Library / Foundations: Chemistry / FCH-35
Foundations: Chemistry
Absorbance from the Beer–Lambert law
Absorbance of a dilute solution: proportional to the molar absorptivity, the path length through the sample and the concentration.
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- FCH-35
- Rev
- v1
- Field
- Foundations: Chemistry
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- A
The Beer–Lambert law says the absorbance of light by a dilute solution rises in direct proportion to the concentration c and the path length ℓ: A = log₁₀(I₀/I) = εℓc, where ε is the molar absorptivity (dm³ mol⁻¹ cm⁻¹) of the species at the wavelength used. A is dimensionless. Here c is entered in µmol/dm³ and converted to mol/dm³. A is 1 when 10 % of the light is transmitted, and 2 when 1 % is.
Absorbance of a dilute solution: proportional to the molar absorptivity, the path length through the sample and the concentration. It is a logarithm, so it has no unit, and the proportionality fails at high concentration.
Calculator
- Transmittance, 10−A0.2512
- Percentage of light transmitted25.12 %
- Concentration that would give an absorbance of 183.33 µmol/dm³
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A solution of 50 µmol/dm³ with ε = 12,000 in a 1 cm cell has an absorbance of 0.6: only 25.1 % of the light gets through.
- Doubling the concentration doubles the absorbance, but the transmitted light falls from 25.1 % to 6.31 %, which is why high absorbances are hard to measure: very little light reaches the detector.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| ℓ Path length | 0.1 to 10 cm | 0.06 | 6 | 5.94 | +10% | +10 % |
| ε Molar absorptivity | 1,000 to 100,000 dm³/(mol cm) | 0.05 | 5 | 4.95 | +10% | +10 % |
| c Concentration | 1 to 200 µmol/dm³ | 0.012 | 2.4 | 2.388 | +10% | +10 % |
Sweep of Concentration
| Concentration (µmol/dm³) | Absorbance |
|---|---|
| 1 | 0.012 |
| 25.88 | 0.3105 |
| 50.75 | 0.609 |
| 75.63 | 0.9075 |
| 100.5 | 1.206 |
| 125.4 | 1.504 |
| 150.3 | 1.803 |
| 175.1 | 2.101 |
| 200 | 2.4 |
Sweep one variable
Variables and how each one contributes
Absorbance
What the equation gives you. Everything else feeds it.
Molar absorptivity
The molar absorptivity at the measurement wavelength, in dm³/(mol cm). Raises the absorbance in direct proportion: a more strongly absorbing species gives a larger signal. Raising it by 1,000 (to 13,000) moves the result by +8.33 %, from 0.6 to 0.65.
Path length
The path length of light through the solution, in cm (1 cm for a standard cuvette). Raises the absorbance in direct proportion: a longer path means more absorbing molecules. Raising it by 0.5 cm (to 1.5 cm) moves the result by +50 %, from 0.6 to 0.9.
Concentration
The concentration of the absorbing species, in µmol/dm³. Raises the absorbance in direct proportion, which is the basis of using it to measure concentration. Raising it by 10 µmol/dm³ (to 60 µmol/dm³) moves the result by +20 %, from 0.6 to 0.72.
Worked example with the values above
- Concentration in mol/dm³: 50 µmol/dm³ = 5×10−5 mol/dm³.
- Multiply: ε ℓ c = 12,000 × 1 × 5×10−5.
- A = 0.6.
Assumptions and limits
- A dilute solution, in which the absorbance rises in direct proportion to concentration, with a single absorbing species.
- Monochromatic light and no scattering, fluorescence or chemical change with concentration.
Common mistakes
- Using a concentration that is too high, which makes the plot of absorbance against concentration curve over.
- Mixing µmol/dm³ and mol/dm³.
- Confusing absorbance with percentage transmittance.
Statistics and uncertainty
Spectrometers are most accurate for absorbances of about 0.2 to 1, where the relative error in concentration is smallest; above about 2 stray light and low signal make the result unreliable. If A is outside that range, dilute the sample or use a shorter cell.
Where it is used
- Measuring concentrations in water analysis, process control and quality checks.
- Calibrating a spectrometer with standard solutions.
Maths you need
- Logarithm: the power to which 10 is raised to give a number
- Concentration: amount of solute per volume of solution