If an electron is moving in the nᵗʰ orbit of the hydrogen atom, then its velocity (vₙ) for the nᵗʰ orbit is given as :

Solution

Physics sir · 3 steps

Combine the force balance with Bohr's quantisation rule, and r cancels to leave v in terms of n alone.

1. Force balance. The Coulomb pull of the nucleus supplies the centripetal force: mv²/r = ke²/r². This gives mv² r = ke².
2. Quantisation rule. Bohr's postulate fixes the angular momentum: mvr = nh/2π. Divide the first result by this one, and m and r both cancel.

   vₙ = (2π k e²)/nh = e²/(2ε₀ n h)

3. So. Everything except n is a constant, so vₙ ∝ 1/n. Numerically, vₙ = (2.19 × 10⁶)/n m s⁻¹ for hydrogen. The answer is (a).

The idea to keep. Remember the Bohr trio for hydrogen-like atoms: r ∝ n²/Z, v ∝ Z/n, E ∝ -Z²/n². Outer orbits are bigger and slower.

Answer from the NTA final answer key. Solution by PrepWiser.