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Question: The volume of nucleus is about: A. \({10^{ - 4}}\) times of an atom B. \({10^{ - 15}}\) times o...

The volume of nucleus is about:
A. 10−4{10^{ - 4}} times of an atom
B. 10−15{10^{ - 15}} times of an atom
C. 10−5{10^{ - 5}} times of an atom
D. 10−10{10^{ - 10}} times of an atom

Explanation

Solution

The volume of the nucleus is a function of the radius of the atom as the atom is considered as a hollow sphere with empty space inside and electron distribution in the orbitals. The protons and neutrons are present inside the nucleus. The ratio of radius of the nucleus and the radius of the atom helps in deciding the volume of the atom.

Complete step by step answer:

The diameter of the atom ranges between 0.1nm−0.5nm0.1nm - 0.5nm. The diameter of the nucleus ranges between 1.5fm−15fm1.5fm - 15fm . As the atom is considered as a hollow sphere, we can apply the formula of the volume of a sphere to determine the volume of an atom. Mathematically, the volume of an atom can be written as:
V=43πr3V = \dfrac{4}{3}\pi {r^3}
Where, r=r = radius of the atom or the distance of the valence electron and the center of the nucleus.
Thus, we can also deduce that the relation between the volume and the radius of an atom is:
Vatom∝r3{V_{atom}} \propto {r^3} ….(i)
Similarly, the volume of the nucleus can be written as:
V=43πR3V = \dfrac{4}{3}\pi {R^3}
Where, R=R = radius of the nucleus
Hence, we can also deduce that the relation between the volume and the radius of the nucleus is:
Vnucleus∝R3{V_{nucleus}} \propto {R^3}….(ii)
The relation between the radius of the atom and the radius of the nucleus is:
r=105R⇒rR=105r = {10^5}R \Rightarrow \dfrac{r}{R} = {10^5} ….(iii)
Thus, combining equation (i) and (ii), we have:
VatomVnucleus=r3R3=(rR)3\dfrac{{{V_{atom}}}}{{{V_{nucleus}}}} = \dfrac{{{r^3}}}{{{R^3}}} = {\left( {\dfrac{r}{R}} \right)^3} …(iv)
Substituting the value of equation (iii) in equation (iv), we have:
VatomVnucleus=(rR)3=(1051)3=1015\dfrac{{{V_{atom}}}}{{{V_{nucleus}}}} = {\left( {\dfrac{r}{R}} \right)^3} = {\left( {\dfrac{{{{10}^5}}}{1}} \right)^3} = {10^{15}}
Thus, the volume of the nucleus is 10−15{10^{ - 15}} times the volume of the atom.
The correct option is B. 10−15{10^{ - 15}} times of an atom.

Note:
Although the atom is being considered a hollow sphere with the nucleus at its center and the electrons revolving around it, it is not actually a perfect sphere. This is due to the uneven distribution of the electrons around the nucleus and due to the difference in the various energy levels and sublevels.