What a spectrophotometer measures
A spectrophotometer sends light of one wavelength through a cuvette比色皿 holding the solution, compares the incoming and transmitted intensities, and reports the absorbance A. Over a certain range, absorbance is proportional to concentration:
Here ε is the molar absorptivity (set by the substance and the wavelength), b is the path length (the cuvette width, usually 1.00 cm), and c is the concentration.
Choosing the wavelength
- A solution’s color is the light it doesn’t absorb. A blue solution absorbs orange and red light; a purple solution absorbs green.
- Measure near the absorbance peak (λmax): the slope εb is largest there, so the measurement is most sensitive and a given reading error causes the smallest concentration error.
The calibration curve
- Zero the instrument with the solvent (the blank) to subtract any absorbance from the solvent and the cuvette.
- Make a set of standards of known concentration and measure their absorbance at the same wavelength.
- Plot A against c to get a straight line through about the origin.
- Measure the unknown’s absorbance and read its concentration from the line. The unknown’s absorbance should fall within the range of the standards; if it’s too high, dilute first and multiply by the dilution factor.
Drag the wavelength in the simulation below and watch how the slope and the concentration uncertainty change:
The top graph is the absorption spectrum of a blue dye solution (5.0 μM). Drag the slider to choose the measuring wavelength; the bottom graph shows five standards and the unknown (orange circle) measured at that wavelength.
Example 1: concentration from a calibration curve, then before dilution
A dye’s calibration curve is A = 0.1048c + 0.002 (c in μM). A student dilutes 10.00 mL of a drink to 50.00 mL and measures an absorbance of 0.457. Find the dye concentration in the drink.
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Concentration in the cuvette. Solve the line’s equation for c:
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Dilution factor. Diluting 10.00 mL to 50.00 mL makes the concentration 1/5 of the original.
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Concentration in the drink.
Example 2: molar absorptivity from the slope
The cuvette in Example 1 has a path length of 1.00 cm. Find the dye’s molar absorptivity ε at the measuring wavelength, and explain what the intercept of 0.002 means.
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The slope is εb. Since A = εbc, the slope of A against c is εb:
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Convert units. 1 μM⁻¹ = 10⁶ M⁻¹; then divide by the 1.00 cm path length:
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The intercept. Ideally the absorbance is 0 at zero concentration. 0.002 is tiny, within reading error, which shows the blank was zeroed properly.
- A yellow food-dye solution
A yellow solution absorbs violet and blue light.
- Blood-red FeSCN²⁺ solution
It absorbs light on the blue–green side and transmits red.
- Purple KMnO₄ solution
Purple is what remains when green light is absorbed.
- Pink phenolphthalein in a basic solution
Pink and magenta solutions absorb green light.
- Orange-red iron–phenanthroline complex
It absorbs blue-green to green light and transmits orange-red.
- A red food-dye solution
A red solution absorbs around the green.
- A blue food-dye solution
A blue solution absorbs orange and red light.
- Deep blue [Cu(NH₃)₄]²⁺ solution
Deep blue means orange and red light are absorbed.
Common mistakes
Mistake 1: “Measure with light the same color as the solution.” The solution’s color is the transmitted light. Measure with the light that is absorbed, the complementary color.
Mistake 2: “The higher the absorbance, the better, so measure concentrated solutions directly.” When the absorbance is very high, too little light gets through, the reading is unreliable, and the line stops being straight. The unknown’s absorbance should fall within the standards’ range; dilute it if it’s too concentrated.
Mistake 3: “The calibration line goes through the origin, so one standard is enough.” A set of standards confirms the straight-line relationship, reduces random error, and reveals any problem with the blank.
Quick check
In A = εbc, b is:
b is the path length; a standard cuvette is 1.00 cm.
Why is the instrument zeroed with a blank before measuring?
After zeroing, the measured absorbance comes only from the substance of interest.
A sample’s absorbance is 1.8, beyond the range of the standards. What should be done?
The result is reliable only within the range of the calibration curve.
If the concentration of a solution doubles, its absorbance:
A is proportional to c.
Which of these does NOT affect the measured absorbance?
Absorbance depends only on the solution in the light path, not on how much is left in the beaker.
Why is the measuring wavelength usually set at the absorbance peak?
With a large slope, a reading error causes only a small concentration error.
AP-style practice
To find the iron content of an iron supplement, a student dissolves one tablet, reduces all the iron to Fe²⁺, and makes the solution up to 250.0 mL in a volumetric flask (solution A). A 1.00 mL portion of solution A is mixed with excess phenanthroline (forming an orange-red complex) and diluted to 100.0 mL; its absorbance at 510 nm is 0.517. Data for the standards (path length 1.00 cm):
| [Fe²⁺] (M) | A |
|---|---|
| 1.0 × 10⁻⁵ | 0.112 |
| 2.0 × 10⁻⁵ | 0.221 |
| 4.0 × 10⁻⁵ | 0.445 |
| 6.0 × 10⁻⁵ | 0.664 |
| 8.0 × 10⁻⁵ | 0.889 |
(a) A solution of Fe²⁺ alone is almost colorless. Why is phenanthroline added before measuring?
(b) Estimate the slope of the calibration curve from the first and last standards, and find the molar absorptivity of the complex at 510 nm.
(c) Calculate the concentration of Fe²⁺ in the solution that was measured.
(d) Calculate the mass of iron in one tablet (Fe 55.85 g/mol), in mg.
(e) When the sample is measured, there are fingerprints on the outside of the cuvette. Is the calculated mass of iron too high or too low? Explain.
Model answer and scoring
(a) Fe²⁺ barely absorbs visible light, so its absorbance would be too small to measure sensitively. The complex is intensely colored with a large ε, and its absorbance is proportional to the Fe²⁺ concentration, so the measurement is sensitive.
(b) Slope = (0.889 − 0.112) ÷ (8.0 × 10⁻⁵ − 1.0 × 10⁻⁵) M = 1.11 × 10⁴ M⁻¹; with b = 1.00 cm, ε = 1.11 × 10⁴ M⁻¹ cm⁻¹.
(c) c = A ÷ (εb) = 0.517 ÷ (1.11 × 10⁴) = 4.66 × 10⁻⁵ M.
(d) Solution A was diluted 100.0 ÷ 1.00 = 100 times, so [Fe²⁺] in A = 4.66 × 10⁻³ M; n(Fe) = 4.66 × 10⁻³ × 0.2500 = 1.16 × 10⁻³ mol; m = 1.16 × 10⁻³ × 55.85 = 0.0650 g = 65.0 mg.
(e) Too high. Fingerprints absorb and scatter some light → the measured absorbance is too high → the calculated concentration and mass of iron are too high.
Scoring: (a) Fe²⁺ barely absorbs while the complex absorbs strongly, giving sensitivity; (b) the slope and ε with units; (c) correct use of A = εbc; (d) the dilution factor and volume conversion; (e) too high, with the chain of reasoning.
Summary
- A = εbc. Measurements are most sensitive near the absorbance peak; a solution’s color is the transmitted light, so measure with the light it absorbs.
- Calibration curve: zero with a blank → standards → plot A vs c → read the unknown from the line; dilute a concentrated sample and multiply by the dilution factor.
- The calibration slope is εb. Mistakes that raise the absorbance (fingerprints, no blank) make the concentration too high; an over-diluted sample makes it too low.
AP tip: Beer-Lambert calculations often include a dilution step. Find the concentration in the cuvette first, then multiply by the dilution factor, and only then convert to mass or moles.
Sources: the blue dye spectrum in the simulation and the data in the examples and practice are illustrative; the atomic mass of Fe is from the periodic table supplied with the AP Chemistry exam.