The radius of 2ⁿᵈ orbit of He⁺ of Bohr's model is r₁ and that of fourth orbit of Be³⁺ is represented as r₂. Now the ratio r₂/r₁ is x: 1. The value of x is ___________.

Solution

Physics sir · 3 steps

Bohr radius goes as n²/Z, so work out n²/Z for each ion and divide.

1. Bohr radius formula. For a hydrogen-like ion, the radius of the nth orbit grows as n² and shrinks as the nuclear charge Z grows. The constant a₀ = 0.529 Å is the same for every ion, so it cancels in a ratio.

   rₙ = a₀ n²/Z

2. Each orbit. For He⁺, Z = 2 and n = 2, so r₁ = a₀ × 4/2 = 2a₀. For Be³⁺, Z = 4 and n = 4, so r₂ = a₀ × 16/4 = 4a₀.
3. So. r₂/r₁ = 4a₀/2a₀ = 2, so the ratio is 2 : 1 and x = 2.

The common slip: 4. Leaving out Z and comparing only n² gives 16/4 = 4, not 2. A larger nuclear charge pulls the orbit inward.

The idea to keep. In any hydrogen-like ion, r ∝ n²/Z. For a ratio, compare n²/Z for the two orbits; the Bohr radius a₀ always cancels.

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