Vardaan Learning Institute
Structure of Atom
Student Quick Revision Focus
High-yield exam revision guide for CBSE Boards, NEET, JEE Main & Advanced. Covers discovery of subatomic particles, Thomson & Rutherford models, Planck's Quantum Theory, Photoelectric effect, Bohr's Model, De Broglie dual nature, Heisenberg Uncertainty Principle, Quantum Numbers, and Electronic Configurations!
1. Discovery of Subatomic Particles
1.1 Discovery of Electron (Cathode Rays)
Discovered by J.J. Thomson (1897) via discharge tube experiment at low pressure and high voltage.
- Cathode rays travel in straight lines from negative cathode to positive anode.
- Deflected toward positive plate in electric field $\rightarrow$ consist of negatively charged electrons.
- Specific Charge ($e/m$): $\frac{e}{m_e} = \mathbf{1.758820 \times 10^{11}\text{ C/kg}}$ (constant for all gases).
- Electron Charge & Mass: Charge $e = -1.602 \times 10^{-19}\text{ C}$ (Millikan Oil Drop), Mass $m_e = \mathbf{9.109 \times 10^{-31}\text{ kg}} = 0.00054\text{ u}$.
1.2 Discovery of Proton & Neutron
- Proton (Canal Rays / Anode Rays): Discovered by Goldstein. Positively charged ions whose $e/m$ ratio depends on the gas (highest for Hydrogen gas $\rightarrow$ bare proton $p^+$). Charge $+1.602 \times 10^{-19}\text{ C}$, Mass $m_p = \mathbf{1.6726 \times 10^{-27}\text{ kg}} = 1.00727\text{ u}$.
- Neutron ($n^0$): Discovered by James Chadwick (1932) by bombarding Beryllium with $\alpha$-particles:
$${}^9_4\text{Be} + {}^4_2\text{He} \longrightarrow {}^{12}_6\text{C} + {}^1_0\text{n}$$
Neutral particle, Mass $m_n = \mathbf{1.6749 \times 10^{-27}\text{ kg}} = 1.00866\text{ u}$.
2. Early Atomic Models
2.1 Rutherford's $\alpha$-Particle Scattering Experiment
- Observations: Most $\alpha$-particles passed undeflected; a few deflected by small angles; $1$ in $20,000$ bounced back by $180^\circ$.
- Conclusions: Atom is mostly empty space. All positive charge and mass are concentrated in an extremely tiny dense Nucleus ($R_{\text{nucleus}} \approx 10^{-15}\text{ m}$ vs $R_{\text{atom}} \approx 10^{-10}\text{ m}$).
- Nuclear Radius Formula: $R = R_0 A^{1/3}$ (where $R_0 = 1.2 \times 10^{-15}\text{ m}$).
- Failure: According to Maxwell's electromagnetic theory, revolving accelerated electron must continuously radiate energy and spiral into nucleus. Cannot explain stability or discrete line spectra.
3. Wave & Particle Dual Nature of Radiation
3.1 Electromagnetic Wave Parameters
$$\mathbf{c = \nu \lambda} \quad \text{and} \quad \mathbf{\bar{\nu} = \frac{1}{\lambda}}$$
3.2 Black Body Radiation Graph
Ideal body absorbing/emitting all frequencies. Intensity vs wavelength peak shifts to shorter wavelength at higher temperature (Planck's quantum concept).
3.3 Planck's Quantum Theory & Photoelectric Effect
Planck's Energy Quantum Formula
Radiant energy is emitted or absorbed in discrete packets called quanta (photons):
$$\mathbf{E = h\nu = \frac{hc}{\lambda}}$$
Where $h = \text{Planck's constant} = \mathbf{6.626 \times 10^{-34}\text{ J s}}$.
Einstein's Photoelectric Equation
Ejection of electrons when light of frequency $\nu > \nu_0$ strikes metal surface:
$$\mathbf{E = W_0 + K.E._{max} \implies h\nu = h\nu_0 + \frac{1}{2} m_e v^2 = e V_0}$$
Where $W_0 = h\nu_0 = \text{Work Function}$, $\nu_0 = \text{Threshold Frequency}$, $V_0 = \text{Stopping Potential}$.
4. Bohr's Model for Hydrogen Atom
Bohr's Postulates & Derived Formulas
1. Quantization of Angular Momentum:
$$\mathbf{m v r = \frac{n h}{2\pi}} \quad (n = 1, 2, 3...)$$
2. Radius of $n^{\text{th}}$ Orbit:
$$\mathbf{r_n = 0.529 \times \frac{n^2}{Z} \text{ \AA} = 52.9 \times \frac{n^2}{Z} \text{ pm}}$$
3. Velocity of Electron:
$$\mathbf{v_n = 2.18 \times 10^6 \times \frac{Z}{n} \text{ m/s}}$$
4. Total Energy of Electron:
$$\mathbf{E_n = -13.6 \times \frac{Z^2}{n^2} \text{ eV/atom} = -2.18 \times 10^{-18} \times \frac{Z^2}{n^2} \text{ J/atom}}$$
4.1 Hydrogen Line Spectrum & Rydberg Equation
Rydberg Formula for Spectral Lines:
$$\mathbf{\bar{\nu} = \frac{1}{\lambda} = R_H Z^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)}$$
Where $R_H = \text{Rydberg Constant} = \mathbf{109,677\text{ cm}^{-1} = 1.097 \times 10^7\text{ m}^{-1}}$.
Spectral Series of Hydrogen Atom
| Series | $n_1$ | $n_2$ | Spectral Region |
| Lyman | 1 | $2, 3, 4...$ | Ultraviolet (UV) |
| Balmer | 2 | $3, 4, 5...$ | Visible |
| Paschen | 3 | $4, 5, 6...$ | Infrared (IR) |
| Brackett | 4 | $5, 6, 7...$ | Far Infrared |
| Pfund | 5 | $6, 7, 8...$ | Far Infrared |
5. Towards Quantum Mechanical Model
5.1 De Broglie Dual Nature of Matter
$$\mathbf{\lambda = \frac{h}{p} = \frac{h}{m v} = \frac{h}{\sqrt{2 m (K.E.)}} = \frac{h}{\sqrt{2 m q V}}}$$
5.2 Heisenberg's Uncertainty Principle
$$\mathbf{\Delta x \cdot \Delta p \ge \frac{h}{4\pi} \implies \Delta x \cdot (m \Delta v) \ge \frac{h}{4\pi}}$$
Conclusion: Impossible to determine position and velocity of microscopic particle simultaneously with absolute accuracy $\Rightarrow$ rules out Bohr's fixed circular orbits!
6. Quantum Mechanical Model & Quantum Numbers
6.1 Radial Probability Distribution Curves
- Total Nodes $= n - 1$
- Angular Nodes $= l$
- Radial Nodes $= n - l - 1$
6.2 Shapes of Atomic Orbitals
The Four Quantum Numbers
| Quantum Number |
Symbol |
Allowed Values |
Significance |
| Principal |
$n$ |
$1, 2, 3, 4...$ |
Main shell, size & energy. Total orbitals $= n^2$, Max $e^- = 2n^2$. |
| Azimuthal |
$l$ |
$0$ to $(n-1)$ |
Subshell shape ($s=0, p=1, d=2, f=3$). Orbital angular momentum $L = \sqrt{l(l+1)}\frac{h}{2\pi}$. |
| Magnetic |
$m_l$ |
$-l \text{ to } +l$ |
Spatial 3D orientation. Total orientations $= (2l + 1)$. |
| Spin |
$m_s$ |
$+\frac{1}{2}, -\frac{1}{2}$ |
Electron spin orientation. Spin angular momentum $= \sqrt{s(s+1)}\frac{h}{2\pi}$. |
7. Rules for Electronic Configuration
7.1 Aufbau Principle
Orbitals are filled in increasing order of energy via $(n + l)$ rule. Order: $1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s...$
7.2 Pauli Exclusion & Hund's Rule of Maximum Multiplicity
- Pauli Principle: No two electrons can have identical set of all 4 quantum numbers. Max 2 $e^-$ per orbital with opposite spins.
- Hund's Rule: Electron pairing in degenerate orbitals ($p, d, f$) does not occur until each orbital is singly occupied with parallel spins.
Exceptional Configurations (High-Yield)
Half-filled & fully-filled subshells possess extra stability due to high exchange energy & symmetry:
• Chromium ($\text{Cr}, Z=24$): $[\text{Ar}] 3d^5 4s^1$ (not $3d^4 4s^2$)
• Copper ($\text{Cu}, Z=29$): $[\text{Ar}] 3d^{10} 4s^1$ (not $3d^9 4s^2$)
Student Quick Revision Self-Test
Q1 (JEE Main): Calculate the energy and radius of electron in 1st orbit of $\text{He}^+$ ($Z=2$).
Solution:
$E_1 = -13.6 \times \frac{2^2}{1^2} = \mathbf{-54.4\text{ eV}} = \mathbf{-8.72 \times 10^{-18}\text{ J}}$.
$r_1 = 0.529 \times \frac{1^2}{2} = \mathbf{0.2645\text{ \AA}} = \mathbf{26.45\text{ pm}}$.
Q2 (NEET): How many radial nodes and angular nodes are present in $4p$ orbital?
Solution: For $4p$: $n=4, l=1$.
• Angular Nodes $= l = \mathbf{1}$.
• Radial Nodes $= n - l - 1 = 4 - 1 - 1 = \mathbf{2}$.
• Total Nodes $= n - 1 = 3$.
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