Question

If the atom \[_{100}F{m^{257}}\] follows the Bohr model and the radius of \[_{100}F{m^{257}}\] is $n$ times the Bohr radius, then find $n$ .
A. \[100\]
B. \[200\]
C. \[4\]
D. $\dfrac{1}{4}$

Answer 1

Hint:In the Bohr atomic model, electrons circle the nucleus in well-defined circular orbits. The quantum number $n$ , an integer, is used to identify the orbits. To find the required value of $n$ we will write the radius of \[_{100}F{m^{257}}\] in terms of Bohr radius and equate with the given value of radius of \[_{100}F{m^{257}}\] .

Formula Used:
Radius of an atom,
$r = \dfrac{{{m^2}{r_o}}}{Z}$

Complete step by step solution:
Given: $r = n{r_o}$...........(1)
Where $r$ is the radius of an atom (here, that of $Fm$ ) and ${r_o}$ is the Bohr radius. The synthetic element fermium has an atomic number of \[100\] and has the symbol $Fm$. Although pure fermium metal has not yet been created, it is an actinide and the heaviest element that can be created by neutron bombardment of lighter elements. As a result, it is the last element that can be synthesised in macroscopic amounts. We know that, radius of an atom can be written as,
$r = \dfrac{{{m^2}{r_o}}}{Z}$...........(2)

Now, electronic configuration of \[_{100}F{m^{257}}\] is \[2,{\text{ }}8,{\text{ }}18,{\text{ }}32,{\text{ }}50\] , that is, there are five number of orbits in \[_{100}F{m^{257}}\] . This implies that $m = 5$.
Putting the known values in equation (2), we get,
$r = \dfrac{{{5^2}{r_o}}}{{100}}$
Solving this we get,
$r = \dfrac{1}{4}{r_o}$
Comparing equation (1) and equation (2), we get,
$\therefore n = \dfrac{1}{4}$

Hence, option D is the answer.

Note: The Bohr model, often known as a planetary model, states that the electrons orbit the atom's nucleus in fixed, permitted paths. The energy of the electrons is fixed when it is in one of these orbits.