-concentration spectrum, the measured absorbance maximum indicates the greatest measured ε and hence the greatest A-versus-c slope at fixed path length. This sensitivity comparison assumes suitable blanking, no interfering absorber and reliable readings; one spectrum cannot establish an entire linear concentration range. Hold wavelength and cuvette path length fixed across standards and unknown. A suitable solvent/matrix blank corrects baseline absorption from non-analyte components. Standards of known c establish a calibration, then interpolate the unknown within the tested linear range. A different wavelength changes ε, so it cannot be swapped silently after calibration.
For ideal blank-corrected data the line goes through zero, but a measured intercept can reflect baseline offset. Do not force a fit through zero merely because the ideal equation has no intercept. Inspect the blank, retain a justified measured intercept A0, or correct the offset transparently. Use c = (A − A0)/slope for that measured calibration. An unknown outside the standard range should be diluted into range and its original concentration recovered with the dilution factor.
Trace an error through the measurement
If a dirty surface, bubble or suspended particle removes additional light from the beam, the instrument can report too large an apparent absorbance; applying a positive-slope clean-standard calibration then overestimates concentration. If an unknown is accidentally diluted, it has lower absorbing-species concentration and absorbance; reporting it without correcting that dilution underestimates the original sample concentration. A longer-path cuvette used only for the unknown increases A and biases a concentration inferred from the shorter-path calibration high.
Use realistic limits. Concentrated samples can depart from the simple linear model because species interactions, changing chemical forms or instrument limitations alter the response. Repeated absorbance readings estimate random spread, while a systematic baseline or dilution error can persist across repeats. For an approximate concentration uncertainty with a fixed known slope s, Δc ≈ ΔA/s. Uncertainty in standards and slope may also matter; do not report more precision than the data support.
Let's work through it
Work it through
An original calibration at fixed wavelength and 1.00 cm path has the following data. An unknown has A = 0.395 ± 0.006; treat slope and intercept as fixed for this simplified uncertainty estimate.
| c / mol L−1 | A |
|---|---|
| 0.0000 | 0.020 |
| 0.0010 | 0.170 |
| 0.0020 | 0.320 |
| 0.0030 | 0.470 |
- Slope = (0.470 − 0.020)/(0.0030 − 0) = 150 L mol−1. The measured line is A = 150c + 0.020.
- Subtract the measured intercept: c = (0.395 − 0.020)/150 = 0.00250 mol L−1. The value lies within the standard range.
- Approximate Δc = 0.006/150 = 0.00004 mol L−1; report (0.00250 ± 0.00004) mol L−1 under the stated assumptions.
- If the calibration was used without subtracting the offset, c would be 0.395/150 = 0.00263 M, biased high. The offset should be investigated or transparently accounted for.
Make the picture move
Interactive concept model
A calibration slope carries both chemistry and geometry.
Predict first. Predict how doubling concentration or path length changes absorbance. Then compare absorbance with the fraction of light transmitted.
A = εℓc = −log₁₀(I/I₀); ε = 1.00 × 10⁴ L mol⁻¹ cm⁻¹
- Ideal calibration at selected path
- Labeled selected or reference state
On a small screen, scroll inside the graph or diagram to read its labels. The values and explanation also appear below.
For c = 5.00 × 10^-5 mol L⁻¹ and ℓ = 1.0 cm, A = 0.500 and 31.6% of incident light is transmitted.
Read the data and comparisons
| Concentration (mol L⁻¹) | Absorbance | Transmitted light (%) |
|---|---|---|
| 0 | 0.000 | 100.0 |
| 2.50 × 10^-5 | 0.250 | 56.2 |
| 5.00 × 10^-5 | 0.500 | 31.6 |
| 7.50 × 10^-5 | 0.750 | 17.8 |
| 1.00 × 10^-4 | 1.000 | 10.0 |
Macroscopic / data. An instrument measures incident and transmitted light; their ratio determines absorbance.
Particle view. More independently absorbing species along the beam path increase the probability of absorption; the model does not assign molecules a visible color.
Symbolic view. A = (10000 L mol⁻¹ cm⁻¹)(1.0 cm)(5.00 × 10^-5 mol L⁻¹).
Why it works. At fixed wavelength and path length, absorbance is proportional to concentration. The calibration slope is εℓ; changing the cuvette path changes the slope. Transmission is exponential: absorbance 1 means 10% transmission, and absorbance 2 means 1% transmission.
Conditions and limits. Synthetic ideal calibration; ε is supplied for a hypothetical solute at one fixed wavelength and 25 °C. Clear dilute solution, independent absorbers, additive volumes, matched blank, no scattering or concentration-dependent speciation. Real data have uncertainty and can depart from linearity. Units require ℓ in cm and c in mol L⁻¹.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
At fixed wavelength, what happens to A if b doubles and c halves?
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
At fixed wavelength, what happens to A if b doubles and c halves?
Worked answer and reason
A stays the same under the linear model.
The product bc remains unchanged.
Retrieval 2
Why hold the wavelength fixed after calibration?
Worked answer and reason
Molar absorptivity depends on wavelength, so changing wavelength can change the calibration slope.
The unknown must be measured under the same response conditions as standards.
Retrieval 3
If an unknown’s cuvette is dirty and attenuates extra light, what likely happens to inferred concentration?
Worked answer and reason
Apparent A rises and concentration inferred from a clean positive-slope calibration is biased high.
Less transmitted light is interpreted as more analyte absorption.
Retrieval 4
Should a nonzero measured intercept always be hidden by forcing the line through zero?
Worked answer and reason
No. Investigate baseline/blank effects, then retain or transparently correct a justified offset.
The ideal relationship does not erase measured experimental evidence.
Catch the tempting mistake
- Absorbance is the percentage of light absorbed.
- It is logarithmic attenuation; A = 0.50 corresponds to about 31.6% transmitted, not 50% absorbed.
- The wavelength maximum is used because photon energy is automatically largest there.
- It is used because the analyte’s absorptivity and calibration sensitivity are large there.
- Repeating a reading removes systematic concentration error.
- Repeated readings reduce uncertainty in random variation but can retain a systematic bias.
- A theoretical zero intercept requires deleting every measured baseline offset.
- Blank and intercept evidence must be handled transparently rather than forced away.
Put it to work
Give the chemical reason as well as the calculation. Worked solutions are on a separate page.
- Unit 3 problem 31 · ap application · symbolic
- Unit 3 problem 32 · ap synthesis · data
- Unit 3 problem 33 · ap synthesis · data
- Unit 3 problem 34 · ap synthesis · data
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For the teacher
- Ask for ε, b and c units and a slope interpretation.
- Use one dilution or path-length error and require both A direction and concentration direction.
- Check interpolation range and whether a measured intercept was used.
Source objectives and scope
Source edition: ap-chemistry-ced-fall2024-local-b5dfe867. See this topic in the course outline →
- 3.13.A
- Explain the amount of light absorbed by a solution of molecules or ions in relationship to the concentration, path length, and molar absorptivity.
- 3.13.A.1
- The Beer-Lambert law relates the absorption of light by a solution to three variables according to the equation: EQN: A = εbc. The molar absorptivity, ε, describes how intensely a chemical species absorbs light of a specific wavelength. The path length, b, and concentration, c, are proportional to the number of light-absorbing particles in the light path.
- 3.13.A.2
- In most experiments the path length and wavelength of light are held constant. In such cases, the absorbance is proportional only to the concentration of absorbing molecules or ions. The spectrophotometer is typically set to the wavelength of maximum absorbance (optimum wavelength) for the species being analyzed to ensure the maximum sensitivity of measurement.
Science references for this teaching
- AP Chemistry Course and Exam Description, effective Fall 2024 (supplied source) — Exact topic-numbered learning objectives, essential knowledge, science practices and AP exclusions; supplied PDF SHA256 b5dfe8677ef3d88c613865d2e2a3e8d6125d652e2b24c71ef1e8ce4e011094f0.
Glossary and deeper reading · 7 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Molarity
Molarity is the amount of a solute in moles divided by the final solution volume in liters. It describes concentration rather than the total amount of solute. For example, 0.10 mole dissolved to make 0.50 liter of solution gives a molarity of 0.20 mole per liter.
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Dilution
Dilution lowers solute concentration by adding solvent while conserving the solute amount when no reaction or loss occurs. Concentration calculations use the final solution volume rather than simply the volume of solvent added. For example, increasing a solution's volume from 100 to 200 milliliters halves its concentration if its solute amount is unchanged.
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Photon
A photon is a quantum of electromagnetic radiation. Its energy increases with frequency and decreases as wavelength increases. For example, a blue-light photon carries more energy than a red-light photon.
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Absorbance
Absorbance is a logarithmic measure of how much a sample reduces transmitted light at a specified wavelength. It depends on the ratio of incident to transmitted intensity rather than simply being the fraction absorbed. For example, absorbance 1 corresponds to transmission of one-tenth of the incident intensity.
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Molar absorptivity
Molar absorptivity describes a species' absorption strength at a specified wavelength in the Beer–Lambert model. It depends on the absorbing species and measurement conditions, while concentration and path length are separate factors. For example, selecting a wavelength near an absorption maximum can give a larger absorbance for the same concentration and path length.
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Analyte
An analyte is the substance being identified or measured in an analysis. Its amount or concentration is the target of the measurement rather than necessarily the largest component of the sample. For example, an unknown acid is the analyte in an acid–base titration.
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Solvent
A solvent is the component used as the dissolving medium in a solution. It is often present in the greater amount, though the designation also depends on the mixture's context. For example, water is the solvent in a dilute aqueous sugar solution.