The wavelength of characteristic X-ray Kα like emitted by hydrogen line atom is 0.32A0. The wavelength of Kβ line emitted by the same element is N×10−2A0 .The value of N is?

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The wavelength of characteristic X-ray $K_{α}$ like emitted by hydrogen line atom is $0.32 \overset{0}{A}$ . The wavelength of $K_{β}$ line emitted by the same element is $N \times 10^{- 2} \overset{0}{A}$ .The value of N is?

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answer is 27.

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Detailed Solution

We use for $K_{α}$ line
$\frac{1}{λ_{1}} = R {(Z - 1)}^{2} [1 - \frac{1}{4}]$
for $K_{β}$ line we use
$\begin{array}{l} \frac{1}{λ_{2}} = R {(Z - 1)}^{2} [1 - \frac{1}{9}] \\ \frac{(1)}{(2)} \Rightarrow \frac{λ_{2}}{λ_{1}} = \frac{3}{4} \times \frac{9}{8} \\ \Rightarrow λ_{2} = \frac{27}{32} λ = \frac{27}{32} \times 0.32 Å \\ = 0.27 Å \end{array}$