Question

An element with atomic number Z = 11 emits X-ray of wavelength λ. The atomic number of the element which emits Kα X-ray of wavelength 4λ is?
A. 11
B. 44
C. 6
D. 5

Answer 1

Hint: As we all know that, the Moseley’s law is an empirical law related to atoms that emit $X$ rays. According to Moseley’s experiment, the square root of the frequencies of various elements is plotted against the position number in the periodic table.

Complete step by step solution:
We all know that in an X-ray spectrum, the frequency of the spectral line is directly proportional to the square root of the atomic number (Z) of the element under study.
$\sqrt v = a\left( {Z - b} \right)$…… (I)
As we can see that here $a$ and $b$ are constants depending upon the subsequent spectral line and $v$ is the frequency
We can substitute $v = \dfrac{c}{\lambda }$ in equation (I) and here $c$ is the speed of light and $\lambda $ is the wavelength of the spectral line. Now after substituting we will now get the following result as,
\[ \Rightarrow \sqrt {\dfrac{c}{\lambda }} = a\left( {Z - b} \right)\]
\[ \Rightarrow \dfrac{c}{\lambda } = {a^2}{\left( {Z - b} \right)^2}\]
So, we can write it for two different elements using 1 and 2 as subscripts.
$ \Rightarrow \dfrac{c}{{{\lambda _1}}} = {a^2}{\left( {{Z_1} - b} \right)^2}$…… (II)
$ \Rightarrow \dfrac{c}{{{\lambda _2}}} = {a^2}{\left( {{Z_2} - b} \right)^2}$…… (III)
Here ${\lambda _1}$ is for the given ${Z_1} = 11$ and ${\lambda _2}$ is for the unknown ${Z_2}$ .
So we can divide equation (I) by equation (II) to get the relationship between
${\lambda _1}$ and ${\lambda _2}$.
$\therefore \dfrac{{{\lambda _1}}}{{{\lambda _2}}} = \dfrac{{{{\left( {{Z_2} - 1} \right)}^2}}}{{{{\left( {{Z_1} - 1} \right)}^2}}}$…… (IV)
We can here now substitute ${\lambda _1} = \lambda $, ${Z_1} = 11$, ${\lambda _2} = 4\lambda $ in equation (IV) to find the value of ${Z_2}$.
$ \Rightarrow \dfrac{\lambda }{{4\lambda }} = \dfrac{{{{\left( {{Z_2} - 1} \right)}^2}}}{{{{\left( {11 - 1} \right)}^2}}}$
$ \Rightarrow {\left( {{Z_2} - 1} \right)^2} = 25$
$\therefore {Z_2} = 6$

$\therefore$ Hence, Option (C) is correct.

Note:
As we all know that using Moseley’s law, gases like krypton and argon, Nickel and cobalt were placed in a proper way in a periodic table. We also know that many new elements like Technetium, Rhodium, etc. were discovered using this law.