Radius Calculation

🔬 Bohr’s Model – Derivation of Atomic Radius

🔹 Key Assumptions

1. Electron revolves in a circular orbit due to Coulombic attraction.
2. Angular momentum is quantized: m × v × r = (n × h) / 2π

🔹 Step 1: Equating Forces

Electrostatic Force = Centripetal Force

(m × v²) / r = (1 / 4πε₀) × (Z × e²) / r²

🔹 Step 2: Angular Momentum Quantization

Given: m × v × r = (n × h) / (2π)

⇒ v = (n × h) / (2π × m × r)

🔹 Step 3: Substitution & Simplification

Substitute v into the force equation:

(m / r) × [ (n × h) / (2π × m × r) ]² = (Z × e²) / (4π × ε₀ × r²)

After simplifying, we get:

rₙ = (n² × h² × ε₀) / (π × m × Z × e²)

🔹 Final Formula

rₙ = (n² × a₀) / Z

Where: a₀ ≈ 0.529 Å is the Bohr radius for hydrogen.