16.2 Atomic Structure, Bohr's Model, Quantum Numbers & Electronic Configurations

Atomic Structure, Bohr's Model, Quantum Numbers & Electronic Configurations

In the Intermediate Chemistry Part I syllabus and the ECAT competitive examination, atomic structure and quantum mechanics bridge physical principles with periodic chemical behavior. Examination questions emphasize quantitative transitions in Bohr's hydrogen model, the identification of valid quantum number sets, application of the $(n+l)$ rule without reference charts, and the rationalization of anomalous transition-metal ground states. Mastering these quantum foundations ensures rapid, accurate problem-solving across both physics and chemistry sections.

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1. Discovery of Subatomic Particles & The Nuclear Atom

Classical Daltonian atomic theory regarded the atom as an indivisible solid sphere. A sequence of cathode ray and radioactivity experiments revealed its complex internal subatomic architecture:

Discharge Tube Experiments and the Electron

• Cathode Rays (William Crookes, 1879): High-voltage electrical discharge ($5,000-10,000\text{ V}$) through a gas at low pressure ($0.01\text{ torr}$) produces luminous rays emanating from the cathode. Cathode rays travel in straight lines perpendicular to the cathode surface, cast sharp shadows of opaque objects, drive small paddle wheels (confirming momentum $p = mv$ and particulate nature), deflect toward positive electric plates and magnetic poles (demonstrating negative charge), produce green fluorescence on glass walls, and heat thin metal foils upon bombardment.
• Charge-to-Mass Ratio ($e/m$) (J.J. Thomson, 1897): By simultaneously balancing perpendicular electric and magnetic fields, Thomson evaluated the specific charge of the electron:

$\frac{e}{m} = 1.7588 \times 10^{11}\text{ C}\cdot\text{kg}^{-1} = 1.7588 \times 10^8\text{ C}\cdot\text{g}^{-1}$

Crucially, Thomson demonstrated that the $e/m$ ratio of cathode rays is completely independent of the gas inside the tube and the cathode material, proving that the electron is a universal constituent of all matter.

• Measurement of Charge ($e$) (R.A. Millikan, 1909): Millikan's oil-drop experiment determined the fundamental elementary charge by balancing gravitational force ($mg$) against electrostatic force ($qE$):

$e = 1.6022 \times 10^{-19}\text{ C}$

$\text{Mass of electron: } m_e = \frac{e}{e/m} = \frac{1.6022 \times 10^{-19}\text{ C}}{1.7588 \times 10^{11}\text{ C}\cdot\text{kg}^{-1}} = 9.10938 \times 10^{-31}\text{ kg} = \frac{1}{1836}\text{ mass of H-atom}$

Discovery of the Proton and Neutron

• Canal Rays / Positive Rays (Eugen Goldstein, 1886): Perforated cathode discharge tubes reveal rays travelling toward the cathode. Unlike cathode rays, their $e/m$ ratio depends fundamentally on the nature of the residual gas inside the tube. The highest $e/m$ value was obtained when hydrogen gas was employed. Rutherford named this lightest positive unit particle the proton ($m_p = 1.6726 \times 10^{-27}\text{ kg} \approx 1836\, m_e$, charge $= +1.6022 \times 10^{-19}\text{ C}$).
• The Neutron (James Chadwick, 1932): Bombarding a beryllium ($^9_4\text{Be}$) target with energetic $\alpha$-particles ($^4_2\text{He}$) produced highly penetrating, electrically neutral radiation capable of knocking protons out of paraffin wax:

$^9_4\text{Be} + ^4_2\text{He} \to ^{12}_6\text{C} + ^1_0\text{n}$

The neutron possesses zero charge and a rest mass of $m_n = 1.6749 \times 10^{-27}\text{ kg} \approx 1840\, m_e$ (slightly greater than a proton). A free neutron is unstable, decaying with a half-life of about $10\text{ minutes}$ (mean lifetime about $14.7\text{ minutes}$) into a proton, an electron, and an antineutrino ($^1_0\text{n} \to ^1_1\text{p} + ^0_{-1}\text{e} + \bar{\nu}_e$).

Rutherford's $\alpha$-Scattering Experiment & Nuclear Model (1911)

Bombardment of a $0.00004\text{ cm}$ thin gold foil with $\alpha$-particles showed:

1. $>99\%$ of $\alpha$-particles traversed undeflected, demonstrating that the vast majority of atomic volume is empty space.
2. A small fraction suffered significant deflections ($>90^\circ$), and a very small fraction (of the order of $1$ in $10{,}000$) rebounded almost straight back. Rutherford concluded that the entire positive charge and virtually all atomic mass are concentrated in an ultra-dense central core termed the nucleus ($r_{\text{nucleus}} \sim 10^{-15}\text{ m}$ vs $r_{\text{atom}} \sim 10^{-10}\text{ m}$).

Classical Defects of Rutherford's Model:

• According to Maxwell's classical electromagnetic theory, an accelerating charged particle (such as an orbiting electron) must continuously radiate electromagnetic energy. Losing energy, the electron must spiral inward and collapse into the nucleus within $\sim 10^{-8}\text{ s}$, making stable atoms impossible.
• A continuously decaying orbit would emit a continuous rainbow spectrum, whereas excited atomic gases strictly emit discrete line spectra.

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2. Bohr's Quantum Atomic Model

In 1913, Niels Bohr resolved these classical contradictions by introducing Max Planck's quantum concept to Rutherford's nuclear model.

Bohr's Three Fundamental Postulates

1. Stationary Non-Radiating Orbits: The electron revolves around the central nucleus only in certain discrete, stable circular orbits termed stationary states. While occupying these orbits, the electron radiates no energy.
2. Quantization of Angular Momentum: The electron can only revolve in orbits for which its orbital angular momentum ($L = mvr$) is an integral multiple of $\frac{h}{2\pi}$:

$mvr = \frac{nh}{2\pi}, \quad n \in \{1, 2, 3, 4, \dots\}$

where $n$ is the principal quantum number and $h = 6.626 \times 10^{-34}\text{ J}\cdot\text{s}$ is Planck's constant. 3. Quantum Energy Transitions: Radiation is absorbed or emitted only when an electron jumps from one stationary orbit to another. The photon energy is given by the Bohr-Einstein frequency condition:

$\Delta E = E_2 - E_1 = h\nu = \frac{hc}{\lambda}$

Absorption occurs during excitation ($n_1 \to n_2$, where $n_2 > n_1$); emission occurs during de-excitation ($n_2 \to n_1$).

Mathematical Formulas for Hydrogen-Like Species ($Z$)

For a single-electron ion with nuclear charge $Z$ (e.g., $\text{H}$, $\text{He}^+$, $\text{Li}^{2+}$, $\text{Be}^{3+}$):

1. Orbit Radius ($r_n$): Balancing electrostatic Coulomb attraction against centripetal acceleration:

$r_n = \frac{\epsilon_0 h^2 n^2}{\pi m e^2 Z} = \frac{0.529 n^2}{Z}\text{ \AA} = \frac{0.0529 n^2}{Z}\text{ nm}$

For hydrogen ($Z=1$): $r_1 = 0.529\text{ \AA}$ (the first Bohr radius, $a_0$). Orbits expand quadratically: $r_1 : r_2 : r_3 : r_4 = 1 : 4 : 9 : 16$.

1. Orbital Velocity ($v_n$):

$v_n = \frac{Z e^2}{2\epsilon_0 n h} \propto \frac{Z}{n}$

Velocity decreases inversely with principal quantum number ($v_1 : v_2 : v_3 = 1 : \frac{1}{2} : \frac{1}{3}$).

1. Total Orbit Energy ($E_n$): The total energy is the sum of kinetic energy ($K = \frac{1}{2}mv^2 = \frac{kZe^2}{2r}$) and electrostatic potential energy ($U = -\frac{kZe^2}{r}$):

$E_n = K + U = -\frac{k Z e^2}{2 r_n} = -\frac{m e^4 Z^2}{8\epsilon_0^2 h^2 n^2}$

$E_n = -\frac{13.6 Z^2}{n^2}\text{ eV} = -\frac{2.18 \times 10^{-18} Z^2}{n^2}\text{ J} = -\frac{1312.36 Z^2}{n^2}\text{ kJ}\cdot\text{mol}^{-1}$

• The negative sign indicates an attractive bound state between nucleus and electron. At $n=\infty$, $E = 0$, representing an ionized, completely free electron at rest.
• Ground state energy of Hydrogen: $E_1 = -13.6\text{ eV}$. Subsequent levels: $E_2 = -3.4\text{ eV}$, $E_3 = -1.51\text{ eV}$, $E_4 = -0.85\text{ eV}$.
• Energy Gap Convergence: Energy gaps between successive orbits shrink progressively as $n$ increases:

$\Delta E_{1 \to 2} = 10.2\text{ eV} > \Delta E_{2 \to 3} = 1.89\text{ eV} > \Delta E_{3 \to 4} = 0.66\text{ eV}$

The Hydrogen Emission Spectral Series

The wave number ($\bar{\nu} = \frac{1}{\lambda}$) of an emitted spectral line is governed by the Rydberg formula:

$\bar{\nu} = \frac{1}{\lambda} = R_H Z^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$

where Rydberg constant $R_H = 1.09678 \times 10^7\text{ m}^{-1} = 1.09678 \times 10^5\text{ cm}^{-1}$.

ECAT Note: The Balmer series is the only series falling in the humanly visible spectrum ($400-700\text{ nm}$). Its first line ($H_\alpha$, $3 \to 2$) emits red light at $\lambda \approx 656.3\text{ nm}$.

Critical Defects and Limitations of Bohr's Model

1. It successfully predicts spectra of 1-electron systems only ($\text{H}, \text{He}^+, \text{Li}^{2+}$) and fails completely for multi-electron atoms.
2. High-resolution spectroscopy shows each spectral line splits into a multiplet of very closely spaced fine lines (fine structure), which Bohr's circular orbits cannot account for.
3. It cannot explain the Zeeman effect (splitting of spectral lines in an external magnetic field) or the Stark effect (splitting in an external electric field).
4. It treats the electron strictly as a localized particle moving in a planar 2D orbit, directly violating the wave nature of matter and Heisenberg's uncertainty principle.

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3. Wave-Particle Duality & Heisenberg's Uncertainty Principle

The de Broglie Wavelength (1924)

Louis de Broglie postulated that moving material particles exhibit wave characteristics, with wavelength inversely proportional to momentum:

$\lambda = \frac{h}{p} = \frac{h}{mv} = \frac{h}{\sqrt{2m(KE)}} = \frac{h}{\sqrt{2mqV}}$

Davisson-Germer and G.P. Thomson confirmed de Broglie's prediction experimentally by demonstrating electron diffraction through nickel crystals and thin gold foils. For macroscopic bodies (e.g., a $0.15\text{ kg}$ cricket ball moving at $30\text{ m/s}$, $\lambda \approx 1.5 \times 10^{-34}\text{ m}$), wave properties are immeasurably small. For microscopic electrons ($m_e \approx 9.1 \times 10^{-31}\text{ kg}$, $v = 10^6\text{ m/s}$), $\lambda \approx 0.73\text{ nm}$, on the order of atomic spacing.

Heisenberg Uncertainty Principle (1927)

Werner Heisenberg proved that it is physically impossible to simultaneously measure both the precise position ($x$) and precise linear momentum ($p$) of a subatomic particle:

$\Delta x \cdot \Delta p \ge \frac{h}{4\pi} = \frac{\hbar}{2}$

$\Delta x \cdot (m \Delta v) \ge \frac{h}{4\pi} \implies \Delta x \cdot \Delta v \ge \frac{h}{4\pi m}$

Alternatively, in terms of energy and time:

$\Delta E \cdot \Delta t \ge \frac{h}{4\pi}$

Because an electron's position cannot be localized to a definite path without imparting unpredictable momentum, Bohr's deterministic planetary "orbits" are discarded in favor of quantum mechanical orbitals.

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4. The Four Quantum Numbers & Atomic Orbitals

An orbital is a three-dimensional region of space surrounding the nucleus where the probability of finding an electron is maximal (typically defined by a $90-95\%$ probability contour). Four quantum numbers fully specify the quantum state of any atomic electron:

1. Principal Quantum Number ($n$)

• Values: $n = 1, 2, 3, 4, \dots$ (historically designated $K, L, M, N, \dots$ shells).
• Physical Significance: Determines the principal energy level, the overall size of the orbital, and the average distance of the electron cloud from the nucleus.
• Capacities:
  • Maximum electrons in shell $= 2n^2$.
  • Total number of subshells in $n$-th shell $= n$.
  • Total number of atomic orbitals in $n$-th shell $= n^2$.

2. Azimuthal (Orbital Angular Momentum) Quantum Number ($l$)

• Values: $l = 0, 1, 2, \dots, (n-1)$ (total $n$ values per shell).
• Physical Significance: Specifies the subshell geometry and shape of the electron density cloud, and defines the orbital angular momentum:

$L_{\text{orbital}} = \sqrt{l(l+1)}\frac{h}{2\pi} = \sqrt{l(l+1)}\hbar$

• Subshell Notations and Shapes:
  • $l = 0$: $s$ subshell (spherical, symmetrical, non-directional; zero angular nodes).
  • $l = 1$: $p$ subshell (dumb-bell shaped with two lobes separated by a nodal plane; directional along $x, y, z$).
  • $l = 2$: $d$ subshell (four cloverleaf/double dumb-bell shaped orbitals $d_{xy}, d_{yz}, d_{zx}, d_{x^2-y^2}$ and one doughnut-torus lobed orbital $d_{z^2}$).
  • $l = 3$: $f$ subshell (complex 8-lobed geometries; 7 degenerate orbitals).
• Electron Capacity of Subshell: Maximum electrons $= 2(2l+1) = 4l + 2$.
  • $s$: $2(2(0)+1) = 2$ electrons.
  • $p$: $2(2(1)+1) = 6$ electrons.
  • $d$: $2(2(2)+1) = 10$ electrons.
  • $f$: $2(2(3)+1) = 14$ electrons.

3. Magnetic Quantum Number ($m_l$)

• Values: $m_l = -l, -(l-1), \dots, 0, \dots, +(l-1), +l$ (total $2l+1$ integer values).
• Physical Significance: Specifies the spatial orientation of degenerate orbitals in three dimensions under an external magnetic field; theoretically explains the Zeeman effect.
  • For $s$ ($l=0$): $m_l = 0$ ($1$ orbital).
  • For $p$ ($l=1$): $m_l = -1, 0, +1$ ($3$ degenerate orbitals: $p_x, p_y, p_z$).
  • For $d$ ($l=2$): $m_l = -2, -1, 0, +1, +2$ ($5$ degenerate orbitals: $d_{xy}, d_{yz}, d_{zx}, d_{x^2-y^2}, d_{z^2}$).
  • For $f$ ($l=3$): $m_l = -3, -2, -1, 0, +1, +2, +3$ ($7$ degenerate orbitals).

4. Spin Quantum Number ($m_s$)

• Values: $m_s = +\frac{1}{2}$ (spin-up, $\uparrow$) or $-\frac{1}{2}$ (spin-down, $\downarrow$).
• Physical Significance: Discovered by Uhlenbeck and Goudsmit (1925) and demonstrated by the Stern-Gerlach experiment. Represents the intrinsic spin angular momentum of the electron ($S = \sqrt{s(s+1)}\hbar = \frac{\sqrt{3}}{2}\hbar$), producing an intrinsic magnetic dipole moment.

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5. Rules Governing Ground-State Electronic Configurations

1. The Aufbau Principle & The $(n+l)$ Rule

• Aufbau Principle: Orbitals are filled in order of strictly increasing orbital energy, with lowest energy orbitals populated first.
• The $(n+l)$ Rule:
  1. Orbitals with a smaller numerical sum of $(n+l)$ possess lower energy and fill first. For example, compare $4s$ and $3d$:
     • For $4s$: $n=4, l=0 \implies n+l = 4$
     • For $3d$: $n=3, l=2 \implies n+l = 5$ Because $4 < 5$, the $4s$ orbital fills before $3d$.
  2. If two orbitals have identical $(n+l)$ values, the orbital with the lower principal quantum number $n$ has lower energy and fills first. For example, compare $3d$ and $4p$:
     • For $3d$: $n=3, l=2 \implies n+l = 5$
     • For $4p$: $n=4, l=1 \implies n+l = 5$ Because $n=3 < n=4$, $3d$ fills before $4p$.
• Filling Sequence: $1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s \dots$

2. Pauli's Exclusion Principle (1925)

"No two electrons in the same atom can possess an identical set of all four quantum numbers ($n, l, m_l, m_s$)."

• Direct Consequence: An atomic orbital can accommodate a maximum of two electrons, and if two electrons share the same orbital ($n, l, m_l$ identical), they must have antiparallel (opposite) spins ($+\frac{1}{2}$ and $-\frac{1}{2}$).

3. Hund's Rule of Maximum Multiplicity

"When degenerate orbitals of equal energy are available, electrons occupy them singly with parallel spins before any orbital is doubly occupied."

• Singly occupied orbitals minimize inter-electronic Coulomb repulsion by keeping electrons spatially separated, while maximizing parallel-spin exchange stabilization energy.

Exceptional Ground-State Configurations: Chromium and Copper

Two prominent first-row transition metals deviate from standard $(n+l)$ expectations:

1. Chromium ($Z=24$):
   • Expected configuration: $[\text{Ar}] 4s^2 3d^4$
   • Actual ground state: $[\text{Ar}] 4s^1 3d^5$
2. Copper ($Z=29$):
   • Expected configuration: $[\text{Ar}] 4s^2 3d^9$
   • Actual ground state: $[\text{Ar}] 4s^1 3d^{10}$

Theoretical Justification:

• Subshell Symmetry: Exactly half-filled ($d^5$) and completely filled ($d^{10}$) subshells possess spherical electron charge distribution, leading to enhanced structural stability.
• Quantum Exchange Energy: Electrons with parallel spins in degenerate orbitals continuously exchange positions. Each possible pair of parallel-spin electrons releases stabilizing exchange energy ($K$):

$\text{Number of possible exchanges: } E_{\text{ex}} = \frac{N(N-1)}{2}$

For a $3d^4$ configuration ($N=4$ parallel electrons), possible exchanges $= \frac{4(3)}{2} = 6$. For the actual $3d^5$ configuration ($N=5$ parallel electrons), possible exchanges $= \frac{5(4)}{2} = 10$. The release of 4 additional units of exchange energy stabilizes the $4s^1 3d^5$ state over $4s^2 3d^4$.