Beer-Lambert Law & UV-Vis Spectrophotometry

Beer-Lambert Law & UV-Vis Spectrophotometry

1. Derivation of the Beer-Lambert Law

Consider a thin layer $dx$ of solution: the fractional decrease in light intensity $I$ is proportional to the number of absorbing molecules encountered:

$$-\frac{dI}{I} = \alpha \cdot N \cdot dx$$

where $\alpha$ is the absorption cross-section per molecule and $N$ is the number density. Integrating over path length $l$:

$$\ln\frac{I_0}{I} = \alpha N l = \varepsilon c l \cdot \ln 10$$

In absorbance units ($\log_{10}$ convention):

$$A = \log_{10}\frac{I_0}{I} = \varepsilon \cdot c \cdot l$$

where $\varepsilon$ (L·mol$^{-1}$·cm$^{-1}$) is the molar absorptivity (molar extinction coefficient), $c$ (mol/L) is concentration, and $l$ (cm) is path length. Transmittance $T = I/I_0$; $A = -\log_{10}T$. %T $= 100T$.

2. Calibration Curve & Linearity

A calibration set of standards at known concentrations $c_1, c_2, \ldots, c_n$ produces absorbances $A_1, \ldots, A_n$. Linear regression of $A$ on $c$ gives slope $= \varepsilon l$ and intercept $= A_{blank}$. Linearity (ICH Q2(R1)): the $R^2$ of the calibration should be $\geq 0.999$ across the linear dynamic range. Useful range: $0.1 \leq A \leq 1.5$ (below 0.1: poor S/N; above 1.5: stray light deviations from BLL).

3. Limits of Detection (LOD) and Quantitation (LOQ)

ICH Q2(R1) definitions:

$$LOD = \frac{3.3 \sigma}{S}, \qquad LOQ = \frac{10 \sigma}{S}$$

where $\sigma$ is the standard deviation of the blank (or the calibration residuals) and $S$ is the calibration slope ($\varepsilon l$). LOD: minimum detectable signal with $P(\text{false positive}) < 5\%$. LOQ: minimum quantifiable concentration with CV $\leq 10\%$.

4. Deviations from Beer-Lambert Law

Real deviations from linearity arise from:

• Chemical deviations: association/dissociation equilibria (e.g., $Fe^{3+}$ hydrolysis), formation of multiple absorbing species — apparent $\varepsilon$ changes with $c$.
• Instrumental deviations: stray light ($I_s$) — a constant background transmitted to the detector — causes underestimation of $A$ at high concentrations: $A_{obs} = \log\frac{I_0 + I_s}{I T + I_s} \to \log(I_0/I_s) < \infty$ as $T \to 0$. Polychromatic radiation — different wavelengths have different $\varepsilon$, leading to a concave calibration curve.

5. HPLC-UV Detector: the Beer-Lambert Law in Practice

HPLC diode-array detectors measure absorbance of column effluent in a 10 mm (or 50 μm for nano-LC) flow cell. Peak area $= \int A(t)\,dt$ is proportional to the amount injected. Response factor $= \varepsilon \times$ peak area / amount. Detector linearity typically holds from LOQ to $\sim 500\,\mu$g/mL; above this, detector saturation or flow-cell stray light causes curvature.