Atomic Models and Quantum Mechanics: A Comprehensive Guide

Rutherford's Atomic Model: The Nuclear Atom

Ernest Rutherford, through his famous gold foil experiment in 1911, proposed a revolutionary model of the atom. Prior to this, J.J. Thomson's 'plum pudding' model suggested a uniform distribution of positive charge with electrons embedded within it. Rutherford's experiment, which involved firing alpha particles at a thin gold foil, yielded unexpected results:

• Most alpha particles passed straight through the foil, indicating that atoms are mostly empty space.
• A small fraction of alpha particles were deflected at large angles.
• A very few alpha particles (about 1 in 8000) were deflected backward, suggesting a highly dense, positively charged center.

Based on these observations, Rutherford concluded that an atom consists of a tiny, dense, positively charged nucleus at its center, with electrons orbiting around it, much like planets orbit the sun. The nucleus contains almost all of the atom's mass.

Limitations of Rutherford's Atomic Model

Despite its groundbreaking insights, Rutherford's model had two significant limitations:

• Stability of the Atom: According to classical electromagnetic theory, an electron orbiting a nucleus is an accelerating charged particle and should continuously radiate energy. This energy loss would cause the electron to spiral inwards and eventually collapse into the nucleus, making atoms unstable. However, atoms are known to be stable.
• Line Spectrum: Rutherford's model could not explain the observed line spectra of elements. If electrons continuously radiated energy, they should produce a continuous spectrum. Instead, atoms emit light only at specific, discrete wavelengths, resulting in line spectra.

Bohr's Atomic Model: Quantized Orbits

Niels Bohr, in 1913, addressed the shortcomings of Rutherford's model by incorporating Planck's quantum theory. His postulates for the hydrogen atom are:

1. Stationary Orbits: Electrons revolve around the nucleus in certain specific, non-radiating orbits called stationary states. While in these orbits, they do not lose energy. Each orbit is associated with a definite amount of energy.
2. Quantization of Angular Momentum: The angular momentum of an electron in a stationary orbit is quantized, meaning it can only take on discrete values that are integral multiples of h/2π (where h is Planck's constant). Mathematically, mvr = nh/2π, where n = 1, 2, 3... (principal quantum number).
3. Energy Transitions: An electron can jump from a higher energy orbit to a lower energy orbit by emitting a photon of light, or from a lower to a higher energy orbit by absorbing a photon. The energy of the photon (ΔE) is equal to the energy difference between the two orbits: ΔE = E_f - E_i = hν (where ν is the frequency of light).

Applications of Bohr's Model: Bohr's model successfully explained the stability of the atom and, most importantly, the line spectrum of hydrogen. It accurately predicted the wavelengths of the spectral lines, given by the Rydberg formula:

1/λ = R (1/n₁² - 1/n₂²)

Where R is the Rydberg constant, and n₁ and n₂ are integers representing the initial and final energy levels. For example, the Balmer series (visible light) corresponds to n₁ = 2 and n₂ = 3, 4, 5... while the Lyman series (UV) corresponds to n₁ = 1 and n₂ = 2, 3, 4...

Defects of Bohr's Theory

Despite its successes, Bohr's model had its own limitations:

• It could only explain the spectra of hydrogen and hydrogen-like ions (e.g., He⁺, Li²⁺) but failed for multi-electron atoms.
• It could not explain the splitting of spectral lines in a magnetic field (Zeeman effect) or an electric field (Stark effect).
• It assumed circular orbits for electrons, which was later found to be incorrect.
• It did not account for the wave nature of electrons.

Elementary Idea of Quantum Mechanical Model

The quantum mechanical model (also known as the wave mechanical model) replaced Bohr's model and provides a more sophisticated and accurate description of atomic structure. It is based on the idea that electrons exhibit both wave-like and particle-like properties (wave-particle duality).

De Broglie Wave Equation

In 1924, Louis de Broglie proposed that particles, including electrons, can also exhibit wave-like behavior. He suggested that a particle of mass 'm' moving with velocity 'v' has an associated wavelength (λ) given by:

λ = h / mv

Where h is Planck's constant. This equation implies that all matter has wave properties, but these properties are only significant for particles with very small masses, like electrons.

Heisenberg's Uncertainty Principle

Werner Heisenberg, in 1927, formulated the Uncertainty Principle, which states that it is impossible to simultaneously determine with perfect accuracy both the position and momentum (or velocity) of a particle. Mathematically:

Δx · Δp ≥ h / 4π

Where Δx is the uncertainty in position, and Δp is the uncertainty in momentum. This principle profoundly impacts our understanding of electrons in atoms; we cannot precisely pinpoint an electron's location and its trajectory simultaneously. Instead, we talk about the probability of finding an electron in a certain region.

Concept of Probability: Orbitals

Due to the Uncertainty Principle, the quantum mechanical model describes the electron's location in terms of probability. An 'orbital' is a three-dimensional region around the nucleus where there is a high probability (typically >90%) of finding an electron. It is not a fixed path like in Bohr's model, but rather a probability distribution.

Quantum Numbers and Orbitals

To describe the energy, shape, and orientation of orbitals, a set of four quantum numbers is used:

1. Principal Quantum Number (n): Describes the main energy level or shell. It can have positive integer values (1, 2, 3,...). Higher 'n' values indicate higher energy levels and larger average distance from the nucleus.
2. Azimuthal or Angular Momentum Quantum Number (l): Describes the shape of the orbital and subshell. Its values range from 0 to n-1. Each 'l' value corresponds to a subshell:
   • l = 0: s subshell (spherical shape)
   • l = 1: p subshell (dumbbell shape)
   • l = 2: d subshell (more complex shapes, typically four-lobed)
   • l = 3: f subshell (even more complex shapes)
3. Magnetic Quantum Number (m_l): Describes the orientation of the orbital in space. Its values range from -l to +l, including 0. For example, for l=1 (p subshell), m_l can be -1, 0, +1, indicating three p orbitals (p_x, p_y, p_z) oriented along the x, y, and z axes.
4. Spin Quantum Number (m_s): Describes the intrinsic angular momentum of an electron, referred to as 'spin'. It can have only two possible values: +1/2 (spin up) or -1/2 (spin down).

Shape of S and P Orbitals

• s-orbitals (l=0): Are spherical in shape. The size increases with increasing principal quantum number (1s < 2s < 3s).
• p-orbitals (l=1): Are dumbbell-shaped. There are three p-orbitals (p_x, p_y, p_z) in each p subshell, oriented perpendicular to each other along the x, y, and z axes.

Rules for Electronic Configuration

The electronic configuration describes the distribution of electrons among various orbitals in an atom. This is governed by three fundamental rules:

1. Aufbau Principle ('Building Up' Principle): Electrons fill atomic orbitals in order of increasing energy. Lower energy orbitals are filled first. The general order of filling is determined by the (n+l) rule, where orbitals with lower (n+l) values are filled first. If two orbitals have the same (n+l) value, the one with the lower 'n' value is filled first. For example, 1s < 2s < 2p < 3s < 3p < 4s < 3d...
2. Pauli Exclusion Principle: No two electrons in an atom can have the same set of all four quantum numbers. This implies that an atomic orbital can hold a maximum of two electrons, and these two electrons must have opposite spins (one +1/2, one -1/2).
3. Hund's Rule of Maximum Multiplicity: For orbitals within the same subshell (degenerate orbitals, e.g., the three p orbitals or five d orbitals), electrons first occupy each orbital singly with parallel spins before any orbital is doubly occupied. This maximizes the total spin of the electrons and leads to greater stability.

Electronic Configuration of Atoms and Ions (up to Atomic No. 30)

Let's apply these rules to write electronic configurations:

Examples for Atoms:

• Hydrogen (Z=1): 1s¹
• Helium (Z=2): 1s²
• Carbon (Z=6): 1s² 2s² 2p²
• Oxygen (Z=8): 1s² 2s² 2p⁴
• Neon (Z=10): 1s² 2s² 2p⁶
• Sodium (Z=11): [Ne] 3s¹
• Aluminum (Z=13): [Ne] 3s² 3p¹
• Sulfur (Z=16): [Ne] 3s² 3p⁴
• Argon (Z=18): [Ne] 3s² 3p⁶
• Potassium (Z=19): [Ar] 4s¹
• Calcium (Z=20): [Ar] 4s²
• Scandium (Z=21): [Ar] 3d¹ 4s² (Note: 4s fills before 3d due to (n+l) rule)
• Chromium (Z=24): [Ar] 3d⁵ 4s¹ (Exception: Half-filled d-subshell provides extra stability)
• Copper (Z=29): [Ar] 3d¹⁰ 4s¹ (Exception: Fully-filled d-subshell provides extra stability)
• Zinc (Z=30): [Ar] 3d¹⁰ 4s²

Examples for Ions:

When forming cations, electrons are removed from the outermost shell (highest 'n' value) first. For transition metals, this often means 4s electrons are removed before 3d electrons, even if 3d was filled after 4s.

• Na⁺ (from Na: [Ne] 3s¹): [Ne]
• O²⁻ (from O: 1s² 2s² 2p⁴): 1s² 2s² 2p⁶ (isoelectronic with Neon)
• Fe²⁺ (from Fe: [Ar] 3d⁶ 4s²): [Ar] 3d⁶ (4s electrons removed first)
• Fe³⁺ (from Fe: [Ar] 3d⁶ 4s²): [Ar] 3d⁵ (4s electrons removed, then one 3d electron removed)
• Cu⁺ (from Cu: [Ar] 3d¹⁰ 4s¹): [Ar] 3d¹⁰ (4s electron removed)
• Zn²⁺ (from Zn: [Ar] 3d¹⁰ 4s²): [Ar] 3d¹⁰ (4s electrons removed)