Bohr Model

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In the Bohr model of a hydrogen atom, an electron orbits the nucleus in the second allowed stationary orbit, and a student wishes to know the radius of that orbit.

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Bohr postulated that only orbits in which the electron's angular momentum is an integral multiple of $\frac{h}{2\pi}$ are allowed, and from this quantisation NCERT derives the orbit radius $r_n=\frac{n^2 h^2 \varepsilon_0}{\pi m e^2}=n^2 a_0$, where $a_0=0.053$ nm is the Bohr radius. The radius therefore scales as the square of the principal quantum number. For the second orbit $n=2$, so $r_2=2^2	imes 0.053=4	imes 0.053=0.212$ nm. The choice 0.053 nm is wrong because that is the ground-state radius for $n=1$ and ignores the $n^2$ scaling. The choice 0.106 nm is wrong because it multiplies the Bohr radius by $n$ rather than $n^2$, a common linear-scaling error. The choice 0.477 nm is wrong because it corresponds to $n=3$ ($9a_0$) and not the second orbit. As stated in NCERT Class 12, Chapter 12 (Atoms), the discrete radii grow rapidly with $n$, so the electron is far weaker bound in outer orbits. A magnitude check confirms that quadrupling the ground radius gives roughly a fifth of a nanometre, consistent with atomic dimensions.

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