The radius of first Bohr orbit of hydrogen atom is 0.529 A. Calculate

1.	The radius of first Bohr orbit of hydrogen atom is 0.529 A. Calculate the radii of (i) the third orbit of He+ ion and (ii) the second orbit of Li2+ ion.
Answer» Radius of n^th Bohr orbit, r_n = \(\frac{n^2b^2}{4 \pi^2 m Ze^2}\) For hydrogen atom Z = 1, first orbit n = 1 r₁ = \(\frac{b^2}{4 \pi^2 m e^2}\) = 0.529 Å (i) For He⁺ ion, Z = 2, third orbit, n = 3 r₃(He⁺) = \(\frac{3^2b^2}{4 \pi^2m \times 2 \times e^2}\) = \(\frac{9}{2}\Big [\frac{b^2}{4 \pi^2me^2} \Big]\) = \(\frac{9}{2} \)x 0.529 = 2.380 Å (ii) For Li²⁺ ion, Z = 3, Second orbit, n = 2 r₂(Li²⁺) = \(\frac{2^2b^2}{4 \pi^2 m \times 3 \times e^2}\) = \(\frac{4}{3} \Big[ \frac{b^2}{4 \pi^2 me^2} \Big]\) = \(\frac{4}{3}\)x 0.529 = 0.7053 Å

The radius of first Bohr orbit of hydrogen atom is 0.529 A. Calculate the radii of (i) the third orbit of He+ ion and (ii) the second orbit of Li2+ ion.

Answer»

Radius of n^th Bohr orbit, r_n = \(\frac{n^2b^2}{4 \pi^2 m Ze^2}\)

For hydrogen atom Z = 1, first orbit n = 1

r₁ = \(\frac{b^2}{4 \pi^2 m e^2}\) = 0.529 Å

(i) For He⁺ ion, Z = 2, third orbit, n = 3

r₃(He⁺) = \(\frac{3^2b^2}{4 \pi^2m \times 2 \times e^2}\)

= \(\frac{9}{2}\Big [\frac{b^2}{4 \pi^2me^2} \Big]\) = \(\frac{9}{2} \)x 0.529 = 2.380 Å

(ii) For Li²⁺ ion, Z = 3, Second orbit, n = 2

r₂(Li²⁺) = \(\frac{2^2b^2}{4 \pi^2 m \times 3 \times e^2}\) = \(\frac{4}{3} \Big[ \frac{b^2}{4 \pi^2 me^2} \Big]\)

= \(\frac{4}{3}\)x 0.529 = 0.7053 Å