Spectrometry & Beer-Lambert Law

When a beam of monochromatic light (a single wavelength, $\lambda$) with initial intensity $I_0$ enters a solution, photons are absorbed by the solute molecules. The transmitted intensity $I$ is always less than $I_0$. Transmittance (T) is defined as $T = I / I_0$, ranging from 0 (no light passes) to 1 (all light passes). A 1.0 mol/L blue copper sulfate solution absorbs strongly in the orange/red region (~700 nm) and transmits blue light, giving the solution its characteristic color.

Transmittance is inconvenient for analysis because it has a nonlinear relationship with concentration. Absorbance (A) is defined as the negative base-10 logarithm of transmittance: $A = -\log_{10}(T) = \log_{10}(I_0/I)$. If 50% of light is transmitted (T = 0.50), then $A = -\log_{10}(0.50) \approx 0.301$. If only 10% is transmitted (T = 0.10), then $A = 1.00$. Absorbance has no units and typically ranges from 0 to about 2 in standard lab measurements, with values above 2 becoming unreliable.

The Beer-Lambert Law states that absorbance is directly proportional to both the concentration of the absorbing species and the path length of light through the sample: $A = \varepsilon l c$. Here, $\varepsilon$ (epsilon) is the molar absorptivity constant (also called the molar extinction coefficient), $l$ is the path length in centimeters, and $c$ is the molar concentration in mol/L (mol dm$^{-3}$). The molar absorptivity is unique to each chemical species at a specific wavelength — for example, $\ce{KMnO4}$ at 525 nm has $\varepsilon \approx 2{,}400 \text{ L mol}^{-1}\text{cm}^{-1}$.

In practice, the Beer-Lambert Law is applied by preparing a series of standard solutions of known concentration, measuring the absorbance of each, and plotting A versus c. This produces a straight line through the origin with slope = $\varepsilon l$. An unknown sample's absorbance is then measured and its concentration read from the graph. This calibration curve method compensates for any instrument-specific factors and is robust as long as absorbance stays in the linear range (roughly A < 1.5). Deviations from linearity at high concentrations indicate the law's limits.

The Beer-Lambert Law holds under specific conditions: the light must be monochromatic (single wavelength), the solution must be homogeneous, and the solute must not undergo chemical changes (e.g., dissociation equilibria that shift with concentration). Deviations occur at high concentrations (> ~0.01 mol/L for some species) due to solute-solute interactions that alter the effective molar absorptivity. Stray light in the spectrophotometer and polychromatic light sources also cause deviations. For best accuracy, measurements are taken at the wavelength of maximum absorbance ($\lambda_{max}$) where the calibration curve is most sensitive.

• A — Absorbance (dimensionless): The amount of light absorbed by the solution; defined as $-\log_{10}(I/I_0)$. Higher values indicate more absorption.
• \varepsilon — Molar absorptivity (molar extinction coefficient) (L mol$^{-1}$ cm$^{-1}$): A constant intrinsic to the absorbing species at a specific wavelength. Larger values mean the substance absorbs light more strongly.
• l — Path length (cm): The distance light travels through the sample inside the cuvette. Standard laboratory cuvettes have a path length of exactly 1.00 cm.
• c — Molar concentration (mol L$^{-1}$ (mol dm$^{-3}$)): The concentration of the light-absorbing solute in the solution. Increasing concentration increases absorbance linearly.

Problem. A student prepares a solution of potassium permanganate ($\ce{KMnO4}$) and places it in a standard 1.00 cm cuvette. The spectrophotometer measures a transmittance of 31.6% at 525 nm. The molar absorptivity of $\ce{KMnO4}$ at 525 nm is $\varepsilon = 2{,}400 \text{ L mol}^{-1}\text{cm}^{-1}$. Calculate (a) the absorbance and (b) the molar concentration of $\ce{KMnO4}$ in the solution.

Answer. The absorbance is 0.500 and the concentration of KMnO₄ is 2.08 × 10⁻⁴ mol L⁻¹.

• Concentration and Molarity — Beer's Law requires molar concentration (c) as an input; understanding how to calculate mol/L is a prerequisite for applying A = εlc correctly.
• Properties of Solutions — Spectrophotometry measures properties of liquid solutions; understanding solubility, homogeneity, and solution preparation underpins valid Beer-Lambert measurements.
• Electromagnetic Spectrum and Electron Transitions — Light absorption occurs because photons promote electrons to higher energy levels; understanding energy transitions explains why different compounds absorb at different wavelengths (λ_max).
• Logarithms in Chemistry — Absorbance is calculated as A = −log₁₀(T); fluency with base-10 logarithms is essential to convert transmittance readings into absorbance values.