Atomic Structure

1. Discovery of Sub-atomic Particles

Thomson's cathode-ray experiments established the electron and measured its charge-to-mass ratio ($e/m_e = 1.758820 \times 10^{11}\ C\,kg^{-1}$). Millikan's oil-drop experiment then fixed the charge, $e = 1.602 \times 10^{-19}\ C$, giving $m_e = 9.1094 \times 10^{-31}\ kg$. Rutherford's $\alpha$-particle scattering showed that almost all the mass and all the positive charge sit in a tiny nucleus, but his model could not explain why an orbiting (accelerating) electron does not radiate energy and spiral into the nucleus.

• Atomic number ($Z$): number of protons = number of electrons in a neutral atom.
• Mass number ($A$): protons + neutrons. Isotopes have the same $Z$, different $A$; isobars have the same $A$, different $Z$.

2. Dual Nature of Radiation and Matter

Planck's quantum theory: energy is emitted or absorbed in packets, $E = h\nu = hc/\lambda$. In the photoelectric effect, $h\nu = h\nu_0 + \frac{1}{2}m_ev^2$; below the threshold frequency $\nu_0$ no electrons are ejected, however intense the light.

de Broglie extended this to matter: $\lambda = \frac{h}{mv} = \frac{h}{p}$. For an electron accelerated through a potential $V$, $\lambda = \frac{h}{\sqrt{2meV}}$.

Heisenberg's uncertainty principle: $\Delta x \cdot \Delta p \ge \frac{h}{4\pi}$. Exact position and momentum cannot both be known, so fixed Bohr orbits are replaced by orbitals (probability regions).

3. Bohr Model of the Hydrogen Atom

Valid only for one-electron species (H, He⁺, Li²⁺):

The wavenumber of an emitted line is given by the Rydberg equation:

• Lyman ($n_1 = 1$, UV), Balmer ($n_1 = 2$, visible), Paschen ($n_1 = 3$), Brackett ($n_1 = 4$), Pfund ($n_1 = 5$) — the last three are infrared.
• Maximum number of spectral lines when an electron falls from level $n$ to the ground state: $\frac{n(n-1)}{2}$.

Interactive diagram: open this page in a browser to use it.

4. Quantum Numbers and Orbitals

• Principal ($n$): shell and size; $n = 1, 2, 3 \ldots$
• Azimuthal ($l$): subshell and shape; $l = 0$ to $n-1$ (s, p, d, f).
• Magnetic ($m_l$): orientation; $-l$ to $+l$, giving $2l+1$ orbitals.
• Spin ($m_s$): $+\frac{1}{2}$ or $-\frac{1}{2}$.

Nodes: radial nodes $= n - l - 1$; angular nodes $= l$; total nodes $= n - 1$. Orbital angular momentum $= \sqrt{l(l+1)}\,\frac{h}{2\pi}$.

5. Filling Electrons

• Aufbau principle and the $(n+l)$ rule: lower $(n+l)$ fills first; if equal, lower $n$ fills first. So 4s (4+0) fills before 3d (3+2).
• Pauli exclusion principle: no two electrons in an atom share all four quantum numbers, so an orbital holds at most two electrons of opposite spin.
• Hund's rule: electrons occupy degenerate orbitals singly, with parallel spins, before pairing.

Practice questions

1. The number of radial nodes in a 3p orbital is:

   1. 0
   2. 1
   3. 2
   4. 3
   Answer

   B. Radial nodes = n - l - 1 = 3 - 1 - 1 = 1.

2. Which electronic configuration of chromium (Z = 24) is correct?

   1. [Ar] 3d⁴ 4s²
   2. [Ar] 3d⁵ 4s¹
   3. [Ar] 3d⁶
   4. [Ar] 4s² 4p⁴
   Answer

   B. A half-filled 3d subshell is extra stable, so one 4s electron moves into 3d: [Ar] 3d⁵ 4s¹.

3. The maximum number of spectral lines when an electron in a hydrogen atom falls from n = 5 to the ground state is:

   1. 4
   2. 5
   3. 10
   4. 15
   Answer

   C. n(n-1)/2 = 5 × 4/2 = 10.

4. The radius of the first Bohr orbit of He⁺ is:

   1. 52.9 pm
   2. 26.45 pm
   3. 105.8 pm
   4. 13.2 pm
   Answer

   B. r = 52.9 n²/Z = 52.9 × ½ = 26.45 pm.

5. Which set of quantum numbers is NOT allowed?

   1. n=2, l=1, mₗ=0
   2. n=3, l=2, mₗ=-2
   3. n=2, l=2, mₗ=1
   4. n=4, l=0, mₗ=0
   Answer

   C. l can only take values up to n-1. For n = 2, l = 2 is impossible.

6. The visible lines of the hydrogen spectrum belong to the:

   1. Lyman series
   2. Balmer series
   3. Paschen series
   4. Pfund series
   Answer

   B. Balmer lines end on n₁ = 2 and fall in the visible region. Lyman is UV; Paschen, Brackett and Pfund are infrared.