Principles of Spectroscopy and the Beer-Lambert Law

By the end you should be able to: Describe molecular energy levels and types of spectra, state and apply the Beer-Lambert law, and explain its limitations.

What spectroscopy measures

The interaction of electromagnetic radiation with matter. A molecule absorbs a photon only if its energy exactly matches the gap between two of the molecule's energy levels:

\Delta E = h\nu = \frac{hc}{\lambda}

Because those gaps are characteristic of the molecule, the pattern of absorption is a fingerprint. That is the whole basis of analytical spectroscopy: identify what is there from which energies are absorbed, and how much is there from how strongly.

Molecular energy levels

A molecule stores energy in three separable ways, on wildly different scales — and that separation is what makes different spectroscopies independent techniques rather than one messy one.

Type	Energy	Region	Technique
Electronic	1–10 eV	UV-visible	UV-Vis spectroscopy
Vibrational	0.05–0.5 eV	infrared	IR spectroscopy
Rotational	<0.01 eV	microwave	microwave spectroscopy

E_{total} = E_{electronic} + E_{vibrational} + E_{rotational}

The scales differ by roughly a factor of a hundred at each step. So an electronic transition inevitably drags vibrational and rotational changes along with it, which is why UV-Vis bands in solution are broad and featureless while gas-phase IR spectra show fine rotational structure.

Types of spectra

Absorption spectrum — radiation passed through the sample; the absorbed wavelengths appear as dark lines or bands. The common analytical case.
Emission spectrum — the sample is excited and the emitted wavelengths recorded. Flame photometry, atomic emission.
Continuous, line and band spectra — a hot solid gives a continuum, atoms give sharp lines, and molecules give bands, because each electronic level carries a ladder of vibrational and rotational sublevels.

The Beer-Lambert law

Two separate observations combined.

Lambert's law — each successive layer of a medium absorbs an equal fraction of the light entering it, so intensity falls exponentially with path length.

Beer's law — absorption is proportional to the number of absorbing molecules, hence to concentration.

Together:

\boxed{\,A = \log_{10}\frac{I_0}{I} = \varepsilon c l\,}

$A$ — absorbance, dimensionless
$I_0, I$ — incident and transmitted intensity
$\varepsilon$ — molar absorptivity (molar extinction coefficient), L mol⁻¹cm⁻¹, a constant of the substance at a given wavelength
$c$ — concentration, mol L⁻¹
$l$ — path length, cm

Transmittance is the alternative measure:

T = \frac{I}{I_0}, \qquad \%T = 100T, \qquad A = -\log_{10}T

Why absorbance and not transmittance. Absorbance is linear in concentration; transmittance is logarithmic in it. A calibration graph of $A$ against $c$ is a straight line through the origin, and that is what makes quantitative analysis straightforward.

Useful landmarks: $A = 0$ means 100% transmitted, $A = 1$ means 10% transmitted, $A = 2$ means 1%. Each unit of absorbance is another factor of ten removed.

Molar absorptivity measures how strongly a species absorbs at that wavelength. Values above about $10^4$ indicate a strongly allowed transition; below $10^3$ , a weak or forbidden one. It is why measurements are made at $\lambda_{max}$ — the wavelength of maximum absorption — where sensitivity is greatest and where the curve is flattest, so small wavelength errors matter least.

Limitations of the law

The law is an idealisation, and it fails predictably:

High concentration, above about 0.01 M. Molecules interact, perturbing each other's energy levels and altering $\varepsilon$ .
Chemical change with concentration — association, dissociation, or a shift in an acid-base equilibrium — because the absorbing species itself changes.
Non-monochromatic light. $\varepsilon$ varies with wavelength, so a broad band averages several values and bends the calibration line.
Scattering by turbid or colloidal samples, which removes light without absorbing it and is counted as absorbance.
Fluorescence from the sample reaching the detector, which under-reports absorbance.

The practical consequence: dilute a strongly absorbing sample into range rather than trusting the law outside it. Working absorbance is usually kept between 0.1 and 1.0, where the relation is reliable and the photometric error is smallest.