The radius of first Bohr orbit of hydrogen atom is 0.529 $˚ A$ . Calculate the radii of $(i)$ the third orbit of $H e^{+}$ ion and $(i i)$ the second orbit of $L i^{2 +}$ ion.

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Solution

Let us consider the $n^{t h}$ bohr orbit,
$r_{n} = \frac{n^{2} h^{2}}{4 π^{2} m Z e^{2}}$

For hydrogen atom $z = 1$ , first orbit $n = 1$

$r_{1} = \frac{h^{2}}{4 π r^{2} m e^{2}} = 0.592 A^{0}$

(i)For $H e^{+} i o n, Z = 2, t h i r d o r b i t, n = 3$

$r_{3} (H e^{+}) = \frac{3^{2} h^{2}}{4 π^{2} m \times 2 \times e^{2}}$

$= \frac{9}{2} [\frac{h^{2}}{4 π^{2} m e^{2}}]$

$= \frac{9}{2} \times 0.592$

$= 2.380 A^{0}$

(ii) For $L i^{2 +} i o n, Z = 3, s e c o n d o r b i t n = 2$

$r_{2} (L i^{2 +}) = \frac{2^{2} h^{2}}{4 π^{2} m \times 3 \times e^{2}}$

$= \frac{4}{3} [\frac{h^{2}}{4 π^{2} m e^{2}}]$

= $0.705 A^{0}$

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