Spectroscopy and the Beer-Lambert Law

A = abc, where A is absorbance, a is the molar absorptivity of the substance, b is the path length of the cuvette in centimetres, and c is the concentration in mol/L. Absorbance itself has no units, and molar absorptivity carries whatever units make the product dimensionless.

The common alternative symbol for molar absorptivity is ε, so A = εbc is the same relationship as A = abc. With b in cm and c in mol/L, ε has units L mol⁻¹ cm⁻¹. Absorbance is defined by A = −log₁₀(T), where transmittance T = I/I₀. T is a fraction, not a percent: 10% transmission means T = 0.10 and A = 1.00, while 1% means A = 2.00. Absorbance is linear in concentration within the applicable range, but percent transmission is not.

Every substance absorbs some wavelengths more than others. Measurements are taken at λ max, the wavelength of maximum absorbance, because the signal is largest there and small errors in wavelength setting have the least effect on the reading. A solution appears the complementary colour to the light it absorbs, so a solution that absorbs red light looks green.

At a wavelength where ε is larger, the calibration slope εb is steeper, so a small concentration change produces a larger absorbance change. Near a broad absorbance maximum the slope of absorbance versus wavelength is small, reducing sensitivity to small wavelength-setting differences. This is a reason to choose λ_max when no interfering species or instrument limitations dictate otherwise. The transmitted mixture of wavelengths gives the observed color; a single absorbed wavelength does not always map to one exact named complementary color.

Prepare a blank and several standards spanning the unknown's expected range, then measure them with the same wavelength, path length, solvent and relevant chemical conditions. Plot A vertically and concentration horizontally. The ideal blank-corrected relation passes through zero, but a measured best-fit line can have an intercept: A = mc + b₀. Use c = (A − b₀)/m rather than forcing the line through the origin without justification.

For A = (250 L/mol)c + 0.020 and A_unknown = 0.520, c = (0.520 − 0.020)/250 = 0.00200 M. Do not confuse the intercept b₀ with the cuvette path-length symbol b. Interpolate inside the standard range; if the unknown lies above it, dilute a measured aliquot, remeasure and multiply the inferred diluted concentration by V_final/V_aliquot to recover the original concentration. A 5.00 mL aliquot made to 50.00 mL has dilution factor ten.

At high concentration the particles begin to interact and the line bends away from the axis, so very concentrated samples should be diluted into the linear range. Cloudiness scatters light and inflates the apparent absorbance, and a smudged or mismatched cuvette does the same.

Deviations can come from concentration-dependent interactions, changing chemical speciation, stray light or detector limitations. Turbidity removes light by scattering rather than only absorption, so it can inflate apparent absorbance and the inferred concentration. Fingerprints, bubbles, inconsistent cuvette orientation and a changed path length can also alter readings. Keep conditions matched and investigate nonlinearity rather than extending the calibration blindly.

Blanking removes the cuvette and solvent background when performed as specified by the instrument. If readings are already blank corrected, subtracting the blank again would introduce an error. Use the same correction convention for standards and unknowns; a regression intercept may already represent residual background.

Suppose every standard is actually diluted to 80% of its labeled concentration while its x-axis label is left unchanged. At each labeled concentration the measured absorbance is too low, so the fitted slope is 80% of the true slope. For an otherwise correctly measured unknown, c_reported = A/slope is then too high by 1/0.80 = 1.25. The instrument is not seeing more analyte; the incorrect calibration makes a given absorbance appear to require a larger concentration. If instead only the unknown is diluted without correcting its dilution factor, its reported original concentration is too low. Name the affected sample before deciding the direction of error.