⚛Molecular Physics Unit 7 Review

The electromagnetic spectrum provides the toolkit for probing molecular structure. Different regions of the spectrum carry different photon energies, and those energies map directly onto the spacing between a molecule's quantized energy levels. Matching the right spectral region to the right type of molecular motion is the foundation of all spectroscopy.

Electromagnetic Radiation Characteristics

Electromagnetic radiation travels as oscillating electric and magnetic fields. You can describe any photon by three interrelated quantities: wavelength ( $\lambda$ ), frequency ( $\nu$ ), and energy ( $E$ ).

The core relationships are:

$c = \lambda \nu$

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

where $h$ is Planck's constant ( $6.626 \times 10^{-34}$ J·s) and $c$ is the speed of light ( $3.00 \times 10^{8}$ m/s).

Energy is directly proportional to frequency: double the frequency, double the energy.
Energy is inversely proportional to wavelength: halve the wavelength, double the energy.

A photon at $6 \times 10^{14}$ Hz carries twice the energy of one at $3 \times 10^{14}$ Hz. Equivalently, a 500 nm photon carries twice the energy of a 1000 nm photon.

The energy of the photon determines which region of the spectrum it belongs to and, therefore, which type of molecular transition it can drive.

Electromagnetic Spectrum and Molecular Energy, Electromagnetic Energy | Chemistry

Molecular Energy Levels and Transitions

Molecules don't have a continuous range of allowed energies. Instead, they occupy discrete, quantized energy levels that emerge as solutions to the Schrödinger equation for the molecular system. These levels fall into three broad categories, each with a characteristic energy scale:

Rotational levels (lowest energy, smallest spacing)
Vibrational levels (intermediate energy)
Electronic levels (highest energy, largest spacing)

A transition between two levels happens in one of two ways:

Absorption: the molecule absorbs a photon whose energy exactly matches the gap $\Delta E$ between a lower and a higher level, promoting the molecule upward.
Emission: the molecule drops from a higher level to a lower one, releasing a photon with energy $\Delta E$ .

The condition for both processes is the same:

$\Delta E = h\nu$

This equation is what links molecular structure to the electromagnetic spectrum. A molecule with closely spaced rotational levels absorbs low-energy microwave photons; one undergoing an electronic transition absorbs high-energy UV or visible photons.

Electromagnetic Spectrum and Molecular Energy, Photon Energies and the Electromagnetic Spectrum | Physics

Regions of the Electromagnetic Spectrum

Each spectral region probes a different type of molecular motion. The table below summarizes the key regions from low to high energy:

Region	Wavelength Range	Transition Type	Spectroscopic Example
Microwave	~1 mm to ~1 m	Rotational	Microwave spectroscopy of small polar molecules (e.g., HCl, OCS)
Infrared	~700 nm to ~1 mm	Vibrational	IR spectroscopy to identify functional groups (e.g., C=O stretch near 1700 cm⁻¹)
Visible	~400 to ~700 nm	Electronic	Absorption spectroscopy of conjugated π-systems and transition-metal complexes
Ultraviolet	~10 to ~400 nm	Electronic (higher energy)	UV-Vis spectroscopy of aromatic compounds and lone-pair → π* transitions
X-ray / Gamma	< ~10 nm	Core-electron / nuclear	X-ray diffraction for crystal structures; gamma spectroscopy for nuclear transitions
A few things to note:

The infrared and microwave regions probe nuclear motion (vibrations and rotations), while the visible and UV regions probe electronic rearrangements.
X-rays and gamma rays carry enough energy to ionize molecules or excite nuclear transitions, which is why they sit at the extreme high-energy end.
The boundaries between regions are approximate conventions, not hard cutoffs.

Quantized Energy Levels in Detail

Quantum Numbers

Each type of molecular energy level is labeled by its own set of quantum numbers:

Electronic states are described by quantum numbers analogous to atomic ones ( $n, l, m_l, m_s$ ), adapted for molecular orbital symmetry. In molecular spectroscopy, electronic states are often labeled by term symbols (e.g., $^1\Sigma_g^+$ ) rather than individual electron quantum numbers.
Vibrational states are labeled by the vibrational quantum number $v = 0, 1, 2, \ldots$ for each normal mode. For a harmonic oscillator, the energy is $E_v = h\nu_0(v + \tfrac{1}{2})$ , where $\nu_0$ is the classical vibration frequency.
Rotational states are labeled by $J = 0, 1, 2, \ldots$ (total rotational angular momentum). For a rigid rotor, $E_J = BJ(J+1)$ , where $B$ is the rotational constant. Symmetric and asymmetric tops introduce additional quantum numbers $K$ and $M$ .

The total energy of a molecular state is approximately the sum of its electronic, vibrational, and rotational contributions:

$E_{\text{total}} \approx E_{\text{elec}} + E_{\text{vib}} + E_{\text{rot}}$

This separation works because the three energy scales differ by roughly orders of magnitude: $E_{\text{elec}} \gg E_{\text{vib}} \gg E_{\text{rot}}$ . This hierarchy is the physical basis of the Born-Oppenheimer approximation.

Spectroscopic Techniques

Different techniques exploit different parts of this energy-level structure:

Absorption spectroscopy measures which wavelengths a sample removes from a broadband light source, revealing $\Delta E$ values directly.
Emission spectroscopy collects light emitted by excited molecules (e.g., after thermal or electrical excitation) to map out energy-level spacings.
Raman spectroscopy measures the shift in wavelength of scattered light. The molecule gains or loses a quantum of vibrational (or rotational) energy during the scattering event, so the frequency shift equals the energy-level spacing.

Not every transition between levels is allowed. Selection rules, derived from the symmetry of the wavefunctions and the transition dipole moment integral, dictate which transitions can occur with appreciable probability. For example, in a molecule with a center of symmetry, transitions between states of the same parity (both gerade or both ungerade) are electric-dipole forbidden. You'll encounter specific selection rules for rotational, vibrational, and electronic spectroscopy throughout the rest of this unit.

Connecting Photon Energy to Spectra

When you look at a molecular spectrum, every absorption or emission line corresponds to a specific $\Delta E$ between two quantized levels. The position of that line on the frequency (or wavelength, or wavenumber) axis tells you the size of the energy gap.

Wavenumber ( $\tilde{\nu}$ , in cm⁻¹) is commonly used in vibrational and rotational spectroscopy because it's directly proportional to energy:

$E = hc\tilde{\nu}$

This makes wavenumber a convenient "energy unit" when reading IR or microwave spectra.

The key takeaway: the electromagnetic spectrum isn't just a classification scheme. It's a direct map between photon properties and the internal energy structure of molecules. Knowing where a transition falls on the spectrum immediately tells you what kind of molecular motion is involved, and measuring the exact frequency gives you quantitative information about bond strengths, bond lengths, and electronic structure.

⚛Molecular Physics Unit 7 Review