JEE Main · 2021 · 24 Feb 2021 · Shift 2 · Atomic Structure

Bohr's model gives the orbital speed as $v\propto \frac{Z}{n}$ and the orbit radius as $r\propto \frac{n^2}{Z}$, and every listed quantity follows from these two. The kinetic energy equals the magnitude of the total energy, $\mathrm{KE}=13.6\frac{Z^2}{n^2}$ eV, so the proportionality $\mathrm{KE}\propto \frac{Z^2}{n^2}$ holds. Since $v\propto Z/n$, the product $vn$ varies as $Z$ and not as $Z^2$, so that proportionality fails. The frequency of revolution is $f=\frac{v}{2\pi r}\propto \frac{Z/n}{n^2/Z}=\frac{Z^2}{n^3}$, not $\frac{Z^3}{n^3}$, so that one fails as well. The electrostatic attraction is $F=\frac{kZ{e}^2}{r^2}\propto \frac{Z}{(n^2/Z{)}^2}=\frac{Z^3}{n^4}$, so that proportionality holds. Only the kinetic energy statement and the Coulombic force statement are correct.