Basics of spectrophotometry and optical measurements

Why it matters

Spectrophotometers measure how much light a sample absorbs at each wavelength. They are the workhorse analysers of water-quality labs, pharmaceutical QC, clinical chemistry and process plants (on-line colour and concentration monitors), and the same building blocks — sources, monochromators, detectors and ratio measurement — appear in optical power meters, spectrum analysers and gas analysers. An instrumentation engineer must be able to turn a transmittance reading into a concentration and judge when that reading can be trusted.

Key ideas

Transmittance and absorbance. A beam of radiant power P₀ enters a sample and P leaves it. Transmittance is T = P/P₀ (often quoted as %T). Absorbance is A = −log₁₀T = log₁₀(P₀/P). A = 1 means 10% transmitted; A = 2 means 1%.

Beer–Lambert law. For a dilute solution of one absorbing species in monochromatic light, A = ε·l·c, where ε is the molar absorptivity (a property of the substance at that wavelength), l the path length and c the concentration. Absorbance, not transmittance, is linear in concentration, so calibration curves plot A against c. Absorbances of several non-interacting species at one wavelength add, which allows mixtures to be analysed by measuring at as many wavelengths as there are components.

Deviations from Beer's law. High concentrations (solute interactions, refractive-index change), chemical equilibria (dissociation, association), polychromatic light (finite bandwidth on a sloping absorption band), stray light reaching the detector, scattering by turbid samples and fluorescence all bend the calibration curve. Stray light is especially damaging at high absorbance, putting a ceiling on measurable A.

Instrument components.

• Sources: deuterium lamp for UV (about 190–350 nm), tungsten-halogen lamp for visible and near IR (about 350–2500 nm), xenon flash lamps, LEDs in simple photometers.
• Wavelength selector: absorption or interference filters (photometers/colorimeters), or a monochromator — entrance slit, collimator, dispersing element (prism or diffraction grating), focusing optics and exit slit. Narrower slits give finer spectral bandwidth but less light.
• Sample cells (cuvettes): glass or plastic for the visible, fused silica (quartz) for the UV, standard path 1 cm.
• Detectors: photomultiplier tubes (high gain, low light), silicon photodiodes, photodiode or CCD arrays (whole spectrum at once in diode-array instruments).
• Readout: a ratio of sample to reference signals, converted to A.

Single-beam and double-beam. A single-beam instrument measures the blank (solvent) and then the sample, so source drift between the two readings appears as error. A double-beam instrument splits or chops the light between sample and reference cells and takes their ratio continuously, cancelling source and detector drift.

Diffraction grating. A reflection grating with groove spacing d sends wavelength λ in order m to the angle satisfying m·λ = d(sin θ_i + sin θ_m); at normal incidence m·λ = d·sin θ. Its resolving power is R = λ/Δλ = m·N, where N is the number of illuminated grooves.

Photometric accuracy. A fixed error in reading T gives the smallest relative concentration error near A ≈ 0.434 (T ≈ 37%); in practice keep A between about 0.2 and 1.0 by diluting or changing path length.

Other optical measurements. Optical power meters (calibrated photodiodes or thermal heads, readings in dBm), optical spectrum analysers (scanning grating monochromators), colorimeters, turbidimeters and nephelometers (scattered light), fluorometers and flame photometers apply the same source–selector–detector chain.

Formulas

T = P/P₀, A = −log₁₀T = log₁₀(P₀/P) P₀ incident and P transmitted radiant power (W) or intensity (W/m²); T and A dimensionless.

A = ε·l·c ε molar absorptivity (L·mol⁻¹·cm⁻¹), l path length (cm), c concentration (mol/L).

A_total = Σ ε_i·l·c_i Additivity for non-interacting species.

m·λ = d·sin θ (normal incidence) m diffraction order, d groove spacing (m), θ diffraction angle.

R = λ/Δλ = m·N R resolving power, Δλ smallest resolvable wavelength difference, N number of illuminated grooves.

T_measured = (T + s)/(1 + s) s stray-light fraction (relative to P₀).

Worked examples

Example 1 (standard). A dye with ε = 1.2 × 10⁴ L·mol⁻¹·cm⁻¹ at its absorption peak shows 40% transmittance in a 1 cm cell. Find the absorbance and concentration.

1. A = −log₁₀(0.40) = 0.398.
2. c = A/(ε·l) = 0.398/(1.2 × 10⁴ × 1).
3. c = 3.32 × 10⁻⁵ mol/L. Answer: A ≈ 0.398, c ≈ 3.3 × 10⁻⁵ mol/L (33 µmol/L).

Example 2 (GATE level). (a) A grating with 600 lines/mm is illuminated at normal incidence over a width of 50 mm. Find the first-order angle for 589 nm and the smallest resolvable Δλ there. (b) A sample has true transmittance 1% (A = 2). The instrument has 1% stray light. What absorbance does it display?

1. (a) d = 1/600 mm = 1.667 µm; sin θ = mλ/d = 0.589/1.667 = 0.3534, so θ = 20.7°.
2. N = 600 × 50 = 30 000; Δλ = λ/(mN) = 589/30 000 = 0.0196 nm — easily resolving the 0.6 nm sodium doublet.
3. (b) T_measured = (0.01 + 0.01)/(1 + 0.01) = 0.0198.
4. A_measured = −log₁₀(0.0198) = 1.70. Answer: (a) 20.7°, 0.020 nm; (b) A reads 1.70 instead of 2.00 — a 15% negative error from only 1% stray light.

Example 3 (dilution). A sample reads A = 2.4, outside the reliable range. It is diluted 1:5 and re-measured. What absorbance is expected?

1. A is proportional to c, so A_new = 2.4/5 = 0.48. Answer: about 0.48, within the 0.2–1.0 range; multiply the result by 5 afterwards.

Common mistakes

• Treating transmittance as linear in concentration. Only absorbance is.
• Using natural log instead of log₁₀ for absorbance.
• Forgetting %T must be converted to a fraction (40% → 0.40).
• Measuring UV absorbance in glass or plastic cuvettes, which themselves absorb below about 320 nm.
• Ignoring the blank: solvent and cell absorbance must be subtracted (zeroed) first.
• Trusting readings above A ≈ 2 when stray light and noise dominate.

For GATE IN

• Transmittance, absorbance and Beer–Lambert concentration numericals, including mixtures at two wavelengths.
• Grating equation and resolving power.
• Effect of stray light and polychromatic light on measured absorbance.
• Instrument blocks: sources and detectors for UV, visible and IR; single- versus double-beam.

Quick check

1. What absorbance corresponds to T = 0.2?
2. If path length doubles at constant concentration, what happens to A?
3. Which source covers the UV region in a UV–visible spectrophotometer?
4. A 1200 lines/mm grating at normal incidence, first order, 500 nm: what is θ? Answers: 1. 0.699. 2. It doubles. 3. Deuterium lamp. 4. 36.9°.