Structure of an Atom

The classical view of the atom as an indivisible sphere, postulated by John Dalton in 1808, was challenged toward the close of the nineteenth century by a succession of electrical discharge experiments proving the existence of subatomic particles.

• The Electron (Cathode Rays): In 1897, J.J. Thomson investigated electrical discharge through rarefied gases in Crookes tubes at low pressure (~0.01 mm Hg) and high voltage (~10,000 V). Cathode rays traveled in straight lines, cast sharp shadows of opaque objects, exerted mechanical force on paddle wheels, and deflected toward the positive plate in electric and magnetic fields. Thomson determined their universal charge-to-mass ratio: \[\frac{e}{m_e} = 1.758820 \times 10^{11}\text{ C/kg}\] Robert A. Millikan (1909) determined the fundamental charge of the electron via the Oil Drop Experiment: \(e = -1.602176 \times 10^{-19}\text{ C}\). Consequently, the rest mass of the electron was calculated as: \[m_e = \frac{e}{e/m_e} = \frac{1.602176 \times 10^{-19}\text{ C}}{1.758820 \times 10^{11}\text{ C/kg}} = 9.109383 \times 10^{-31}\text{ kg} \approx \frac{1}{1837}\text{ u}\]
• The Proton (Anode / Canal Rays): E. Goldstein (1886) utilized a perforated cathode in discharge tubes and observed canal rays traveling toward the cathode. Unlike cathode rays, their \(e/m\) ratio depended upon the residual gas, reaching a maximum value when hydrogen was used: \(e/m = 9.58 \times 10^7\text{ C/kg}\). The particle was identified as the proton: charge \(+1.6022 \times 10^{-19}\text{ C}\), mass \(m_p = 1.67262 \times 10^{-27}\text{ kg} = 1.00727\text{ u}\).
• The Neutron: James Chadwick (1932) bombarded thin sheets of beryllium with high-velocity \(\alpha\)-particles and detected neutral penetrating radiation consisting of neutral particles with mass slightly greater than a proton: \[^9_4\text{Be} + ^4_2\alpha \rightarrow ^{12}_6\text{C} + ^1_0\text{n}\] Mass of neutron \(m_n = 1.67493 \times 10^{-27}\text{ kg} = 1.00866\text{ u}\).

Thomson (1898) proposed that an atom is a positively charged spherical cloud (~\(10^{-10}\text{ m}\) radius) with electrons embedded like raisins in a pudding. This model could not explain large-angle scattering of charged particles.

In 1911, Ernest Rutherford, Hans Geiger, and Ernest Marsden directed a collimated beam of \(5.5\text{ MeV}\) \(\alpha\)-particles (\(^4_2\text{He}^{2+}\)) onto an ultra-thin gold foil (~\(100\text{ nm}\) thick) surrounded by a fluorescent zinc sulfide (ZnS) screen:

• Observations: Most \(\alpha\)-particles (~99.9%) passed through undeflected; a small fraction (~1 in 8,000) was deflected by angles \(\theta > 90^\circ\); and extremely rarely (~1 in 20,000), an \(\alpha\)-particle bounced almost directly backward (\(\theta \approx 180^\circ\)).
• Conclusions & Nuclear Model: Most of the atom is empty space. All positive charge and virtually all mass are concentrated in an extraordinarily minute, dense central core termed the nucleus (radius \(r_N \sim 10^{-15}\text{ m}\) compared to atomic radius \(r_A \sim 10^{-10}\text{ m}\); volume fraction \(\sim 10^{-15}\)). Electrons orbit the nucleus in circular paths, held by electrostatic Coulombic attraction.
• Limitations of Rutherford Model:
  1. Instability under Classical Electrodynamics: According to Maxwell electromagnetic theory, an accelerating charged particle (in circular orbit) must continuously radiate energy. The electron would lose kinetic energy, spiral inward in \(\sim 10^{-8}\text{ s}\), and collapse into the nucleus.
  2. Failure to Explain Atomic Spectra: If the spiral is continuous, the emitted radiation must form a continuous spectrum; experimentally, atoms emit sharp, discrete line spectra.

To overcome the limitations of classical mechanics, physicists investigated the interaction of electromagnetic radiation with matter, uncovering the dual wave-particle nature of light.

James Clerk Maxwell (1870) demonstrated that accelerating electric charges generate oscillating electric and magnetic fields perpendicular to each other and to the direction of propagation. All electromagnetic waves travel through vacuum at the speed of light \(c = 2.997925 \times 10^8\text{ m/s} \approx 3.0 \times 10^8\text{ m/s}\).

• Wavelength (\(\lambda\)): Distance between two adjacent crests or troughs. Units: m, cm, nm (\(10^{-9}\text{ m}\)), or Ångström (\(1\text{ \AA} = 10^{-10}\text{ m} = 0.1\text{ nm}\)).
• Frequency (\(\nu\)): Number of wave cycles passing a fixed point per second. Units: Hertz (\(\text{Hz} = \text{s}^{-1}\)). \[c = \nu \lambda \implies \nu = \frac{c}{\lambda}\]
• Wavenumber (\(\bar{\nu}\)): Number of wavelengths per unit length. Units: \(\text{m}^{-1}\) or \(\text{cm}^{-1}\). \[\bar{\nu} = \frac{1}{\lambda} = \frac{\nu}{c}\]

Classical wave theory failed to explain the spectral distribution of thermal radiation emitted by an ideal black body—an ideal absorber and radiator of all electromagnetic frequencies. Classical physics (Rayleigh-Jeans law) predicted infinite energy emission at high frequencies (the ultraviolet catastrophe).

In 1900, Max Planck resolved this discrepancy by proposing that radiant energy is not emitted or absorbed continuously, but in discrete packets called quanta (photons for light). The energy \(E\) of a quantum is directly proportional to its frequency:

where \(h = 6.62607015 \times 10^{-34}\text{ J}\cdot\text{s}\) is Planck constant. Total radiant energy emitted or absorbed is quantized in integer multiples: \(E_{\text{total}} = n h \nu\) (where \(n = 1, 2, 3, \dots\)).

Heinrich Hertz (1887) observed that when ultraviolet light illuminates clean metal surfaces (e.g., Cs, K, Na), electrons are instantly ejected. Classical wave theory predicted that increasing light intensity would increase kinetic energy and that a time lag should occur for low-intensity light. Observations contradicted this:

1. Ejection of electrons is instantaneous (time lag \(< 10^{-9}\text{ s}\)).
2. For each metal, there exists a characteristic threshold frequency (\(\nu_0\)) below which no photoelectrons are ejected, regardless of light intensity.
3. The number of ejected electrons (photocurrent) is proportional to light intensity (photon flux).
4. The maximum kinetic energy (\(E_{k,\max}\)) of ejected electrons depends linearly on the frequency of incident radiation and is independent of intensity.

Albert Einstein (1905) explained this phenomenon by treating light as a stream of photons. When a photon of energy \(h\nu\) collides with a bound electron, a fixed portion of energy, termed the work function (\(W_0\) or \(\Phi\)), is consumed to liberate the electron from the surface lattice. The surplus energy is converted into electron kinetic energy:

where \(\nu_0\) is the threshold frequency, \(\lambda_0 = c/\nu_0\) is the threshold wavelength, and \(V_0\) is the stopping potential.

Niels Bohr (1913) merged Rutherford nuclear model with Planck quantum concept to construct a quantitative model for single-electron species (\(\text{H}, \text{He}^+, \text{Li}^{2+}, \text{Be}^{3+}\)).

• Postulate I (Stationary Orbits): An electron revolves around the nucleus only in certain stable, non-radiating circular orbits called stationary states. While in these orbits, the electron does not emit electromagnetic radiation.
• Postulate II (Quantization of Angular Momentum): The orbital angular momentum \(L = m_e v r\) of the revolving electron is an integral multiple of \(h / (2\pi)\): \[m_e v r = n \frac{h}{2\pi} = n \hbar \quad (n = 1, 2, 3, \dots)\] where \(n\) is the principal quantum number.
• Postulate III (Quantum Energy Transitions): Radiation is emitted or absorbed only when an electron jumps from one stationary state to another. The frequency \(\nu\) of the photon is given by Bohr frequency condition: \[\Delta E = E_2 - E_1 = h \nu = \frac{h c}{\lambda}\]

Balancing the Coulombic electrostatic attraction with the centrifugal requirement: \(\frac{Z e^2}{4\pi \varepsilon_0 r^2} = \frac{m_e v^2}{r}\). Combining this with \(v = \frac{n h}{2\pi m_e r}\) yields:

• Radius of \(n\)-th Orbit: \[r_n = \frac{\varepsilon_0 n^2 h^2}{\pi m_e Z e^2} = \frac{n^2}{Z} a_0 = 0.529 \times \frac{n^2}{Z} \text{ \AA} = 52.9 \times \frac{n^2}{Z} \text{ pm}\] where \(a_0 = 0.529\text{ \AA}\) is the Bohr radius for hydrogen (\(n=1, Z=1\)).
• Velocity in \(n\)-th Orbit: \[v_n = \frac{Z e^2}{2 \varepsilon_0 n h} = 2.188 \times 10^6 \times \frac{Z}{n} \text{ m/s}\]
• Total Energy of \(n\)-th Orbit: Total energy is the sum of kinetic energy (\(E_k = \frac{1}{2}m_e v^2 = -\frac{1}{2}E_p\)) and electrostatic potential energy (\(E_p = -\frac{Z e^2}{4\pi \varepsilon_0 r}\)): \[E_n = -\frac{m_e Z^2 e^4}{8 \varepsilon_0^2 n^2 h^2} = -R_H \left(\frac{Z^2}{n^2}\right)\] \[E_n = -2.18 \times 10^{-18} \times \left(\frac{Z^2}{n^2}\right) \text{ J/atom} = -13.6 \times \left(\frac{Z^2}{n^2}\right) \text{ eV/atom} = -\frac{1312}{n^2} \times Z^2 \text{ kJ/mol}\]

  Significance of the Negative Sign: The negative sign indicates that the electron is electrostatically bound to the nucleus. When the electron is infinitely distant (\(n = \infty\)), attraction is zero and \(E_\infty = 0\). As it approaches the nucleus, potential energy drops, releasing energy.

When atomic hydrogen gas is excited in an electrical discharge tube, emitted photons produce discrete line series described by Johannes Rydberg empirical formula:

where \(R_H = 109,677\text{ cm}^{-1} = 1.09677 \times 10^7\text{ m}^{-1}\) is the Rydberg constant.

The classical concept of a particle having a definite trajectory was replaced by quantum mechanics through two revolutionary physical principles.

Louis de Broglie proposed that if electromagnetic radiation exhibits both wave and particle characteristics, material particles (such as electrons, protons, and atoms) must also possess wave-like properties. Equating Planck photon energy \(E = h \nu = h c / \lambda\) with Einstein mass-energy equivalence \(E = m c^2\) for a photon of momentum \(p = m c\):

For a particle of mass \(m\) and kinetic energy \(E_k = \frac{1}{2}m v^2 = \frac{p^2}{2m} \implies p = \sqrt{2m E_k}\):

When a charged particle of charge \(q\) is accelerated through a potential difference \(V\) volts, \(E_k = q V\):

For an electron (\(m = 9.11 \times 10^{-31}\text{ kg}, q = 1.602 \times 10^{-19}\text{ C}\)):

• Davisson and Germer Experiment (1927): Directed a beam of electrons accelerated at 54 V onto a nickel crystal. The scattered electrons formed distinct diffraction rings matching Bragg X-ray diffraction, verifying \(\lambda_{\text{exp}} = 0.165\text{ nm}\) in agreement with de Broglie prediction (\(0.167\text{ nm}\)).
• Macroscopic vs Microscopic: A macroscopic object (e.g., a 0.1 kg cricket ball at 20 m/s) has \(\lambda = \frac{6.63 \times 10^{-34}}{0.1 \times 20} \approx 3.3 \times 10^{-34}\text{ m}\), which is smaller than nuclear dimensions and physically undetectable. For an electron (\(m \sim 10^{-30}\text{ kg}\)), \(\lambda \sim 10^{-10}\text{ m}\), comparable to atomic spacing and measurable via electron diffraction.

Werner Heisenberg formulated that it is fundamentally impossible to measure simultaneously and accurately both the position (\(x\)) and linear momentum (\(p\)) of a subatomic particle:

where \(\Delta x\) is the uncertainty in position and \(\Delta p_x\) (or \(\Delta v_x\)) is the uncertainty in momentum (or velocity). Alternative formulations include energy-time uncertainty: \(\Delta E \cdot \Delta t \ge \frac{h}{4\pi}\).

In 1926, Erwin Schrödinger formulated the quantum mechanical wave equation for matter waves, replacing deterministic orbits with probabilistic 3D electron clouds called orbitals.

For an electron moving in three dimensions under a potential energy field \(V(x,y,z)\):

where \(\hat{H} = -\frac{\hbar^2}{2m}\nabla^2 + V\) is the Hamiltonian operator, \(E\) is the total energy eigenvalue, and \(\psi\) is the wavefunction.

• Physical Significance of \(\psi\): The amplitude \(\psi\) has no direct physical meaning and can be positive, negative, or complex.
• Physical Significance of \(|\psi|^2\) (Max Born Interpretation): The square of the absolute value, \(|\psi|^2\) (or \(\psi^*\psi\)), represents the probability density of locating the electron per unit volume around a specific point \((x,y,z)\).
• Atomic Orbital: A three-dimensional boundary surface in space around the nucleus within which the probability of finding the electron is approximately 90% to 95%.

• \(s\)-Orbitals (\(l = 0\)): Spherically symmetrical; probability density depends only on distance \(r\) from nucleus, non-directional.
• \(p\)-Orbitals (\(l = 1\)): Dumbbell-shaped consisting of two lobes separated by a nodal plane passing through the nucleus. Three mutually perpendicular degenerate orbitals: \(p_x, p_y, p_z\).
• \(d\)-Orbitals (\(l = 2\)): Cloverleaf / double dumbbell shape. Five degenerate orbitals: four with four lobes in planes (\(d_{xy}, d_{yz}, d_{zx}\) between axes; \(d_{x^2-y^2}\) along axes) and one donut-shaped orbital (\(d_{z^2}\) with a central collar ring in the xy-plane).

A node is a region where the probability density \(|\psi|^2\) drops to zero.

• Radial Nodes (Spherical Nodes): Spherical surfaces around the nucleus where \(|\psi|^2 = 0\): \[\text{Radial Nodes} = n - l - 1\]
• Angular Nodes (Nodal Planes / Cones): Planar or conical surfaces passing through the nucleus where \(|\psi|^2 = 0\): \[\text{Angular Nodes} = l\]
• Total Nodes: \[\text{Total Nodes} = \text{Radial} + \text{Angular} = (n - l - 1) + l = n - 1\]

The ground-state distribution of electrons among available orbitals is governed by three fundamental quantum principles.

The word Aufbau is German for 'building up'. In the ground state of an atom, electrons occupy orbitals in order of increasing orbital energies:

Bohr-Bury \((n + l)\) Rule:

• Rule 1: An orbital with a lower value of \((n + l)\) has lower energy and is filled first. For example, for \(4s\), \(n+l = 4+0 = 4\); for \(3d\), \(n+l = 3+2 = 5\). Hence, \(4s\) fills before \(3d\).
• Rule 2: If two orbitals possess identical \((n + l)\) values, the orbital with the smaller value of \(n\) has lower energy and fills first. For example, for \(3d\) (\(3+2 = 5\)) and \(4p\) (\(4+1 = 5\)), \(3d\) fills before \(4p\).

Statement: No two electrons in an atom can have the same set of all four quantum numbers (\(n, l, m_l, m_s\)).

If two electrons have identical \(n, l,\) and \(m_l\) (meaning they reside in the same orbital), they must have different spin quantum numbers \(m_s\) (one \(+1/2\) and the other \(-1/2\)). Consequently, an orbital can accommodate a maximum of two electrons, and they must have antiparallel (opposite) spins (\(\uparrow\downarrow\)).

Statement: Pairing of electrons in degenerate orbitals (orbitals belonging to the same subshell, such as \(p, d, f\)) does not occur until each available orbital is singly occupied with electrons having parallel spins.

Physical Rationale: Electrons repel each other electrostatically. By occupying separate spatial orbitals with parallel spins, electrons maximize their average distance, minimizing inter-electronic repulsion. Parallel spins also maximize exchange energy.

• Chromium (\(Z = 24\)):
  • Expected: \([\text{Ar}]\, 3d^4 4s^2\)
  • Actual Ground State: \([\text{Ar}]\, 3d^5 4s^1\)
• Copper (\(Z = 29\)):
  • Expected: \([\text{Ar}]\, 3d^9 4s^2\)
  • Actual Ground State: \([\text{Ar}]\, 3d^{10} 4s^1\)

Underlying Causes for Extra Stability:

1. Symmetrical Distribution of Charge: Spherically symmetrical and completely degenerate half-filled (\(d^5, p^3, f^7\)) and fully-filled (\(d^{10}, p^6, f^{14}\)) subshells minimize inter-electronic shielding and repulsion.
2. Maximum Exchange Energy: Electrons with parallel spins in degenerate orbitals can exchange positions. Each exchange releases stabilizing energy. The number of exchange pairs for \(n\) electrons with parallel spins is \(K = \frac{n(n-1)}{2}\). For \(d^4\), \(K = \frac{4(3)}{2} = 6\); for \(d^5\), \(K = \frac{5(4)}{2} = 10\). The 4 extra exchange interactions substantially lower the ground-state energy.