Enter the orbit number n and the atomic number Z to get the radius rₙ and the energy Eₙ of that Bohr orbit, for hydrogen or any hydrogen-like ion.
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From the article Bohr Model — Radius and Energy of the nth Orbit.
Question: For a hydrogen atom in the n = 3 orbit, find the radius, the energy and the electron's speed.
Hydrogen, so Z = 1, and n = 3, so n² = 9.
Radius: r3 = 0.529 × 9 ÷ 1 = 4.761 Å = 476.1 pm
Energy: E3 = −13.6 × 1 ÷ 9 = −1.5111 ≈ −1.51 eV
Speed: v3 = 2.18 × 106 × 1 ÷ 3 = 7.27 × 105 m/s
Check with the quantisation condition. The angular momentum should equal
3h/2π. Using me = 9.11 × 10−31 kg,
r = 4.761 × 10−10 m:
m v r = (9.11 × 10−31)(7.27 × 105)(4.761 ×
10−10) = 3.153 × 10−34 J·s
3h / 2π = 3 × (6.626 × 10−34) ÷ 6.2832 =
3.164 × 10−34 J·s. ✔ (agreeing to the rounding in the
three-figure constants)
Worked in full in Bohr Model — Radius and Energy of the nth Orbit.
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