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The Beer-Lambert law relates absorbance $A$ to an absorbing species' concentration $c$ and path length $l$ by $A=\epsilon lc$ in a homogeneous medium using sufficiently narrow-band light. The coefficient $\epsilon$ depends on the species, wavelength and chemical solution conditions. This makes Ellman's reagent useful for quantitative thiol assays once its released colored species has been calibrated. ^[ref]

Here $c$ is molar concentration, $l$ is the geometric length through the sample, and $\epsilon$ is the molar absorption coefficient. The ideal beam has parallel light rays, a single wavelength, and a medium with uniform composition and isotropy; finite spectral width must be small relative to features in the optical spectrum.^[ref]

For a spectrophotometer, distinguish the source's output from the reference that reaches the photodetector. Let $P_0$ be the received radiant power through a reference blank and $P$ the received radiant power through the sample, after subtracting the photodetector's dark response. Both refer to the same selected wavelength band and collection geometry. Define the blank-normalized transmittance and absorbance byThe reference blank includes the solvent and other assay ingredients except the absorbing target. Matching cuvettes and conditions makes their common losses divide out. Thus this $T$ is relative to the reference blank; it is not the absolute transmittance of the entire cuvette and its contents. Both radiant powers can be expressed in watts, while their ratio, $T$, and its logarithm, $A$, are dimensionless. An instrument displaying $40\%$ transmission means $T=0.40$, not $T=40$.

Why take a logarithm? Consider two consecutive identical absorbing layers. Each transmits a fraction $t$ of the radiant power entering it, so together they transmit $t^2$. Their absorbances add because $-{\log }_{10}(t^2)=2(-{\log }_{10}t)$. Doubling the sample length doubles absorbance while squaring transmittance. This turns multiplicative loss into an additive scale and makes a straight line useful for inferring molar concentration.

A short model derivation makes the assumption explicit. Suppose a thin layer of thickness $dx$ removes the fraction $kcdx$ of its incoming radiant power, where $k$ is a fixed constant of proportionality and $c$ is a uniform molar concentration. The corresponding ordinary differential equation, initial condition, and solution areTaking the base-ten logarithm gives $A=kcl/\ln 10$, so $\epsilon =k/\ln 10$. The exponential decay follows from equal layers removing equal fractions of what remains. Proportionality to molar concentration additionally assumes that the absorbing chemical species retains the same response as its molar concentration changes; the geometric argument alone does not establish that.

An actual teaching experiment prepares a Kool-Aid stock by weighing powder, transferring it quantitatively into a volumetric flask, and filling to the mark with water. A Genesys 30 spectrophotometer, blanked with water in a cuvette, scans for the peak absorbance wavelength. A Genesys 10 then measures absorbance at that wavelength, with a water reference blank. Each serial dilution transfers $50\mathrm{mL}$ of the preceding chemical solution to a $100\mathrm{mL}$ volumetric flask and fills to the mark; the protocol records the blank and sample readouts and repeats the halving sequence. These are prescribed experimental steps, not a published numerical data set.^[ref]

The experiment tests whether absorbance tracks the prepared concentration. Its concentration is powder mass per final volume, so the calibration does not by itself supply a single dye's molar absorption coefficient. The manual identifies loss of discrimination for very concentrated, very dilute, or very similar samples.^[ref]

To interpret such a calibration, hold cuvette orientation fixed with its clear faces in the light path, keep fingerprints and bubbles out of that path, and keep the geometric length unchanged. Mix each dilution completely. Re-read the reference blank during the sequence and re-read one standard to test drift. A separate preparation of a standard checks errors propagated through serial dilution; repeating only its instrumental reading does not test the transferred volumes. These controls follow from keeping $P_0$, $l$, and $c$ comparable between readings.

Three mechanisms explain why a straight line can fail. First, scattering of light can divert radiant power away from the photodetector, so cloudiness can raise apparent absorbance without the intended absorbing solute. A reference blank removes only losses shared with the sample. Second, stray light reaching the photodetector can make a strongly absorbing sample appear too transmissive. In an original model with extra detected radiant power $s{P}_0$ in both readings,As the ideal transmittance approaches zero, the reported absorbance approaches ${\log }_{10}((1+s)/s)$ for $s>0$. This finite ceiling is a model prediction, not measured instrument performance.

Third, chemical equilibrium can redistribute the absorbing chemical species when dilution, pH, or temperature changes. Even if each chemical species separately follows the Beer-Lambert law, their combined absorbance is $A=l{\sum }_i{\epsilon }_i{c}_i$, where $c_i$ is each species' molar concentration; changing those fractions need not preserve one effective molar absorption coefficient. Thus an unknown should be compared with standards under matching chemical solution conditions, within the tested concentration range. A good linear regression alone does not establish chemical selectivity.

The decadic molar absorption coefficient specifies absorbance per geometric length and molar concentration. With $l$ in centimetres and $c$ in $\mathrm{mol}{\mathrm{L}}^{-1}$, $\epsilon$ has units $\mathrm{L}\mathrm{mo}{\mathrm{l}}^{-1}\mathrm{c}{\mathrm{m}}^{-1}$; its SI unit is ${\mathrm{m}}^2\mathrm{mo}{\mathrm{l}}^{-1}$.^[ref]

The units cancel in $\epsilon lc$. Since $1\mathrm{L}=10^{-3}{\mathrm{m}}^3$ and $1\mathrm{cm}=10^{-2}\mathrm{m}$, one $\mathrm{L}\mathrm{mo}{\mathrm{l}}^{-1}\mathrm{c}{\mathrm{m}}^{-1}$ equals $0.1{\mathrm{m}}^2\mathrm{mo}{\mathrm{l}}^{-1}$. Specify the wavelength and chemical solution conditions when giving a molar absorption coefficient: changing the absorbing chemical species can change it even at fixed total analytical concentration.