Q. The wavelength of characteristic $X$-rays $K_{\alpha}$ line emitted by hydrogen like atom is 0.32$\ \mathring{\mathrm{A}}$. The wavelength of $K_{\beta}$ line emitted by the same element is
A) 0.21 $\mathring{\mathrm{A}}$
B) 0.27 $\mathring{\mathrm{A}}$
C) 0.33 $\mathring{\mathrm{A}}$
D) 0.40 $\mathring{\mathrm{A}}$
Explanation
(b) $\frac{1}{{\lambda_{K}}_{\alpha}} = R(Z - b)^{2}\left\lbrack \frac{1}{1^{2}} - \frac{1}{2^{2}} \right\rbrack$
$\frac{1}{{\lambda_{K}}_{\beta}} = R(Z - b)^{2}\left( \frac{1}{1^{2}} - \frac{1}{3^{2}} \right)$
$\frac{{\lambda_{K}}_{\beta}}{{\lambda_{K}}_{\alpha}} = \frac{(3/4)}{(8/9)} = \frac{3}{4} \times \frac{9}{8} = \frac{27}{32}$
or ${\lambda_{K}}_{\beta} = \frac{27}{32} \times {\lambda_{K}}_{\alpha} = \frac{27}{32} \times 0.32 = 0.27\mathring{\mathrm{A}}$.
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.