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Q.
The radius of first Bohr orbit of hydrogen atom is $0.529 \AA$. Calculate the radii of (i) the third orbit of $He ^{+}$ion and (ii) the second orbit of $Li ^{2+}$ ion.
Let us consider the $n ^{\text {th }}$ bohr orbit,
$
r_{n}=\frac{n^{2} h^{2}}{4 \pi^{2} mZe ^{2}}
$
For hydrogen atom $z =1$, first orbit $n =1$
$
r _{1}=\frac{ h ^{2}}{4 \pi r ^{2} me ^{2}}=0.592 A ^{0}
$
(i)For $He ^{+}$ion, $Z =2$, third orbit, $n =3$
$
r _{3}\left( He ^{+}\right)=\frac{3^{2} h ^{2}}{4 \pi^{2} m \times 2 \times e ^{2}}
$
$
=\frac{9}{2}\left[\frac{ h ^{2}}{4 \pi^{2} me ^{2}}\right]
$
$
=\frac{9}{2} \times 0.592
$
$
=2.380 A ^{0}
$
(ii) For $Li ^{2+}$ ion, $Z =3$, second orbit $n =2$
$
\begin{array}{l}
r _{2}\left( Li ^{2+}\right)=\frac{2^{2} h ^{2}}{4 \pi^{2} m \times 3 \times e ^{2}} \\
=\frac{4}{3}\left[\frac{ h ^{2}}{4 \pi^{2} me ^{2}}\right] \\
=0.705 A ^{0}
\end{array}
$