Spectroscopy: fundamentals of rotational, vibrational & electronic spectra; Beer-Lambert law

Spectroscopy Fundamentals & Beer–Lambert Law

1. What spectroscopy is

Molecules absorb/emit electromagnetic radiation at discrete energies matching gaps between quantized levels:

where $\bar{
u} = 1/\lambda$ is the wavenumber (cm⁻¹, the unit of IR spectra).

Energy ladder: electronic (largest gap) > vibrational > rotational (smallest). So UV-Vis photons drive electronic jumps, IR drives vibrations, microwaves drive rotations.

2. The three regions compared

Type	Region	Transition	Energy order	Selection rule / condition
Rotational (microwave)	$\lambda \sim$ mm–cm	$J \to J+1$ rotation levels	smallest	molecule must have permanent dipole (HCl yes, N₂/O₂ no)
Vibrational (IR)	4000–400 cm⁻¹	$v = 0 \to 1$ stretching/bending	medium	dipole moment must change during vibration (CO₂ asymmetric stretch active, symmetric stretch inactive)
Electronic (UV-Vis)	200–800 nm	HOMO → LUMO ($\sigma\to\sigma^\*$, $n\to\pi^\*$, $\pi\to\pi^\*$)	largest	spin-allowed ($\Delta S = 0$); conjugation shifts $\lambda_{max}$ up

Rotational lines are equally spaced: separation $= 2B$, giving bond length. Vibrational frequency follows Hooke's law: $\bar{
u} = \frac{1}{2\pi c}\sqrt{k/\mu}$— stronger bond (bigger$k$) and lighter atoms (smaller reduced mass$\mu$) absorb at higher wavenumber.

3. Beer–Lambert law (quantitative analysis)

Absorbance is proportional to concentration and path length:

• $A$ = absorbance (unitless), $I_0, I$ = incident/transmitted intensity.
• $\varepsilon$ = molar absorptivity (L·mol⁻¹·cm⁻¹), $c$ = concentration (mol/L), $l$ = path length (cm, usually 1).
• Transmittance $T = I/I_0$; $A = -\log_{10}T$.

Numericals:

• $\varepsilon = 5000$, $c = 2 \times 10^{-5}$ M, $l = 1$ cm → $A = 0.10$.
• $A = 0.699$ means $T = 10^{-0.699} \approx 0.20$ (20% transmitted).
• Calibration curve: plot $A$ vs known $c$, read unknown from line — standard exam numerical.

Deviations: law fails at high concentration (> 0.01 M, molecular interactions), if light isn't monochromatic, or if the analyte dissociates/associates.

Exam Points

• Energy-order + region + selection rule table (3–5 marks, frequently asked as "distinguish rotational, vibrational, electronic spectra").
• Beer–Lambert statement, formula, one numerical + calibration-curve method (5 marks favourite); remember $A$ uses log base 10, and units of $\varepsilon$.