Structure of Atom

1. Subatomic Particles & Cathode Ray Mechanics

• Cathode Rays (Discovery of Electron):
  • Emitted from cathode towards anode under high electrical potential ($10\text{ kV}$) and low gas pressure ($10^{-2}\text{ mm Hg}$) in discharge tubes.
  • Consist of negatively charged particles whose charge-to-mass ratio ($e/m_e$) is independent of the nature of the gas or the electrode metal used:

    $$\frac{e}{m_e} = 1.758820 \times 10^{11}\text{ C kg}^{-1}$$

  • Millikan's Oil Drop Experiment determined charge: $e = -1.6022 \times 10^{-19}\text{ C}$. Combined with Thomson's ratio, $m_e = 9.1094 \times 10^{-31}\text{ kg}$.
• Canal Rays (Protons):
  • Positively charged streams originating as residual gaseous ions.
  • Unlike cathode rays, their $e/m$ ratio strictly depends on the gas present in the tube (highest value achieved for hydrogen).
• Neutrons: Discovered by James Chadwick (1932) by bombarding a thin sheet of beryllium with $\alpha$-particles:

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

2. Electromagnetic Radiation & Planck's Quantum Theory

Wave Mechanics

• Frequency ($\nu$), wavelength ($\lambda$), and speed of light ($c = 3.0 \times 10^8\text{ m s}^{-1}$) are connected by:

  $$c = \nu \lambda \quad \implies \quad \bar{\nu} = \frac{1}{\lambda} = \frac{\nu}{c}$$

• Electromagnetic Spectrum Energy Order:

  $$\text{Radio} < \text{Microwave} < \text{Infrared (IR)} < \text{Visible} < \text{UV} < \text{X-Rays} < \gamma\text{-Rays}$$

Planck's Quantum Postulate

• Energy emitted or absorbed by an oscillator is quantized into discrete packets called quanta (or photons):

  $$E = h\nu = \frac{hc}{\lambda} = hc\bar{\nu} \quad (h = 6.626 \times 10^{-34}\text{ J s})$$

• For $n$ photons: $E_{\text{total}} = n h\nu = n \frac{hc}{\lambda}$.
• Energy of 1 Einstein (1 mole of photons): $E_{\text{mole}} = N_A \frac{hc}{\lambda}$.

3. Photoelectric Effect (Einstein's Formulation)

When light of frequency $\nu$ strikes a clean metal surface, electrons are ejected instantaneously provided $\nu \ge \nu_0$ (threshold frequency):

$$h\nu = h\nu_0 + \text{KE}_{\text{max}} = W_0 + \frac{1}{2}m_e v^2$$

Where:

• Work Function ($W_0 = h\nu_0 = \frac{hc}{\lambda_0}$): Minimum energy required to overcome surface attraction. Alkali metals ($\text{Cs}, \text{K}$) have the lowest work functions and are used in photoelectric cells.
• Kinetic Energy: Depends linearly on the frequency ($\nu$) of incident light above $\nu_0$ and is independent of intensity.
• Number of Ejected Photoelectrons: Proportional strictly to the intensity (brightness) of incident radiation.

4. Bohr's Model & Line Spectrum of Hydrogen

Bohr's postulates apply strictly to single-electron species ($\text{H}, \text{He}^+, \text{Li}^{2+}, \text{Be}^{3+}$):

1. Quantization of Angular Momentum:

   $$m_e v r = n \frac{h}{2\pi} \quad (n = 1, 2, 3, \dots)$$

2. Orbit Radius:

   $$r_n = \frac{a_0 n^2}{Z} = \frac{52.9\,n^2}{Z}\text{ pm} = 0.529 \frac{n^2}{Z}\text{ \AA}$$

3. Orbit Energy:

   $$E_n = -2.18 \times 10^{-18} \left(\frac{Z^2}{n^2}\right)\text{ J} = -13.6 \left(\frac{Z^2}{n^2}\right)\text{ eV}$$

Rydberg Spectral Formula

$$\bar{\nu} = \frac{1}{\lambda} = R_H Z^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \quad (R_H = 109,677\text{ cm}^{-1} \approx 1.097 \times 10^7\text{ m}^{-1})$$

• Total Spectral Lines Emitted from State $n$: $N = \frac{n(n-1)}{2}$.
• Transitions between two levels ($n_2 \rightarrow n_1$): $N = \frac{(n_2 - n_1)(n_2 - n_1 + 1)}{2}$.

5. Dual Nature of Matter & Heisenberg Uncertainty

de Broglie Relationship

Matter exhibits wave-particle duality:

$$\lambda = \frac{h}{p} = \frac{h}{m v} = \frac{h}{\sqrt{2 m \cdot \text{KE}}} = \frac{h}{\sqrt{2 m q V}}$$

Heisenberg Uncertainty Principle

It is impossible to determine simultaneously the precise position and momentum of a subatomic particle:

$$\Delta x \cdot \Delta p \ge \frac{h}{4\pi} \quad \implies \quad \Delta x \cdot \Delta v \ge \frac{h}{4\pi m}$$

6. Quantum Numbers & Rules for Filling Orbitals

• Orbital Angular Momentum: $\mu_l = \sqrt{l(l+1)} \frac{h}{2\pi}$ (for s-orbitals, $l=0 \implies \mu_l = 0$).
• Spin-Only Magnetic Moment: $\mu_s = \sqrt{n(n+2)}\text{ BM}$ (where $n$ is number of unpaired electrons).

Governing Principles

1. Aufbau Principle: Orbitals fill in order of increasing $(n + l)$ energy. If $(n + l)$ is identical, lower $n$ fills first.
2. Pauli Exclusion Principle: No two electrons in an atom can have an identical set of four quantum numbers. An orbital holds at most 2 electrons with opposite spins.
3. Hund's Rule of Maximum Multiplicity: Degenerate orbitals must be singly occupied with parallel spins before electron pairing begins.

7. Nodes & Exceptional Electronic Configurations

Node Formulas

• Radial Nodes (Spherical Nodes): $= n - l - 1$
• Angular Nodes (Nodal Planes): $= l$
• Total Nodes: $= n - 1$

Stability of Half-Filled and Fully-Filled Subshells

Chromium and Copper exhibit anomalous ground-state configurations:

• $\text{Cr}\,(Z=24)$: $[\text{Ar}]\,3d^5\,4s^1$ (not $3d^4\,4s^2$)
• $\text{Cu}\,(Z=29)$: $[\text{Ar}]\,3d^{10}\,4s^1$ (not $3d^9\,4s^2$)

Stability arises from:

1. Symmetrical Electron Distribution: Minimizes mutual inter-electronic shielding and repulsion.
2. Maximum Exchange Energy: Electrons with parallel spin in degenerate orbitals can exchange positions. Number of possible exchanges for $d^5$:

   $$K = 4 + 3 + 2 + 1 = 10\text{ exchanges}$$