First slide

X-rays

Question

X-rays emitted from a copper target and a molybdenum target are found to contain a line of wavelength 22.85 nm attributed to the $K_{α}$ line of an impurity element. The $K_{α}$
lines of copper (Z = 29) and molybdenum (Z = 42) have wavelengths 15.42 nm and 7 .12 nm, respectively. The atomic number of the impurity element is

Difficult

A

22
B

23
C

24
D

25

Solution

$Using Moseley's law, v^{1 / 2} = a (z - b)$
${(\frac{c}{λ_{1}})}^{1 / 2} = a (z_{1} - b) and {(\frac{c}{λ_{2}})}^{1 / 2} = a (z_{2} - b)$
$\begin{array}{rcl} {(\frac{λ_{2}}{λ_{1}})}^{\frac{1}{2}} & = & \frac{a (z_{1} - b)}{a (z_{2} - b)} \Rightarrow {(\frac{7.12}{15.42})}^{\frac{1}{2}} = \frac{(29 - b)}{(42 - b)} \\ (42 - b) & = & 1.47 (29 - b) \Rightarrow b = 1.44 \\ {(\frac{λ_{1}}{λ})}^{\frac{1}{2}} & = & \frac{(z - 1.44)}{(z_{1} - 1.44)} \end{array}$
${(\frac{15.42}{22.85})}^{\frac{1}{2}} (27.56) = z - 1.44 \Rightarrow z = 24$

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