Bohr Model - Orbit Radius

In the Bohr model the radius of a permitted orbit is quantised and given by $r_n=0.53n^2$ Å, so the orbital size scales with the square of the principal quantum number rather than linearly. This square dependence follows from combining the angular-momentum quantisation $mvr=n\hslash$ with the Coulomb attraction supplying the required centripetal force. Setting $n=3$ gives $r_3=0.53\times 3^2=0.53\times 9=4.77$ Å. The value 1.59 Å is wrong because it multiplies by 3 instead of $3^2$, ignoring the square dependence. The value 0.53 Å is the ground-state radius for $n=1$ and does not describe the third orbit. The value 14.31 Å incorrectly assumes an $n^3=27$ scaling. This is precisely the standard NCERT Bohr-orbit relation. A quick plausibility check confirms that higher orbits must be physically larger than the ground state, consistent with 4.77 Å clearly exceeding 0.53 Å.