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