How a spectrophotometer works
A lamp gives white light (a tungsten lamp for the visible, a deuterium lamp for the ultraviolet). A monochromator, a prism or diffraction grating with a slit, picks out a narrow band of wavelengths. That light passes through the sample in a cuvette and lands on a detector. First a blank, the cuvette with only the solvent, is measured to give I₀; then the sample gives I. Dividing one by the other cancels the lamp's brightness, the cuvette's reflections and the solvent's own absorption.
Transmittance and absorbance
T = I ÷ I₀, A = −log₁₀ T
Transmittance is the share of the light that gets through. Absorbance is its negative logarithm, which turns the way light dies away in a sample, by the same fraction in every millimeter, into a number that grows in a straight line with the amount of absorber:
- A = 0: all the light gets through.
- A = 1: 10% gets through.
- A = 2: 1%. A = 3: 0.1%.
The Beer-Lambert law
A = ε l c
- ε is the absorptivity of the substance at that wavelength: how strongly it absorbs. In M⁻¹ cm⁻¹ it is the molar absorptivity, or extinction coefficient.
- l is the path length through the sample, 1 cm in a standard cuvette.
- c is the concentration.
Double the concentration or the path and the absorbance doubles. The law lets a measured absorbance be turned straight into a concentration when ε is known.
Worked examples
- Potassium permanganate has ε = 2,455 M⁻¹ cm⁻¹ at 525 nm. An 80 µM solution in a 1 cm cuvette: A = 2,455 × 1 × 0.000080 = 0.196, T = 10−0.196 = 63.7%.
- Protein: bovine serum albumin absorbs 0.667 per mg/mL per cm at 280 nm, from its tryptophan and tyrosine. A reading of 0.40 means 0.40 ÷ 0.667 = 0.60 mg/mL.
- DNA: double-stranded DNA at 50 µg/mL gives A260 = 1.0. A sample diluted 1 in 10 that reads 0.25 holds 0.25 × 50 × 10 = 125 µg/mL. Pure DNA has A260 ÷ A280 of about 1.8; a lower ratio suggests protein in it.
Absorption spectra
Plotting absorbance against wavelength gives the substance's absorption spectrum. Measure at its peak, λmax: the reading is largest, so most sensitive, and the curve is flat there, so small errors in wavelength matter least. A solution looks the color of the light it does not absorb: permanganate absorbs green (around 525 nm) and looks purple; copper sulfate absorbs red and near infrared (around 800 nm) and looks pale blue.
Standard curves
Often ε is not known for the exact sample and conditions, or the instrument needs checking. Then a series of standards of known concentration is measured, absorbance is plotted against concentration and a straight line A = m c + b is fitted. An unknown's concentration is c = (A − b) ÷ m.
Example: permanganate standards of 0, 25, 50 and 100 µM read 0.000, 0.061, 0.123 and 0.245. The fitted slope is 0.00245 per µM with an intercept near zero. An unknown reading 0.150 is 0.150 ÷ 0.00245 = 61 µM. Trust the answer only inside the range of the standards, and only if R² is close to 1.
Using the simulation
In the Spectrophotometer and Beer-Lambert Law Simulator, things to try:
- Permanganate at its peak: double the concentration and the absorbance doubles, while the transmittance falls from 64% to 41%.
- Halve the path length: the absorbance halves.
- Push the concentration to the top: the measured absorbance falls below the Beer-Lambert line, flattening near 3 because of stray light. Untick stray light and the two agree.
- Widen the slit to 20 nm on permanganate's narrow peaks: the peak absorbance falls and the spectrum smooths out.
- Choose DNA at 260 nm in a plastic cuvette: the cuvette absorbs nearly all the ultraviolet and the reading becomes noise.
- Add four standards, measure the unknown, and compare the answer with the actual value shown.
What the model assumes
- Absorption spectra built from smooth peaks, scaled to published values at one wavelength: permanganate 2,455 M⁻¹ cm⁻¹ at 525 nm, copper sulfate about 12.5 M⁻¹ cm⁻¹ at 810 nm, albumin 0.667 per mg/mL at 280 nm, DNA 0.020 per µg/mL at 260 nm. The shapes elsewhere are approximate.
- A triangular slit band: the reading averages the transmittance over ± the slit width.
- Stray light of 0.1% that reaches the detector without passing through the absorber at the chosen wavelength.
- Cuvettes: quartz transmits everywhere from 200 nm; polystyrene stops transmitting below about 300 nm.
- A little detector noise, so readings wobble in the last digit and become unreliable when almost no light gets through.
- The blank cancels the cuvette and solvent exactly, and the unknown is a new random concentration for each solution after Reset.
Edge cases
- Zero concentration reads A = 0, apart from noise, because the blank is the same cuvette and solvent.
- Wavelengths where nothing absorbs give A ≈ 0 at any concentration; copper sulfate barely absorbs in the green.
- Very high absorbance reaches a ceiling near −log₁₀ of the stray light: about 3 for 0.1%.
- Standards at one wavelength, unknown at another give a meaningless answer; the simulation warns.
Where the model stops being right
- Concentrated solutions. Above about 10 mM, molecules sit close enough to affect each other's absorption, and the refractive index changes; the line curves even with a perfect instrument.
- Chemistry in the cuvette. Dyes that pair up, acids and bases whose color depends on pH, and complexes that dissociate on dilution all change ε with concentration.
- Scattering. Cloudy samples, cells and precipitates scatter light away from the detector, which reads as absorbance that is not there; this is what an OD600 reading of a bacterial culture measures on purpose.
- Fluorescence. A sample that re-emits some of the light can send it to the detector and read low.
- Proteins differ. The 0.667 figure is for albumin; other proteins have different amounts of tryptophan and tyrosine and can absorb half or twice as much per mg/mL.
Calculators
Solve A = εlc for any of its terms with the Beer-Lambert Law Calculator, fit your own readings with the Standard Curve Calculator, and work out nucleic acid and protein concentrations with the DNA Concentration Calculator and the Protein Concentration Calculator (A280). The guides How to Make a Standard Curve and How to Measure Protein Concentration at A280 go further.