Unit 1: Atomic Structure

Bohr's Theory, Limitations, and Hydrogen Spectrum

Bohr's Theory of the Hydrogen Atom

Niels Bohr proposed a model for the hydrogen atom in 1913, successfully explaining its line spectrum. The key postulates are:

1. Fixed Orbits: Electrons revolve around the nucleus in fixed, circular paths called "orbits" or "stationary states," each with a definite amount of energy.
2. Quantization of Angular Momentum: The electron can only revolve in those orbits for which its angular momentum (mvr) is an integral multiple of (h) / (2π).

   Formula: mvr = n(h) / (2π)
   Where: n = 1, 2, 3, ... (Principal Quantum Number), h = Planck's constant

3. No Radiation in Orbits: As long as an electron remains in a particular orbit, it does not lose or gain energy (i.e., it does not radiate energy).
4. Energy Transitions: An electron can "jump" from one orbit to another by absorbing or emitting a photon of energy. The energy of this photon is exactly equal to the energy difference (Δ E) between the two orbits.

   Formula: Δ E = E_final - E_initial = hν = (hc) / (λ)
   Where: ν = frequency of radiation, λ = wavelength of radiation

Key Results from Bohr's Model:

• Radius of orbit (r_n): r_n = 0.529 × (n²) / (Z) \AA (for H-like species, where Z is atomic number)
• Energy of orbit (E_n): E_n = -13.6 × (Z²) / (n²) eV/atom

Atomic Spectrum of Hydrogen

When an electron in an excited state (higher n) drops to a lower state (lower n), it emits a photon, creating a spectral line. The wavelength of this line is given by the Rydberg formula:

Formula: (1) / (λ) = R_H Z² ( (1) / (n₁²) - (1) / (n₂²) )
Where: R_H = Rydberg constant (109,677 cm^-1), n₁ = lower energy level, n₂ = higher energy level

Limitations of Bohr's Theory

• Fails for Multi-electron Atoms: It could not explain the spectra of atoms other than hydrogen and H-like species (e.g., He⁺, Li²⁺).
• Zeeman and Stark Effects: It could not explain the splitting of spectral lines in the presence of a magnetic field (Zeeman effect) or an electric field (Stark effect).
• Violates Heisenberg's Principle: It defines a fixed path (orbit) and velocity for the electron, which contradicts the Uncertainty Principle.
• Fine Structure: It could not explain the "fine structure" of spectral lines (i.e., single lines that are actually composed of multiple, very closely spaced lines).
• 3D Model: It did not explain the 3D, wave-like nature of the electron or the concept of orbitals.

Wave Mechanics (Quantum Model)

de Broglie Equation

Louis de Broglie proposed that all matter (like electrons) has wave-particle duality. A particle with momentum p has an associated wavelength λ.

Formula (de Broglie Relation): λ = (h) / (p) = (h) / (mv)
Where: h = Planck's constant, p = momentum, m = mass, v = velocity

Heisenberg's Uncertainty Principle

This principle states that it is impossible to simultaneously measure or know both the exact position (Δ x) and the exact momentum (Δ p) of a microscopic particle (like an electron).

Formula: Δ x · Δ p ≥ (h) / (4π)

Significance: This principle fundamentally refutes Bohr's idea of fixed orbits. If an electron were in a fixed orbit, we would know its position and momentum precisely, which is impossible. This led to the concept of probability and orbitals.

Schrödinger's Wave Equation

Erwin Schrödinger developed a mathematical equation that describes the wave-like behavior of an electron in an atom. The solutions to this equation are the wave functions (ψ) and their corresponding energies (E).

The time-independent Schrödinger equation is often written as:

Formula: Ĥψ = Eψ
Where: Ĥ = Hamiltonian operator (represents the total energy of the system), ψ = wave function, E = Energy eigenvalue (a specific, allowed energy value).

The full form (for one particle in 3D) is: ∇² ψ + (8π² m) / (h²)(E-V)ψ = 0

Significance of ψ and ψ²

• ψ (Wave Function): This is the mathematical solution to the Schrödinger equation. By itself, it has no direct physical meaning. It is an "amplitude function" and can have positive, negative, or complex values.
• ψ² (Probability Density): The square of the wave function (ψ² or |ψ|²) at any point in space gives the probability of finding the electron at that point.
  • If ψ² is high, there is a high probability of finding the electron.
  • If ψ² is zero, there is zero probability of finding the electron (this is a node).

An orbital is a 3D region in space where the probability of finding the electron (ψ²) is maximum (typically > 90%).

Quantum Numbers and Wave Functions

Quantum Numbers and Their Significance

When the Schrödinger equation is solved for the hydrogen atom, it yields solutions that are characterized by three quantum numbers (n, l, m). A fourth (s) was added to describe the electron itself.

Normalized and Orthogonal Wave Functions

• Normalized: A wave function ψ is "normalized" if the total probability of finding the electron *somewhere* in the universe is exactly 1 (or 100%).

  Mathematical Condition: ∫_{all space} ψ² dτ = 1

• Orthogonal: Two different wave functions, ψ_A and ψ_B, (e.g., 1s and 2s) are "orthogonal" if their net overlap in space is zero. This ensures they represent distinct, independent states.

  Mathematical Condition: ∫_{all space} ψ_A ψ_B dτ = 0

Sign of Wave Functions

The sign (+ or -) of the wave function ψ refers to its phase (like the crest or trough of a wave). This is crucial for chemical bonding:

• When two orbitals of the same phase (+) and (+) overlap, they interfere constructively to form a bonding molecular orbital.
• When two orbitals of opposite phase (+) and (-) overlap, they interfere destructively to form an antibonding molecular orbital (with a node in between).

Radial and Angular Wave Functions

The total wave function ψ can be separated into two parts:

ψ(r, θ, φ) = R_n,l(r) × Y_l,m(θ, φ)

• R(r) - Radial Part: Depends on n and l. It describes how the wave function (and probability) changes with distance (r) from the nucleus. It determines the size of the orbital.
• Y(θ, φ) - Angular Part: Depends on l and m_l. It describes how the wave function changes with direction (angles θ, φ). It determines the shape of the orbital.

Nodes

Nodes are regions where ψ² = 0 (zero probability of finding the electron).

• Radial Nodes: Spherical nodes at a certain distance r from the nucleus. Number of radial nodes = n - l - 1.
• Angular Nodes: Planar or conical nodes that pass through the nucleus. Number of angular nodes = l.
• Total Nodes: n - 1

Example: A 3p orbital (n=3, l=1) has (3-1-1) = 1 radial node and (l) = 1 angular node. Total nodes = (3-1) = 2.

Shapes of s, p, d, and f- orbitals

• s-orbitals (l=0): Spherically symmetrical. l=0, so 0 angular nodes.
• p-orbitals (l=1): Dumbbell-shaped. l=1, so 1 angular (planar) node. There are three p-orbitals: p_x, p_y, p_z, oriented along the axes.
• d-orbitals (l=2): Cloverleaf-shaped (mostly). l=2, so 2 angular nodes. There are five d-orbitals:
  • d_xy, d_yz, d_xz (lobes between the axes)
  • d_x²-y² (lobes on the axes)
  • d_z² (dumbbell shape with a "donut" ring)
• f-orbitals (l=3): Complex, multi-lobed shapes. l=3, so 3 angular nodes. There are seven f-orbitals.

Principles of Electron Filling

Aufbau's Principle

From German "Aufbau" meaning "building up." This principle states that electrons fill atomic orbitals starting from the lowest available energy level before moving to higher levels.

The order of filling is generally given by the (n+l) rule:

1. Orbitals with a lower (n+l) value are filled first.
2. If two orbitals have the same (n+l) value, the one with the lower n value is filled first.

Order of filling: 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d ...

Example: 4s (n+l = 4+0=4) is filled before 3d (n+l = 3+2=5).

Limitations of Aufbau Principle

The principle is a guideline and has notable exceptions due to the extra stability of half-filled and completely-filled subshells.

• Chromium (Cr, Z=24): Expected: [Ar] 4s² 3d⁴. Actual: [Ar] 4s¹ 3d⁵ (for half-filled d-subshell).
• Copper (Cu, Z=29): Expected: [Ar] 4s² 3d⁹. Actual: [Ar] 4s¹ 3d¹⁰ (for full d-subshell).

Pauli's Exclusion Principle

Principle: No two electrons in an atom can have the same set of all four quantum numbers.

Consequence: An orbital can hold a maximum of two electrons, and they must have opposite spins (m_s = +(1) / (2) and m_s = -(1) / (2)).

Hund's Rule of Maximum Multiplicity

Rule: For degenerate orbitals (orbitals of the same energy, like p_x, p_y, p_z), electron pairing will not begin until all orbitals in the subshell are occupied by at least one electron (half-filled).

Consequence: The most stable (lowest energy) configuration is the one with the maximum number of unpaired electrons (maximum "total spin multiplicity").

Example: Filling 3 electrons into a p-subshell (p³):

• Correct (Hund's Rule): [ ↑ ] [ ↑ ] [ ↑ ] (Maximum multiplicity)
• Incorrect: [ ↑↓ ] [ ↑ ] [   ]