परमाणु संरचना संक्षेप

Overview

लेक्चर का विषय: Structure of Atom (पूर्ण रिवीजन, चार हिस्सों में विभाजित)
उद्देश्य: प्रमुख कॉन्सेप्ट्स और फॉर्मुले संक्षेप में दोहराना (exam-oriented)
चार मुख्य हिस्से: discovery of subatomic particles, atomic models, spectra & quantum ideas, quantum numbers और orbitals

इलेक्ट्रॉन: कैथोड रे एक्सपेरिमेंट (Thomson) → कैथोड से अनोड की दिशा में invisible rays
कैथोड रे की विशेषताएँ: negatively charged particles, straight-line travel, heating और mechanical effects, गैस/कैथोड मेटरियल से स्वतंत्र
E/m (specific charge) of electron: Thomson ने निकाला → 1.75 × 10^11 C kg^-1 (शॉर्टकट)
चार्ज (e): Millikan oil drop experiment → e = 1.6 × 10^-19 C
द्रव्यमान (m_e): m = 9.1 × 10^-31 kg
प्रोटॉन: perforated cathode/anode experiments → canal (anode) rays, positively charged particles; E/m निर्भर करता है गैस पर, hydrogen के लिए अधिकतम
न्यूट्रॉन: James Chadwick — बीरिलियम पर α-bombardment से neutral particles की खोज

Thomson (Plum pudding): positive sphere में embedded electrons; atom की neutrality; radius ~10^-10 m (सीमाएँ: electron distribution का व्यवहार नहीं समझाया)
Rutherford (Planetary): α-scattering on thin gold foil → अधिकांश atom खाली, केन्द्र में छोटा positive nucleus; कुछ α-rays bounce back → nucleus का प्रमाण
- Distance of closest approach ∝ atomic number, ∝ 1/velocity^2
- Radius of nucleus ≈ 10^-15 m order
- समस्या: accelerated electron radiation (Maxwell) → classical instability
Bohr model:
- Postulates: angular momentum = n h / 2π; electrons fixed orbits की energy orbit-specific; energy absorption/emission discrete
- Radius of nth orbit: r_n = 0.529 n^2 / Z Å ( = 52.9 n^2 / Z pm)
- Energy of nth orbit: E_n = -13.6 Z^2 / n^2 eV
- Relations: total energy = - (kinetic energy), kinetic = -½ (potential)
- Ionisation: n → ∞ पर E → 0 (ionisation energy = -E_n)
- सफल: hydrogen-like atoms; असफल: spectra line splitting, multi-electron systems
De Broglie: matter waves, λ = h / p = h / mv (wave nature of particles)
- For kinetic energy form: λ ∝ 1/√(KE)
- De Broglie validated Bohr quantisation: m v r = n h / 2π
Heisenberg Uncertainty Principle:
- Δx · Δp ≥ h / 4π (या Δx · m Δv = h / 4π for equality in questions)
- ΔE · Δt ≥ h / 4π (energy-time uncertainty)

EM संबंध: ν λ = c (c = 3.0 × 10^8 m s^-1)
Wave number = 1/λ; time period = 1/ν
EM spectrum: radio → microwave → IR → visible → UV → X-ray → γ
Planck’s Quantum Theory:
- Energy of photon: E = h ν ; h = 6.626 × 10^-34 J·s
- E = h c / λ ; n photons → E_total = n h ν
- Units: 1 eV = 1.6 × 10^-19 J
Photoelectric effect:
- Experiment: Hertz; Explanation: Einstein
- Condition: incident energy ≥ threshold energy (h ν ≥ h ν_0)
- Kinetic energy of emitted electron: KE_max = h ν - h ν_0
- Graph KE vs ν: slope = h ; intercept = -h ν_0
- Intensity affects number of photoelectrons, not KE
- Stopping potential V_s = KE_max / e

Rydberg formula (wavelength): 1/λ = R_H (1/n1^2 - 1/n2^2), R_H = 1.09677 × 10^5 cm^-1 (≈ 1/912 Å for series limits)
Series:
- Lyman: n1 = 1 → UV
- Balmer: n1 = 2 → visible
- Paschen: n1 = 3 → IR
- Brackett, Pfund, ... continue (n1 = 4,5 ...)
Number of spectral lines between n1 and n2: (n2 - n1)(n2 - n1 + 1)/2
Limiting line: transition from n = ∞ to specific n1

Schrödinger equation → wavefunction ψ (ψ^2 gives probability density)
Solutions give three quantum numbers from wave equation: n, l, m
- Principal quantum number n: 1 → ∞ (size, energy level)
- Azimuthal (angular momentum) quantum number l: 0 → n-1 (subshell type: s,p,d,f)
- Magnetic quantum number m_l: -l → +l (orientation of orbital)
Fourth quantum number (spin) m_s: ±½ (not from Schrödinger; intrinsic property)

Principal (n): shell (K,L,M...), radius ∝ n^2, energy ∝ -1/n^2
Azimuthal (l): sub-shells
- l = 0 → s ; l = 1 → p ; l = 2 → d ; l = 3 → f
- number of orbitals in a subshell = 2l + 1
- max electrons in subshell = 4l + 2
Magnetic (m_l): orientation values -l ... +l (gives number of orbitals)
Spin (m_s): +½ or -½ ; spin angular momentum = √(s(s+1)) ħ
Angular momentum magnitude for given l: √(l(l+1)) ħ

s-orbital: spherical (1 orbital)
p-orbitals: dumbbell-shaped (3 orbitals: m = -1,0,+1 → px, py, pz)
d-orbitals: 5 orbitals (m = -2,-1,0,+1,+2); d_z^2 unique (donut + lobes); others double-dumbbell
Nodes:
- Radial (or spherical) nodes = n - l - 1
- Angular nodes = l (nodal planes where probability = 0)
- S: angular nodes = 0 ; P: one nodal plane ; D: two nodal planes (except special d_z^2)

Aufbau principle (filling order): fill lower energy orbitals first (n + l rule)
- Higher (n + l) → higher energy; tie → higher n → higher energy
Hund’s rule: maximum multiplicity — orbitals of same subshell singly occupied with parallel spins first
Pauli exclusion: no two electrons in an atom have same set of four quantum numbers
For hydrogen-like (one-electron) atoms: subshells with same n are degenerate (energy equal)
For multi-electron atoms: energy order modified (e.g., 4s filled before 3d, etc.)
Shortcuts for filling (exam tip): S starts at n=1, P at n=2, D at n=3, F at n=4; inert gases mark closed shells

Paramagnetic: attracted by magnetic field; at least one unpaired electron
Diamagnetic: all electrons paired; weakly repelled
Magnetic moment (spin-only approximation): μ = √(n(n+2)) BM , where n = number of unpaired electrons