Question

A target element $A$ is bombarded with electrons and the wavelengths of the characteristic spectrum are measured. A second characteristic spectrum is also obtained because of an impurity in the target. The wavelength of the $K_{\alpha }$ lines are $196\text{pm}$ (element $A$) and $169\text{pm}$ (impurity). If the atomic number of impurity is $z=(10x-1)$. Find the value of $x$.
(atomic number of element $A$ is 27).

Answer 1

From Moseley’s law,
$\sqrt{v}=a(z-b)$
$\because v=\frac{C}{\lambda };b=1,a=\sqrt{\frac{3Rc}{4}}$
$z-1=\sqrt{\frac{4v}{3cR}}$
$z-1\propto \sqrt{\frac{1}{\lambda }}$
Let $z_2$ be atomic number of impurity, $z_1$ be atomic number of $A$
$\frac{z_2-1}{z_1-1}=\sqrt{\frac{{\lambda }_1}{{\lambda }_2}}=\sqrt{\frac{196}{169}}=\frac{14}{13}$
$z_2-1=26\times \frac{14}{13}=28$
$z_2=29=10x-1$
$x=3$