Premed · Premed · Physics 2
Lecture 22: Atomic Structure and Spectra
Physics II — Electromagnetism, Optics & Modern Physics
Learning Objectives
By the end of this lecture, students will be able to:
- Describe the historical models of the atom (Thomson, Rutherford, Bohr)
- Apply the Bohr model to calculate energy levels, radii, and spectral lines of hydrogen
- Explain the quantum mechanical model of the atom and the meaning of quantum numbers
- Describe atomic spectra (emission and absorption) and their relationship to energy levels
- Explain the Pauli exclusion principle and its role in determining electron configurations
Lecture Content
I. Early Atomic Models
Thomson's "plum pudding" model (1897) envisioned the atom as a sphere of positive charge with electrons embedded throughout, like plums in a pudding. While it accounted for the existence of electrons, it explained little else.
Rutherford's nuclear model (1911) emerged from the famous gold foil experiment, in which alpha particles were directed at a thin gold foil. Most alpha particles passed straight through, but a few were deflected at large angles, and some even bounced back. Rutherford concluded that the atom has a tiny, dense, positively charged nucleus surrounded by distant electrons. However, classical physics predicted that orbiting electrons should continuously radiate energy and spiral into the nucleus within about 10^-11 seconds, contradicting the obvious stability of atoms.
Bohr's model (1913) resolved this stability problem by introducing bold new postulates.
II. The Bohr Model of Hydrogen
Bohr proposed three postulates. First, electrons orbit the nucleus in specific allowed circular orbits without radiating energy. Second, the angular momentum of these orbits is quantized: L = n hbar = nh/(2 pi), where n = 1, 2, 3, ... Third, electrons emit or absorb photons only when transitioning between allowed orbits, with the photon energy equal to the difference between the energy levels: E_photon = |E_f - E_i| = hf.
For hydrogen (Z = 1), these postulates yield precise predictions. The orbital radii are r_n = n^2 a_0, where a_0 = 0.0529 nm is the Bohr radius. The energy levels are E_n = -13.6 eV / n^2. The ground state (n = 1) has E_1 = -13.6 eV, the first excited state (n = 2) has E_2 = -3.4 eV, and at n = infinity the electron is free with E = 0. The ionization energy of hydrogen is therefore 13.6 eV.
The wavelengths of photons emitted during transitions follow the formula 1/lambda = R_H (1/n_f^2 - 1/n_i^2), where R_H = 1.097 x 10^7 m^-1 is the Rydberg constant. Different spectral series correspond to different final states. The Lyman series (n_f = 1) falls in the ultraviolet. The Balmer series (n_f = 2) produces visible lines, including the red H-alpha line at 656 nm. The Paschen series (n_f = 3) lies in the infrared, with the Brackett and Pfund series at even longer wavelengths.
The Bohr model has important limitations. It works only for hydrogen-like atoms with a single electron. It cannot explain fine structure in spectral lines, the Zeeman effect, or the behavior of multi-electron atoms, and it provides no fundamental explanation for why angular momentum should be quantized.
<image>Panel A: The Bohr model energy level diagram for hydrogen. Horizontal lines represent energy levels: n = 1 (E = -13.6 eV) at the bottom, n = 2 (E = -3.4 eV), n = 3 (E = -1.51 eV), n = 4 (E = -0.85 eV), and n = infinity (E = 0, ionization) at the top. Downward arrows represent photon emission during transitions: the Lyman series (to n = 1, UV, shown in purple), the Balmer series (to n = 2, visible, shown in different colors: red for H-alpha 3→2, blue-green for H-beta 4→2, etc.), and the Paschen series (to n = 3, IR, shown in red). Panel B: The corresponding emission spectrum showing discrete bright lines against a dark background, with the Balmer series lines visible at their respective wavelengths (656 nm, 486 nm, 434 nm, 410 nm).</image>
III. The Quantum Mechanical Model
The Bohr model was superseded by the full quantum mechanical treatment based on the Schrodinger equation (1926). In this framework, electrons are described not by precise orbits but by wave functions (psi). The square of the wave function, |psi|^2, gives the probability density of finding the electron at a given location. Solving the Schrodinger equation for hydrogen yields the same quantized energy levels as the Bohr model (E_n = -13.6/n^2 eV) but additionally provides spatial probability distributions that describe the shapes of electron orbitals.
IV. Quantum Numbers
Each electron state is described by four quantum numbers. The principal quantum number (n = 1, 2, 3, ...) determines the energy for hydrogen and the average distance from the nucleus, defining the shell (K, L, M, N, ...). The orbital angular momentum quantum number (l = 0, 1, 2, ..., n-1) determines the shape of the orbital: l = 0 gives an s orbital (spherical), l = 1 gives a p orbital (dumbbell-shaped), l = 2 gives a d orbital, and l = 3 gives an f orbital. The magnetic quantum number (m_l = -l, -l+1, ..., 0, ..., l-1, l) determines the orientation of the orbital in space, with (2l + 1) possible values for each l. The spin quantum number (m_s = +1/2 or -1/2) represents the intrinsic angular momentum of the electron (spin up or spin down), discovered through the Stern-Gerlach experiment. The total number of electron states in shell n is 2n^2.
V. The Pauli Exclusion Principle and Electron Configurations
The Pauli exclusion principle states that no two electrons in an atom can have the same set of four quantum numbers. This means each orbital (defined by n, l, and m_l) can hold at most 2 electrons with opposite spins.
Electron configurations describe how electrons fill orbitals. The Aufbau principle dictates that electrons fill the lowest energy orbitals first. Hund's rule states that within a subshell, electrons occupy different orbitals with parallel spins before pairing up. The filling order is 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, and so on. The shell capacities are 2 for n = 1, 8 for n = 2, 18 for n = 3, and 32 for n = 4.
The periodic table is organized according to electron configuration. Elements in the same column have similar outer electron configurations, which gives them similar chemical properties. The table is divided into the s-block, p-block, d-block (transition metals), and f-block (lanthanides and actinides).
<image>Panel A: Shapes of atomic orbitals. The 1s orbital is a sphere centered on the nucleus. The 2p orbitals are three dumbbell-shaped lobes oriented along the x, y, and z axes (p_x, p_y, p_z). The 3d orbitals show four-lobed cloverleaf shapes in various orientations. Each orbital is shown as a probability density surface (boundary surface containing ~90% of the electron probability). Panel B: An energy level diagram for a multi-electron atom showing the filling order. The 1s level is lowest, followed by 2s, 2p, 3s, 3p, 4s, 3d, 4p, etc. Arrows (up and down) represent electrons filling each orbital according to the Aufbau principle, Hund's rule, and the Pauli exclusion principle. The electron configuration of iron (Fe, Z = 26) is used as an example: 1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^6.</image>
VI. Emission and Absorption Spectra
An emission spectrum is produced when excited atoms de-excite and emit photons. It appears as bright lines on a dark background, and each element has a unique emission spectrum that serves as a fingerprint for identification. The photon energy for each line is E = hf = E_i - E_f.
An absorption spectrum is produced when light passes through a gas of atoms. The atoms absorb photons at exactly the same frequencies they would emit, creating dark lines (missing wavelengths) in an otherwise continuous spectrum. The Fraunhofer lines in the solar spectrum reveal the composition of the sun's atmosphere.
These spectra have wide-ranging applications. Spectroscopic identification of elements is used in astronomy, forensics, and chemistry. Flame tests produce characteristic colors from atomic emission (sodium gives yellow, copper gives green, lithium gives red). In medicine, pulse oximetry uses the absorption spectra of oxygenated and deoxygenated hemoglobin. Astronomical redshift, the Doppler shifting of spectral lines, reveals the velocities of stars and galaxies. X-ray spectra arise from high-energy transitions in inner electron shells. Characteristic X-rays are specific to each element (Moseley's law) and are exploited in X-ray fluorescence (XRF) for elemental analysis.
