3.2–3.13: Electromagnetic Radiation to The Bohr Atom and Beyond

3.2 Electromagnetic Radiation

Electromagnetic (EM) radiation is radiant energy transmitted through space as oscillating electric and magnetic fields. In vacuum, all EM radiation travels at the speed of light, denoted by $c\approx 2.998\times 10^8\ \text{m s}^{-1}.$
EM radiation spans a spectrum that includes radio waves, microwaves, infrared, visible light, ultraviolet, X‑rays, and gamma rays. These forms differ only in wavelength and frequency since they share the same speed in vacuum.
Wave properties (general):
- Wavelength, $\lambda$ (meters): distance between successive peaks
- Frequency, $\nu$ (Hz, s⁻¹): number of oscillations per unit time
- Amplitude: vertical height of the wave (half of peak‑to‑trough height for a simple sinusoid)
- Speed, $v$ : distance traveled per unit time; for waves, $v=\lambda\nu$
For EM radiation in vacuum: $c=\lambda\nu$ . When using $c\approx 3.00\times 10^8\ \text{m s}^{-1}$ as the constant, the product of wavelength and frequency is fixed.
Energy relation for EM radiation: photons carry energy given by $E=h\nu=\frac{hc}{\lambda}$ where $h=6.626\times 10^{-34}\ \text{J s}$ is Planck’s constant.
Energy scales with frequency and inversely with wavelength: $E\propto\nu\quad\text{and}\quad E\propto\frac{1}{\lambda}.$ Thus higher frequency (shorter wavelength) photons carry more energy.
Practical examples and data points:
- Radio vs. microwave vs. infrared vs. visible vs. ultraviolet vs. X‑rays vs. gamma: wavelengths span many orders of magnitude.
- Visible spectrum: perceived by the eye, wavelengths roughly from about $7\times 10^{-7}\,\text{m}$ (700 nm, red) to $4\times 10^{-7}\,\text{m}$ (400 nm, violet).
- Ultraviolet (UV) radiation has wavelengths shorter than ~ $350\,\text{nm}$ ; ozone layer absorbs some UV (<~350 nm) protecting life.
Demonstrative examples:
- FM radio station example: a station at $101.1\text{ MHz}$ has wavelength $\lambda = c/\nu\approx 2.97\times 10^{-1}\text{ m}$ (about 29.7 cm).
- Radar gun example: radar at $35.5\ \text{GHz}$ corresponds to a wavelength of roughly $8.45\ \text{mm}$ .
Visualization relationships:
- As frequency increases, wavelength decreases, and vice versa, with speed c remaining constant for EM radiation in vacuum.
Common relationships (summarized):
- $c=\lambda\nu$ , $E=h\nu=\dfrac{hc}{\lambda}$ , $E\propto\nu\propto\dfrac{1}{\lambda}.$
Practical consequence: higher frequency/shorter wavelength EM radiation can deposit more energy into matter and cause different interactions (heating, ionization, imaging, etc.).
Connections to atomic structure: understanding EM radiation is essential to interpreting atomic spectra, which arise from transitions between quantized energy levels in atoms.

3.3 Atomic Spectra

Light emitted by atoms can form a line spectrum (discrete wavelengths) rather than a continuous spectrum when atoms are excited (e.g., hydrogen discharge). A prism reveals individual lines at characteristic wavelengths.
Balmer series (visible region): transitions to n=2 produce lines in the visible; Balmer’s empirical formula predicted the wavelengths of these lines.
- Balmer (1885) found a simple equation for the visible lines of hydrogen: frequencies follow a simple pattern.
Rydberg equation (general): given by
- $\frac{1}{\lambda}=R\left(\frac{1}{n1^2}-\frac{1}{n2^2}\right)$
- where $n1$ is the lower energy level, n2>n_1 is the upper level, and $R\approx 1.097\times 10^7\ \text{m}^{-1}$ is the Rydberg constant.
Bohr’s model (1900s) introduced energy quantization to explain line spectra:
- Energy of an electron in a hydrogen-like atom: $En = -\frac{RH c h}{n^2} = -\frac{Z^2 R\infty h c}{n^2}$ where $n=1,2,3,\dots$ and $Z$ is the nuclear charge (1 for H). Here $RH$ is the Rydberg constant for hydrogen and $R_\infty = 1.09737\times 10^7\ \text{m}^{-1}$ .
- Transitions between energy levels produce photons with energies matching the energy difference: $\Delta E = E{nf}-E{ni} = -\frac{RH c h}{nf^2}+\frac{RH c h}{ni^2} = -RH c h\left(\frac{1}{nf^2}-\frac{1}{n_i^2}\right)$
- Wavelength of emitted/absorbed photon from Bohr transitions: $hc\,/\,\Delta E=\lambda$ or equivalently
 \Delta E = h c\left(\frac{1}{\lambda}
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Hydrogen emission lines and series:
- Lyman series: transitions to n_f=1 (ultraviolet region).
- Balmer series: transitions to nf=2 (visible region). Notable lines: 656 nm (red, ni=3→2); 434 nm (blue, ni=6→2); 410 nm (violet, ni=6→2).
- Pfund, etc., correspond to other final levels (e.g., to n_f=5 for Pfund).
Absorption vs emission:
- Atoms absorb light at specific energies, producing an absorption spectrum with dark lines at the same wavelengths as emission lines.
Important historical figures:
- Balmer (Balmer series for hydrogen in the visible range)
- Rydberg (reformulated Balmer with the Rydberg equation)
- Bohr (quantized circular orbits for hydrogen)
Significance:
- The line spectra of atoms are directly related to the electronic structure and energy level spacings; line spectra provide a fingerprint for elemental identity.
Summary connection to quantum theory:
- The line spectra provided early, compelling evidence for energy quantization in atoms and the need for quantum descriptions beyond classical physics.

3.4 Quantum Theory: Blackbody Radiation and the Birth of Quantum Mechanics

Blackbody radiation problem:
- A blackbody is an idealized emitter/absorber with a continuous spectrum depending only on temperature.
- Classical physics predicted a problem known as the ultraviolet catastrophe: at short wavelengths, predicted intensities diverge, which contradicts experimental data.
Planck’s hypothesis (1900): energy is quantized; energy is emitted or absorbed in discrete units called quanta with energy $E=h\nu$ (or $E=\dfrac{hc}{\lambda}$ ). This quantization resolves the ultraviolet catastrophe.
Planck’s constant: $h=6.626\times 10^{-34}\ \text{J s}$
The concept of quanta leads to the broader idea that energy exchange occurs in discrete steps, not a smooth continuum.
The photoelectric effect (Einstein, 1905):
- Light can eject electrons from a metal surface when photons have energy above a threshold: E=h\nu>\phi, where $\phi$ is the work function (threshold energy).
- If a photon’s energy exceeds the threshold, the excess energy becomes kinetic energy of the ejected electron: $K.E. = h\nu - \phi$
- For sub-threshold frequencies, no electrons are ejected regardless of light intensity.
- Intensity affects the number of emitted electrons (photon flux), while frequency affects their kinetic energy.
Einstein’s photon model unified Planck’s quantum concept with light behaving as particles (photons) with energy $E=h\nu$ .
Consequences:
- Photons carry energy and momentum; energy scales with frequency; the particle-like behavior of light is established.
- The quantum view extends to matter waves (de Broglie) and leads to quantum mechanics.
Summary takeaway:
- Energy quantization explains blackbody radiation and the photoelectric effect, establishing the dual wave–particle nature of light and the foundation for quantum theory.

3.5 The Bohr Atom

Bohr’s model (1913) for the hydrogen atom: electron moves in circular orbits with only certain allowed radii (quantized orbits).
Key energy equation for the electron in a hydrogen-like atom:
- $En = -\frac{RH c h}{n^2} = -\frac{Z^2 R_\infty h c}{n^2}$ , where $n=1,2,3,…\,$ ; the ground state is $n=1$ ; as $n\to\infty$ , the electron is ionized (detached).
Orbital radius:
- $rn = \frac{a0 n^2}{Z}$ with the Bohr radius $a_0\approx 0.529\ \text{Å}$ and $Z$ the nuclear charge.
Energy transitions and line spectra:
- Transitions from a higher-energy orbit (larger radius) to a lower-energy orbit (smaller radius) emit photons with energy equal to the difference in energy between the two levels, giving line spectra.
- Balmer series (to $nf=2$ ) explains visible hydrogen lines; Lyman (to $nf=1$ ) is ultraviolet; Paschen (to $n_f=3$ ) is infrared.
The Balmer series wavelengths (example lines):
- $ni=3\to nf=2$ gives 656 nm (red);
- $ni=4\to nf=2$ gives 486 nm (blue‑green);
- Additional lines at 434 nm and 410 nm correspond to higher levels.
Bohr’s contribution: provided a successful explanation for hydrogen’s emission spectrum and the concept of quantized energy levels.
Limitations: Bohr’s model could not explain spectra of atoms with more than one electron; more advanced quantum theories were required.

3.6 Two Ideas Leading to a New Quantum Mechanics

Wave–particle duality of matter and energy:
- Light shows particle-like properties (photons) and wave-like properties (interference, diffraction).
- Louis de Broglie proposed that particles (e.g., electrons) have wave-like characteristics with wavelength
- $\lambda = \frac{h}{mv}$ for a particle of mass m and velocity v.
Heisenberg uncertainty principle (matrix form):
- $\Delta x\, \Delta p \ge \frac{h}{4\pi}$ where $\Delta p = \Delta(mv)$ is the uncertainty in momentum.
- This bound has profound implications for the simultaneous knowledge of position and momentum.
Standing waves and Bohr’s quantization connection:
- De Broglie argued that Bohr’s allowed orbits can be viewed as standing waves around the nucleus, requiring that the circumference be an integral multiple of the wavelength:
- $2\pi r = n\lambda$ (for circular orbits).
The wave nature of the electron provides a natural basis for quantization and the later development of Schrödinger’s wave mechanics.
The Stern–Gerlach experiment (1926) demonstrated electron spin as a quantum property and motivated the need for a fourth quantum number (spin) to describe electrons.

3.7 Wave Mechanics (Schrödinger)

Wavefunction concept:
- A wavefunction $\Psi(x,y,z,t)$ describes the quantum state of an electron; its magnitude relates to the probability amplitude.
- The magnitude of the wavefunction, or its square, is physically meaningful:
- The probability density is $P(x,y,z,t) = |\Psi(x,y,z,t)|^2$ (for stationary states, time dependence can be separated).
- Wavefunctions are generally complex; the probability is given by $|\Psi|^2 = \Psi\Psi^*$ (multiplication by the complex conjugate).
Quantum numbers and orbitals:
- Schrödinger’s approach uses three quantum numbers to describe orbitals: $n,\ l,\ ml$ , where $n\ge 1$ , $l=0,1,2,…,n-1$ , and $ml=-l,-l+1,…,l-1,l$ .
- Each set (n,l,m_l) defines a spatial distribution called an atomic orbital; the energy is associated with the wavefunction.
Principal, azimuthal, and magnetic quantum numbers:
- Principal: $n$ controls average distance from the nucleus and energy level structure.
- Azimuthal: $l$ shapes the orbital (s, p, d, f designations correspond to $l=0,1,2,3$ ).
- Magnetic: $m_l$ orients the orbital in space relative to a magnetic field.
Shapes of orbitals (brief):
- s orbitals: spherical with no angular nodes (l=0).
- p orbitals: dumbbell-shaped with one nodal plane through the nucleus (l=1).
- d orbitals: more complex shapes with multiple nodal surfaces (l=2).
- f orbitals: even more complex (l=3).
Degeneracy and hydrogen-like atoms:
- In hydrogen (one electron), orbitals with the same principal quantum number n are degenerate (e.g., 2s and 2p have the same energy in hydrogen).
- For multi-electron atoms, orbital energies split due to electron–electron interactions and shielding, so degeneracy is lifted.

3.8 Electron Spin: A Fourth Quantum Number

Stern–Gerlach experiment demonstrated the existence of a quantum property called spin, which takes on two possible orientations for an electron when placed in a magnetic field.
Spin quantum number (m_s): can take values of +\frac{1}{2} (up) or −\frac{1}{2} (down).
Pauli exclusion principle:
- No two electrons in an atom can have the same set of four quantum numbers (n, l, ml, ms).
- This principle explains why an orbital can hold at most two electrons with opposite spins.
Practical consequence: spin pairing governs electron configurations and chemical behavior; Hund’s rule (next section) governs distribution in degenerate orbitals.

3.9 The Shape of Atomic Orbitals

Orbital shapes by angular momentum quantum number l:
- s orbitals (l=0): spherical symmetry; maximum electron density at the nucleus.
- p orbitals (l=1): three degenerate orientations (px, py, pz) with one nodal plane through the nucleus; each has lobes with phase differences.
- d orbitals (l=2): five degenerate orbitals with more complex shapes; typically two nodal surfaces.
- f orbitals (l=3): seven orbitals with even more complex shapes; less central to chemistry but important for heavier elements.
Orbitals per subshell:
- Each subshell with angular momentum quantum number l contains 2l+1 orbitals.
- Example: 2p has 3 orbitals; 3d has 5 orbitals.
Orbital energies and degeneracy:
- In hydrogen, orbitals with the same n are degenerate; in multi‑electron atoms, degeneracy is lifted due to shielding effects and electron–electron repulsions.
Visualizations:
- Orbitals are often drawn as surfaces enclosing ~90% of the electron probability density; nodes indicate regions of zero probability.
Summary designations:
- s, p, d, f designate the orbital types corresponding to l=0,1,2,3 respectively.
Relationship to quantum numbers:
- The combination (n, l, m_l) defines a subshell/orbital; the energy ordering across the periodic table is determined by the Aufbau principle and orbital energies.

3.10 Multielectron Atoms

Aufbau principle (building up):
- Electrons are added to the lowest‑energy available orbital without violating Pauli exclusion.
- Each orbital holds up to two electrons with opposite spins (↑↓ pairing).
Electron configuration notation and noble gas shorthand:
- Example: Na (Z=11) is [Ne]3s¹; Ne (Z=10) has [Ne] as a closed core; the valence configuration of Na is 3s¹.
- This shorthand highlights valence electrons responsible for chemical behavior.
Hund’s rule:
- In degenerate orbitals (same energy), electrons occupy empty orbitals with parallel spins before pairing.
- Example: Carbon (Z=6) has 1s² 2s² 2p²; with Hund’s rule, the 2p electrons singly occupy degenerate 2p orbitals with parallel spins before pairing.
Anomalies and exceptions:
- Chromium (Cr) and Copper (Cu) show deviations from the simple Aufbau order (e.g., [Ar] 4s¹ 3d⁵ for some states, or [Ar] 4s¹ 3d¹⁰ in others) due to extra stabilization from half‑ or fully‑filled subshells.
Multielectron shell and subshell energetics:
- The general filling order across the periodic table follows the diagonal rule, with s < p < d < f in many cases, but irregularities occur due to stability of half‑filled or filled subshells and interelectronic interactions.
Common multielectron configuration patterns (highlights):
- 1s, 2s, 2p; 3s, 3p; 4s, 3d, 4p; 5s, 4d, 5p; 6s, 4f, 5d, 6p, etc.; followed by 7s, 5f, 6d, 7p in heavier elements.
Complete electron configuration for mercury (Hg, Z=80) as an example:
- 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d¹⁰ 4p⁶ 5s² 4d¹⁰ 5p⁶ 6s² 4f¹⁴ 5d¹⁰ 6p⁶.
- Using noble gas shorthand: [Xe]6s² 4f¹⁴ 5d¹⁰ 6p⁶.
Practical takeaway:
- Electron configurations explain chemical periodicity, valence electron counts, and many properties of elements.

3.11 Key Terms (glossary-style highlights)

Actinide, alkali metal, alkaline earth metal, halogen, noble gas, transition metal, inner transition metal, covalent bond, covalent radius, electron affinity, ionization energy, ion, isoelectronic, ligand, lattice, octet rule (where relevant), noble gas core notation, valence electrons, etc.
Quantum numbers: n (principal), l (azimuthal), ml (magnetic), ms (spin).
Wavefunction notation: $\Psi$ , probability density $|\Psi|^2$ , and the idea that orbitals are regions of space where electrons are likely to be found.

3.12 Key Equations

Electromagnetic relations:
- $c=\lambda\nu$
- $E=h\nu=\dfrac{hc}{\lambda}$
Hydrogen/ion energy levels (Bohr and Rydberg context):
- $En = -\frac{Z^2 R\infty h c}{n^2}$
- $\Delta E = E{nf}-E{ni} = -\frac{Z^2 R\infty h c}{nf^2}+\frac{Z^2 R\infty h c}{ni^2} = -\frac{Z^2 R\infty h c}{nf^2}+\frac{Z^2 R\infty h c}{ni^2}$
- $\lambda = \frac{hc}{\Delta E}$
- Rydberg form (alternative): $\frac{1}{\lambda}=R\infty\left(\frac{1}{nf^2}-\frac{1}{n_i^2}\right)$
Planck's quantization and the constant:
- $E=h\nu$
- Planck’s constant: $h=6.626\times 10^{-34}\ \text{J s}$
De Broglie relation:
- $\lambda = \frac{h}{mv}$
Heisenberg uncertainty principle:
- $\Delta x\, \Delta p \ge \frac{h}{4\pi}$
Bohr–Rydberg relationships for radii and energy (summary):
- $rn = \dfrac{a0 n^2}{Z},\quad a_0\approx 0.529\text{ Å}$
Orbital counts and degeneracy:
- In a given subshell with angular momentum $l$ , there are $2l+1$ orbitals and each orbital can hold up to 2 electrons.
Electron configuration shorthand:
- For a full shell, use the noble gas preceding the element, e.g., [Ne] for neon.

3.13 Summary and Periodic Trends

The four chemically important types of atomic orbitals: s (l=0), p (l=1), d (l=2), f (l=3).
Quantum numbers provide a framework for understanding orbital energies and spatial distributions; the four quantum numbers are n, l, ml, ms.
The Aufbau principle, Hund’s rule, and the Pauli exclusion principle govern how electrons fill orbitals to form ground-state configurations.
Valence electrons are those in the outermost shell and largely determine chemical behavior.
The periodic table organizes elements by similar valence configurations; groups and periods reflect recurring chemical properties.
Anomalies in electron configurations (e.g., Cr and Cu) illustrate extra stability from half-filled or filled subshells, complicating simple filling order.
Multielectron atoms introduce shielding and electron–electron repulsion that split degeneracies and alter orbital energies compared with hydrogen.
The connection between atomic structure and spectra underpins much of spectroscopy, astronomy, and materials science, including practical applications (e.g., lasers, lighting, fireworks).

Helpful notes and quick references

Visible wavelengths: roughly $400-700\text{ nm}$ ; red ~ $656\text{ nm}$ (Balmer line), violet ~ $410\text{ nm}$ (Balmer line).
Lyman series wavelength example: the lowest-energy Lyman line corresponds to the transition from $ni=2\to nf=1$ ; wavelength around $121-122\text{ nm}$ (ultraviolet region).
Pfund series wavelength example and region: infrared.
For EM radiation interactions with matter, higher frequency (shorter wavelength) generally means higher energy interactions (ionization, excitation).
Key constants to remember (approximate):
- $c\approx 2.998\times 10^8\ \text{m s}^{-1}$
- $h=6.626\times 10^{-34}\ \text{J s}$
- $R_\infty\approx 1.0973\times 10^7\ \text{m}^{-1}$
- $a_0\approx 5.29\times 10^{-11}\ \text{m}$ (Bohr radius)
Practical applications and examples mentioned:
- Radio (101.1 MHz) and radar (35.5 GHz) demonstrate real‑world uses of EM radiation in the radio/microwave regime.
- Atomic spectra underpin spectroscopy, astronomical analysis, and identification of elements in distant objects.
Connections across topics:
- The study of EM radiation leads to Planck’s quantum hypotheses, which explain blackbody radiation and the photoelectric effect.
- Quantum theory and wave mechanics (Schrödinger) provide a comprehensive framework for predicting and understanding the shapes and energies of atomic orbitals, as well as the arrangement of electrons in atoms ( Aufbau, Hund’s rule, Pauli exclusion).
- The distribution of electrons in atoms explains observed emission and absorption spectra, which in turn reveal information about atomic and molecular structure and bonding.