First slide

X-rays

Question

A cobalt (atomic no. = 27) target is bombarded with electrons, and the wavelengths of its characteristic X-ray spectrum are measured. A second weak characteristic spectrum is also found, due to an impurity in the target. The wavelengths of the $K_{α}$ lines are 225.0 pm (cobalt) and 100.0 pm (impurity). If the atomic number of the impurity is 10 N. The value of N is _________(take b = 1)

Moderate

Solution

Using Mosely's law for both cobalt and impurity
$\sqrt{f} = K (Z - b) \Rightarrow \sqrt{\frac{c}{λ}} = K (Z - b) \sqrt{f} = K (Z - 1) \Rightarrow \sqrt{\frac{c}{λ}} = K (Z - 1) (∵ b = 1)$
$\begin{array}{l} \Rightarrow \sqrt{\frac{c}{λ_{co}}} = K (Z_{Co} - 1) and \sqrt{\frac{c}{λ_{x}}} = K (Z_{x} - 1) \\ \Rightarrow \sqrt{\frac{λ_{co}}{λ_{x}}} = \frac{Z_{x} - 1}{Z_{co} - 1} \\ \sqrt{\frac{225}{100}} = \frac{Z_{x} - 1}{27 - 1} \\ \Rightarrow Z_{X} = 40 = 4 \times 10 \\ N = 4 \end{array}$

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