Atomic Radius Trend Calculator Formula

Understand the math behind the atomic radius trend calculator. Each variable explained with a worked example.

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Formulas Used

Relative Atomic Radius (n²/Z_eff)

relative_radius = pow(n_quantum, 2) / z_eff

Approximate Radius

approx_radius_pm = (pow(n_quantum, 2) / z_eff) * 53

Variables

VariableDescriptionDefault
n_quantumPrincipal Quantum Number (n)3
z_effEffective Nuclear Charge (Z_eff)2.2

How It Works

Atomic Radius Trend

Approximate Formula

r ∝ n² / Z_eff

In Bohr model terms: r ≈ (n² / Z_eff) × a₀

Where a₀ = 53 pm (Bohr radius)

Where:

  • n = principal quantum number (shell number)
  • Z_eff = effective nuclear charge

    • Atomic radius decreases across a period (Z_eff increases, same n) and increases down a group (n increases faster than Z_eff).

    Worked Example

    Sodium: n = 3 (3rd period), Z_eff ≈ 2.2.

    n_quantum = 3z_eff = 2.2
    1. 01r ∝ n²/Z_eff
    2. 02r ∝ 9/2.2
    3. 03r ∝ 4.09
    4. 04Approximate radius = 4.09 × 53 pm ≈ 217 pm
    5. 05Actual Na radius ≈ 186 pm (this is an estimate)

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