What absorbance is and why it is a logarithm
Shine monochromatic light through a coloured solution and some of it comes out the other side. Transmittance T is the surviving fraction: 0.10 means a tenth got through. Absorbance is the negative base-10 logarithm of that fraction, A = −log₁₀T, so 10% transmittance is A = 1, 1% is A = 2, and 0.1% is A = 3.
The logarithm is not a cosmetic choice. Picture the cuvette as a stack of thin slabs. Each slab removes the same fraction of whatever light reaches it, not the same amount, because the number of molecules the light meets is the same in every slab. Repeated fractional removal is exponential decay, and taking a logarithm converts it into something proportional to path length and concentration. Absorbance is the quantity that behaves linearly; transmittance is what the detector actually measures.
The proportionality constant is the molar absorptivity ε — a property of the molecule at that wavelength, not of your instrument or your sample. It tells you how strongly the species absorbs. Values run from a few tens for weakly absorbing species to hundreds of thousands for intensely coloured dyes, and they are large exactly where an electronic transition is fully allowed.
Put together, A = εlc is the workhorse of quantitative UV-Vis spectroscopy. It underlies enzyme assays that follow NADH at 340 nm, nucleic acid quantitation at 260 nm, protein assays at 280 nm and every colorimetric method in water and environmental testing. Because the equation has four variables, it also runs backwards: measure a standard of known concentration and you obtain ε.
The three terms, and where the units have to agree
ε must match the units of c and l. The standard combination is M⁻¹cm⁻¹ with c in mol/L and l in cm, which makes A dimensionless as it must be. You will also meet the specific absorptivity a, quoted in L·g⁻¹·cm⁻¹, which pairs with a concentration in g/L, and the biochemist's A(1%, 1 cm) — the absorbance of a 1% w/v solution in a 1 cm cell. These are different constants for the same molecule and cannot be substituted for each other. Converting between molar and specific absorptivity is a multiplication or division by the molar mass.
l is the internal width of the cell, not the outside. The overwhelming default is 1.00 cm. Microvolume instruments that pedestal a droplet use path lengths near 0.1 cm or 0.05 cm and then normalise the reported absorbance to a 1 cm equivalent — check whether your instrument has already done that before you divide again.
c is the concentration of the absorbing species, in the cuvette. If you diluted the sample to bring the reading into range, the cuvette concentration is not the sample concentration; multiply by the dilution factor, which the optional field above does. Getting this backwards — dividing instead of multiplying — is the single most common arithmetic error in a routine assay, and it produces an answer wrong by the square of the dilution.
A is a blanked reading. The instrument first measures a reference cuvette containing everything except the analyte, and reports the difference. If your blank does not match the sample matrix — the same buffer, the same salt, the same cuvette — the difference includes something other than your analyte.
One consequence of the logarithmic definition is worth internalising: absorbances add. If two species absorb at your wavelength, the total absorbance is the sum of their individual contributions, which is what makes multi-component analysis possible from readings at several wavelengths, and what makes a turbid or contaminated sample read high.
Worked example: quantifying NADH in an enzyme assay
NADH absorbs at 340 nm with a published molar absorptivity of 6220 M⁻¹cm⁻¹, which is the basis of most dehydrogenase assays. You dilute a stock 1-in-10, read it in a 1.00 cm cuvette against a buffer blank, and the instrument shows A = 0.622.
- Form the denominator. ε × l = 6220 M⁻¹cm⁻¹ × 1.00 cm = 6220 M⁻¹.
- Divide. c = 0.622 ÷ 6220 = 1.000 × 10⁻⁴ M in the cuvette, or 100 µM.
- Undo the dilution. The stock was diluted tenfold, so multiply: 1.000 × 10⁻⁴ × 10 = 1.000 × 10⁻³ M, or 1.00 mM.
- Convert to mass if you need it. NADH has M = 663.43 g/mol, so 1.000 × 10⁻³ mol/L × 663.43 g/mol = 0.663 g/L = 663 mg/L.
- Check the reading was in range. A = 0.622 gives %T = 100 × 10^(−0.622) = 23.9% — comfortably inside the 0.1 to 2.0 absorbance window where a spectrophotometer is linear and precise.
Now suppose you had read the undiluted stock. Its absorbance would have been 6220 × 1.00 × 1.000 × 10⁻³ = 6.22, meaning about 0.00006% of the light reaches the detector. No routine instrument measures that: stray light alone would dominate, and the reading would come back far too low and non-linear. The dilution was not optional, and this is exactly why dilution is the standard response to a high reading.
Working backwards to find ε: if you prepared a 1.000 × 10⁻⁴ M standard of an unknown chromophore and measured A = 0.435 in a 1.00 cm cell, then ε = 0.435 ÷ (1.00 × 1.000 × 10⁻⁴) = 4350 M⁻¹cm⁻¹. In practice you would do this at five concentrations and take the slope of the calibration line, which averages out errors in any one standard.
The absorbance window, and why the law goes non-linear
Aim for an absorbance between about 0.1 and 1.0, and treat 2.0 as a hard ceiling. Both ends of that range are set by measurement error rather than by chemistry.
At the low end, the signal approaches the instrument's noise floor and the difference between sample and blank becomes comparable to cuvette-to-cuvette variation, fingerprints and bubbles. At the high end, almost no light reaches the detector, so the small amount of stray light leaking through the monochromator becomes a large fraction of the measured signal — and because stray light is not absorbed by your analyte, it makes the sample look more transparent than it is. The calibration line bends downwards.
There is a precise optimum in between. Differentiating the relationship between absorbance and the relative error in concentration, assuming a constant uncertainty in transmittance, gives a minimum relative error at A = 1/ln(10) = 0.4343, corresponding to 36.8% transmittance. That is the target if you have the freedom to choose the concentration.
Four chemical effects break linearity even when the optics are perfect. Association or dissociation at high concentration changes the absorbing species itself — dimers absorb differently from monomers. Refractive index changes above roughly 0.01 M shift the effective absorptivity. Acid-base or complexation equilibria convert the analyte into a form with different ε, which is why buffered conditions matter. Polychromatic light — a bandwidth wide compared with the absorption band — averages several absorptivities together and always bends the plot towards lower absorbance.
Two physical effects do the same. Turbidity scatters light out of the beam, and the detector cannot distinguish scattering from absorption, so a cloudy sample reads artificially high. Fluorescence from the sample sends extra photons towards the detector and reads artificially low. Filter or centrifuge turbid samples; choose a wavelength away from the emission band for fluorescent ones.
The remedy for all of this is a calibration curve prepared in the same matrix, spanning your working range, rather than a single-point calculation from a literature ε. Use the literature value to check that your slope is sensible, not to replace the measurement. When the analyte is part of a reaction being followed over time, the concentration-time data feeds directly into a rate law and, through several temperatures, into the Arrhenius equation.
Molar absorptivities in common use
| Species | Wavelength | ε (M⁻¹cm⁻¹) | Used for |
|---|---|---|---|
| NADH / NADPH | 340 nm | 6 220 | Dehydrogenase and coupled enzyme assays |
| Tryptophan (in protein) | 280 nm | 5 500 | Protein concentration from sequence |
| Tyrosine (in protein) | 280 nm | 1 490 | Protein concentration from sequence |
| Cystine (disulfide) | 280 nm | 125 | Protein concentration from sequence |
| p-Nitrophenolate | 405 nm | 18 500 | Phosphatase and glycosidase assays, alkaline pH |
| Cyanmethaemoglobin | 540 nm | 11 000 | Reference method for haemoglobin, per haem |
Nucleic acids are conventionally quantified by mass rather than by molar absorptivity: an A₂₆₀ of 1.00 in a 1 cm cell corresponds to roughly 50 µg/mL of double-stranded DNA, 40 µg/mL of RNA and 33 µg/mL of single-stranded DNA.
Mistakes that produce a wrong concentration
- Dividing by the dilution factor instead of multiplying. The cuvette holds the diluted solution, so the original sample was more concentrated. Multiply.
- Using a literature ε at the wrong wavelength, pH or solvent. Absorptivity is a property of a species in an environment. A shift of a few nanometres near a peak edge can change it by tens of percent.
- Reading above A = 2. Stray light makes high readings non-linear and biased low. Dilute and multiply back.
- Blanking against water when the sample is in buffer. The blank must match the sample matrix in everything except the analyte, including the cuvette itself.
- Ignoring turbidity. Scattered light is indistinguishable from absorbed light at the detector, so a cloudy sample reads high. Filter or centrifuge, or subtract a scattering baseline measured away from the absorption band.
- Confusing molar with specific absorptivity. ε is per mole per litre; a is per gram per litre. They differ by the molar mass, which for a protein is a factor of tens of thousands.
- Assuming a microvolume instrument used a 1 cm path. Pedestal instruments use 0.05–0.1 cm and usually normalise the display to a 1 cm equivalent. Dividing again by the real path length double-counts the correction.
- Applying a single-analyte calculation to a mixture. Absorbances add, so any second absorbing species at your wavelength inflates the result.
Where this sits among quantitative methods
UV-Vis absorbance is fast, cheap, non-destructive and requires no separation, which makes it the default for a coloured or chromophoric analyte at micromolar to millimolar concentrations. Its weaknesses are selectivity and range. Anything else absorbing at your wavelength contributes, and below about 10⁻⁶ M the absorbance falls under the noise floor of a bench instrument.
When selectivity is the problem, separate first: HPLC with a UV detector applies exactly this law to each eluting peak, with the interferences removed in time rather than in wavelength. When sensitivity is the problem, fluorescence detection gains two to three orders of magnitude because it measures light against a dark background rather than a small difference between two bright signals. When the analyte is an ion rather than a chromophore, potentiometry with an ion-selective electrode reads it directly through the Nernst equation, and classical titrimetry through the acid-base titration calculator remains the most traceable route of all for major components.
Within spectroscopy, the same law has close relatives. Absorbance measurements at two wavelengths where two species have different ε values give the concentration of each by solving a pair of simultaneous equations — the basis of the A₂₆₀/A₂₈₀ purity ratio for nucleic acids. Kinetic assays follow absorbance against time and convert the slope directly into a rate using the same ε and l. And difference spectroscopy uses the change in absorptivity between two forms of a molecule, which is why some published values are quoted as Δε rather than ε.
For the sample preparation that precedes any of this, the concentration arithmetic connects to the pH calculator when the chromophore is pH-dependent, and to the buffer calculator for keeping it in a single protonation state throughout the measurement.