Concept: Niels Bohr derived the radius of atomic orbits by combining classical mechanics (centripetal force matching electrostatic force) with his quantum postulate regarding angular momentum.
Formula:$$ r_n = \frac{n^2 h^2}{4\pi^2 k m e^2} $$
Solution:- In the formula above, all terms except \( n \) (Planck's constant \( h \), Coulomb's constant \( k \), electron mass \( m \), and elementary charge \( e \)) are fundamental constants for a given atom.
- Grouping these constants together yields: \( r_n = (\text{constant}) \times n^2 \).
- For hydrogen, this simplifies to \( r_n = 0.529\text{\AA} \times n^2 \).
- Therefore, the radius scales quadratically; it is strictly proportional to \( n^2 \).
Why other options are incorrect:Options A, B, and D misrepresent the algebraic derivation. If the radius was simply proportional to \( n \), the quantum spacing of atoms would be strictly linear rather than expanding vastly outward.
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