Q1(a). Probability of the 1s electron at a₀ and at 10a₀

Answer in brief

P(a₀)/P(10a₀) = e¹⁸/100 = 6.57 × 10⁵, using the radial probability P(r) = 4πr²ψ².

Key points

• “At a distance r” means anywhere on the sphere of radius r, so the quantity to compare is the radial probability 4πr²ψ² dr, not ψ² alone.
• For the 1s orbital ψ = (1/√(πa₀³)) e^−r/a₀, so P(r) = (4r²/a₀³) e^−2r/a₀.
• In the ratio the constants cancel: a₀²e⁻² / (100a₀²e⁻²⁰) = e¹⁸/100 = 6.57 × 10⁵.
• The shell at 10a₀ is a hundred times larger, but the exponential fall (e¹⁸) outweighs it; a₀ is also the most probable distance.
• If “at a point” is meant, the ratio of probability densities is ψ²(a₀)/ψ²(10a₀) = e¹⁸ = 6.57 × 10⁷.