Calculate the wavelength of the $k_a$ line for $z=31$, when $a=5 \times 10^7 \mathrm{~Hz}^{1 / 2}$ for a…

Calculate the wavelength of the $k_a$ line for $z=31$, when $a=5 \times 10^7 \mathrm{~Hz}^{1 / 2}$ for a characteristic X-ray spectrum.
  1. $1.33 Ã…$
  2. $1.33 \mathrm{~nm}$
  3. $133 \times 10^{-10} \mathrm{~m}$
  4. $133 \mathrm{~nm}$

Solution

Given, $z=31$ and $a=5 \times 10^7 \mathrm{~Hz}^{1 / 2}$ $\begin{aligned} \sqrt{v} & =a(z-1) \\ \text { or } \quad v & =a^2(z-1)^2 \\ & =\left(5 \times 10^7\right)^2(31-1)^2 \\ & =25 \times 10^{14} \times 30 \times 30 \\ & =2.25 \times 10^{18} \mathrm{~Hz} \\ \therefore \quad & =\frac{c}{\lambda} \Rightarrow \lambda=\frac{c}{v} \\ & =\frac{3 \times 10^8 \mathrm{~ms}^{-1}}{2.25 \times 10^{18} \mathrm{~Hz}} \\ & =1.33 \times 10^{-10} \mathrm{~m} \\ & =1.33 Ã… \end{aligned}$

Asked in: AP EAMCET 2011

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