1. Atomic Structure: Overview

An atom contains a tiny, positively charged nucleus surrounded by electrons. The development of atomic theory moved from classical nuclear models to the modern quantum-mechanical description.

2. Subatomic Particles and Atomic Notation

For a neutral atom, number of electrons = number of protons = Z.

3. Rutherford’s Alpha-Particle Scattering Experiment

Experimental Arrangement

• A radioactive source emitted alpha particles.
• A narrow beam was directed at a very thin gold foil.
• A fluorescent zinc sulfide screen detected scattered particles.

Main Observations

1. Most alpha particles passed through with little or no deflection.
2. Some particles were deflected through small angles.
3. A very small fraction were deflected through large angles or nearly backward.

Conclusions

• Most of the atom is empty space.
• Almost all positive charge and most atomic mass are concentrated in a very small nucleus.
• The nucleus is much smaller than the atom.

4. Rutherford’s Atomic Model

Rutherford proposed a nuclear model in which:

1. The atom has a tiny central nucleus.
2. The nucleus contains positive charge and almost all atomic mass.
3. Electrons move around the nucleus.
4. Most of the atomic volume is empty space.

5. Limitations of Rutherford’s Model

5.1 Stability Problem

According to classical electrodynamics, an accelerating charged particle should radiate energy. An electron moving in a circular path is accelerated toward the center, so it should lose energy, spiral inward and collapse into the nucleus.

5.2 Atomic Spectrum Problem

The classical model could not explain why atoms such as hydrogen produce discrete line spectra instead of a continuous range of frequencies.

5.3 Electron Arrangement

The model gave no satisfactory rule for the allowed energies or arrangement of electrons.

6. Bohr’s Atomic Model

Main Postulates

1. Electrons can move around the nucleus only in certain permitted stationary orbits.
2. An electron in a stationary orbit does not continuously radiate energy.
3. Each permitted orbit has a definite energy.
4. The electron angular momentum is quantized:

Radiation is emitted or absorbed only when an electron changes between allowed energy levels:

For emission, the electron falls from a higher energy level to a lower one. For absorption, it gains energy and moves upward.

7. Bohr Radius, Energy and Applications

For a hydrogen-like species containing one electron and nuclear charge +Ze, Bohr’s model gives:

For hydrogen (Z = 1), the first orbit radius is the Bohr radius a₀.

Energy of nth Orbit

The negative sign means the electron is bound to the nucleus when the zero of energy is taken as a free electron infinitely far away.

8. Origin of the Hydrogen Line Spectrum

Hydrogen emits light at only certain wavelengths because the electron can occupy only discrete energies in the Bohr model.

For a transition from higher level n₂ to lower level n₁:

Combining the Bohr energies gives the Rydberg equation:

For a hydrogen-like ion:

where R ≈ 1.097 × 10⁷ m⁻¹.

9. Spectral Series of Hydrogen

10. Defects / Limitations of Bohr’s Theory

• Works best for one-electron species such as H, He⁺ and Li²⁺; it does not accurately describe many-electron atoms.
• Cannot fully explain fine structure and relative intensities of spectral lines.
• Does not adequately explain Zeeman splitting in magnetic fields or Stark splitting in electric fields.
• Fixed classical orbits conflict with the wave nature of electrons and the uncertainty principle.
• Does not provide the modern probability-based orbital description.

11. Planck’s Quantum Idea

Planck proposed that energy exchange occurs in discrete packets called quanta. For electromagnetic radiation:

where h = Planck constant and ν = frequency.

This quantization idea is central to Bohr transitions and the later quantum-mechanical model.

12. de Broglie’s Wave Equation

For an electron accelerated through potential difference V from rest:

Therefore:

13. Heisenberg’s Uncertainty Principle

Since p = mv for non-relativistic motion:

14. Concept of Probability and the Quantum-Mechanical Model

The modern model describes an electron by a wavefunction ψ. The quantity |ψ|² is related to the probability density for finding the electron in a region of space.

15. Quantum Numbers

Four quantum numbers describe an electron in an atom.

15.1 Principal Quantum Number, n

• Values: n = 1, 2, 3, …
• Related to principal shell, energy and orbital size.
• Maximum electrons in shell n = 2n².

15.2 Azimuthal / Angular-Momentum Quantum Number, l

• Values: l = 0 to n−1.
• Identifies subshell and orbital shape.

15.3 Magnetic Quantum Number, m_l

Values run from −l through 0 to +l:

Thus a p subshell (l=1) has m_l = −1, 0, +1, corresponding to three p orbitals.

15.4 Spin Quantum Number, m_s

Two electrons in the same orbital must have opposite spin quantum numbers.

16. Orbitals and Shapes of s and p Orbitals

16.1 s Orbital

An s orbital has spherical symmetry around the nucleus. Every principal shell n ≥ 1 contains one s orbital.

16.2 p Orbitals

For n ≥ 2, a p subshell contains three mutually perpendicular orbitals commonly labeled p_x, p_y and p_z. Each has two lobes separated by a nodal plane through the nucleus.

17. Aufbau Principle

n + l Rule

1. The orbital with lower (n+l) fills first.
2. If two orbitals have equal (n+l), the orbital with lower n fills first.

Common filling order:

18. Pauli Exclusion Principle

Therefore:

• One orbital holds at most two electrons.
• If two electrons share an orbital, they must have opposite spins.

19. Hund’s Rule of Maximum Multiplicity

Example: 2p³

Example: 2p⁴

20. Electronic Configurations of Atoms up to Z = 30

Apply Aufbau, Pauli and Hund together.

21. Chromium and Copper: Important Ground-State Exceptions

A simple Aufbau prediction would suggest Cr as [Ar] 3d⁴4s² and Cu as [Ar] 3d⁹4s². Experimentally, their ground states are:

The actual ordering reflects the close energies and electron interactions in 3d and 4s subshells; half-filled and filled d subshell arrangements are especially favorable in these cases.

22. Electronic Configurations of Ions

Main-Group Ions

• Na: [Ne]3s¹ → Na⁺: [Ne]
• Mg: [Ne]3s² → Mg²⁺: [Ne]
• Cl: [Ne]3s²3p⁵ → Cl⁻: [Ar]
• O: 1s²2s²2p⁴ → O²⁻: [Ne]

Transition-Metal Cations

• Fe: [Ar]3d⁶4s² → Fe²⁺: [Ar]3d⁶
• Fe: [Ar]3d⁶4s² → Fe³⁺: [Ar]3d⁵
• Cu: [Ar]3d¹⁰4s¹ → Cu⁺: [Ar]3d¹⁰
• Cu²⁺: [Ar]3d⁹
• Zn²⁺: [Ar]3d¹⁰

23. Worked Examples and Numericals

24. High-Yield Formula and Rule Sheet

25. Common Exam Mistakes

26. Important Exam Questions

Very Short / Short Questions

1. State the major observations of Rutherford’s alpha-scattering experiment.
2. Write the conclusions of Rutherford’s experiment.
3. State Rutherford’s atomic model.
4. Give two limitations of Rutherford’s model.
5. State the postulates of Bohr’s atomic model.
6. Write Bohr’s angular-momentum quantization condition.
7. Write the expression for energy of the nth Bohr orbit.
8. Define ground state and excited state.
9. Explain how hydrogen line spectra arise in Bohr’s model.
10. Write the Rydberg equation.
11. Name the Lyman, Balmer and Paschen series and their final n values.
12. State two defects of Bohr’s theory.
13. State Planck’s quantum relation.
14. State de Broglie’s hypothesis and equation.
15. State Heisenberg’s uncertainty principle.
16. Explain the concept of probability in the quantum model.
17. Differentiate orbit and orbital.
18. Define the four quantum numbers and give their allowed values.
19. How many orbitals are present in s, p, d and f subshells?
20. State the shapes of s and p orbitals.
21. State the Aufbau principle and n+l rule.
22. State Pauli’s exclusion principle.
23. State Hund’s rule.
24. Write electronic configurations of atoms up to Z=30.
25. Why are Cr and Cu exceptions to the simple Aufbau pattern?
26. Write configurations of Fe²⁺, Fe³⁺, Cu⁺, Cu²⁺ and Zn²⁺.

Long / Derivation Questions

1. Describe Rutherford’s alpha-particle scattering experiment with observations, conclusions and diagram.
2. Explain Rutherford’s atomic model and its limitations.
3. State Bohr’s postulates and explain the stability of permitted orbits.
4. Explain the origin of the hydrogen spectrum using Bohr’s model.
5. Discuss the spectral series of hydrogen.
6. Explain the limitations of Bohr’s atomic theory.
7. Explain de Broglie’s matter-wave hypothesis and its importance.
8. Explain Heisenberg’s uncertainty principle and its significance.
9. Explain all four quantum numbers with their allowed values.
10. Explain s and p orbital shapes.
11. Explain Aufbau, Pauli and Hund rules with orbital-box examples.
12. Write and explain electronic configurations up to atomic number 30, including Cr and Cu.

Numerical Practice

1. Calculate energy of an electron in a specified hydrogenic Bohr level.
2. Calculate excitation or ionization energy.
3. Calculate wavelength/frequency of a hydrogen spectral transition.
4. Calculate de Broglie wavelength from particle mass and velocity.
5. Calculate de Broglie wavelength of an electron accelerated through a potential difference.
6. Use the uncertainty relation to estimate minimum momentum/velocity uncertainty.
7. Find maximum electrons or orbitals from quantum-number rules.

27. One-Minute Revision

• Unit 3: Atomic Structure — 8 teaching hours.
• Rutherford: atom is mostly empty space with a tiny dense positive nucleus.
• Rutherford could not explain classical atomic stability or line spectra.
• Bohr allowed only stationary quantized orbits.
• mvr = nh/2π.
• Photon transition energy: ΔE = hν.
• Hydrogen energy: Eₙ = −13.6/n² eV.
• Hydrogen orbit radius: rₙ = a₀n².
• Rydberg: 1/λ = R(1/n₁² − 1/n₂²).
• Lyman ends at n=1; Balmer at n=2; Paschen at n=3.
• Bohr works best for one-electron species.
• Planck: E = hν.
• de Broglie: λ = h/p.
• Heisenberg: ΔxΔp ≥ h/4π.
• Quantum model uses probability, not a fixed electron trajectory.
• n = shell; l = subshell; mₗ = orbital; mₛ = spin.
• Allowed l = 0 to n−1.
• s,p,d,f correspond to l = 0,1,2,3.
• s has 1 orbital; p has 3; d has 5; f has 7.
• s orbital is spherical.
• p orbitals are two-lobed and occur as pₓ, pᵧ, p_z.
• Aufbau: lower-energy orbitals fill first.
• Pauli: max 2 electrons per orbital with opposite spins.
• Hund: singly occupy degenerate orbitals before pairing.
• 4s fills before 3d in the usual neutral-atom sequence.
• Cr = [Ar]3d⁵4s¹.
• Cu = [Ar]3d¹⁰4s¹.
• Transition-metal cations lose 4s electrons before 3d electrons.
• Fe²⁺ = [Ar]3d⁶; Fe³⁺ = [Ar]3d⁵.

28. Diagram Practice

1. Development of atomic models.
2. Nuclide notation.
3. Rutherford scattering experiment.
4. Rutherford stability problem.
5. Bohr quantized orbits.
6. Hydrogen energy-level transitions.
7. de Broglie matter wave.
8. Bohr orbit vs quantum orbital.
9. Quantum-number hierarchy.
10. s and p orbital shapes.
11. Aufbau filling-order diagram.
12. Hund’s rule orbital boxes.