4.1 Beer-Lambert law and absorption spectroscopy

Beer-Lambert Law and Absorption Spectroscopy

The Beer-Lambert law connects how much light a sample absorbs to the sample's concentration and the distance light travels through it. This relationship is the foundation of quantitative absorption spectroscopy, letting you figure out how much of a substance is present just by measuring how light passes through a solution.

Beer-Lambert Law Components

The core equation is:

$A = \varepsilon b c$

Each variable has a specific physical meaning:

• A (Absorbance) — a dimensionless number that measures how much light the sample absorbs. Higher A means more light absorbed.
• $\varepsilon$ (Molar attenuation coefficient) — a constant that describes how strongly a substance absorbs light at a particular wavelength. Units are $\text{L mol}^{-1} \text{cm}^{-1}$. Every compound has its own $\varepsilon$ value at each wavelength, so this acts like a fingerprint for absorption strength.
• $b$ (Path length) — the distance light travels through the sample, usually set by the width of the cuvette. Measured in cm (typically 1 cm for standard cuvettes).
• $c$ (Concentration) — the molar concentration of the absorbing species in solution, in $\text{mol L}^{-1}$.

Two other quantities show up constantly alongside this equation:

• $I_0$ — the intensity of light entering the sample (incident light)
• $I$ — the intensity of light that makes it through the sample (transmitted light)
• Transmittance (T) — the fraction of light that passes through: $T = I / I_0$. A value of 1 means all light passes through; 0 means none does.

Absorbance, Transmittance, and Concentration

Absorbance and transmittance are related logarithmically:

$A = -\log_{10}(T)$

This means that as absorbance increases linearly, transmittance drops exponentially. For example, $A = 1$ corresponds to $T = 0.1$ (only 10% of light transmitted), while $A = 2$ corresponds to $T = 0.01$ (only 1%).

When the Beer-Lambert law holds, absorbance is directly proportional to concentration. Double the concentration, double the absorbance. But this linear relationship breaks down at high concentrations (roughly above $A \approx 1.0$). Common causes of deviation include:

• Molecular interactions at high concentrations (solute molecules start affecting each other's absorption behavior)
• Stray light reaching the detector without passing through the sample
• Polychromatic radiation — if the monochromator isn't selecting a narrow enough wavelength band, the law's assumption of monochromatic light is violated

Concentration Calculations with Beer-Lambert Law

To find an unknown concentration from an absorbance measurement, rearrange the equation:

$c = \frac{A}{\varepsilon b}$

Here's the step-by-step process:

1. Choose the wavelength where your analyte absorbs most strongly (its $\lambda_{\text{max}}$). This maximizes sensitivity.
2. Look up or determine $\varepsilon$ for your substance at that wavelength.
3. Measure the path length $b$ of your cuvette (usually 1 cm).
4. Record the absorbance $A$ using a spectrophotometer, after blanking against a reference.
5. Plug values into $c = A / (\varepsilon b)$ and solve.

For reliable results, keep $A$ in the range of roughly 0.1 to 1.0. Below 0.1, noise dominates. Above 1.0, deviations from linearity become significant. If your reading falls outside this range, dilute (or concentrate) and re-measure.

Always check that your units are consistent: $\varepsilon$ in $\text{L mol}^{-1} \text{cm}^{-1}$, $b$ in cm, and $c$ in $\text{mol L}^{-1}$.

Absorption Spectroscopy Setup

A typical UV-Vis spectrophotometer has four main components arranged in sequence:

1. Light source — a tungsten-halogen lamp covers the visible range (~350–800 nm), while a deuterium lamp covers the UV range (~190–350 nm). Many instruments switch between the two automatically.
2. Monochromator — isolates a narrow band of wavelengths from the source. This is what lets you select a specific wavelength for measurement.
3. Sample holder (cuvette) — holds the solution in the light path. Glass cuvettes work for visible wavelengths; quartz cuvettes are needed for UV work because glass absorbs UV light.
4. Detector — measures the intensity of transmitted light. Common types include photomultiplier tubes (very sensitive) and photodiode arrays (can measure many wavelengths simultaneously).

To collect good data:

• Prepare a blank (solvent only, no analyte) and use it to zero the instrument. This corrects for absorption by the solvent and cuvette.
• Handle cuvettes by the frosted sides only; fingerprints on the optical faces scatter light and introduce error.
• For quantitative work with unknowns, build a calibration curve by measuring absorbance for several standards of known concentration. Plot $A$ vs. $c$, confirm linearity, then read your unknown's concentration from the curve.
• Control temperature if your analyte's absorption is temperature-sensitive.
• Watch for interferences from other species in the sample that absorb at the same wavelength.