Question

Two spherical nuclei have mass numbers 216 and 64 with their radii ${{R}_{1}}\text{ }and\text{ }{{R}_{2}}$ respectively. The ratio of $\dfrac{{{R}_{1}}}{{{R}_{2}}}$ is equal to?
a)3:2
b)1:3
c)1:2
d)2:3

Answer 1

In the above question the mass number of two different nuclei is given to us with their respective mass numbers. First we will consider the relation between the mass number and the radius of the nuclei. Further using this relation, we will consider the same for the above two different nuclei, take the ratio of the two and obtain the correct answer from those provided to us in the alternatives.

Formula used:
$R={{R}_{\circ }}{{A}^{1/3}}$

Complete step by step answer:
Let us say there is a spherical nuclei with radius R. If the mass number of the nuclei is denoted by A, the relation between the mass number and the atomic and its radius is given by,
$R={{R}_{\circ }}{{A}^{1/3}}$ ${{R}_{\circ }}$ is the constant of proportionality.
Let us say the nuclei with mass number 216 has radius equal to ${{R}_{1}}$. Therefore from above equation we get,
${{R}_{1}}={{R}_{\circ }}{{(216)}^{1/3}}....(1)$
Similarly, let us say the radius of the nuclei with mass number 64 be ${{R}_{2}}$. Hence using the relation between the radius of the nuclei and mass number we get,
${{R}_{2}}={{R}_{\circ }}{{(64)}^{1/3}}....(2)$
Dividing equation 1 by 2 we get,
$\begin{aligned} & \dfrac{{{R}_{1}}}{{{R}_{2}}}=\dfrac{{{R}_{\circ }}{{(216)}^{1/3}}}{{{R}_{\circ }}{{(64)}^{1/3}}} \\\ & \Rightarrow \dfrac{{{R}_{1}}}{{{R}_{2}}}={{\left( \dfrac{216}{64} \right)}^{1/3}}=\dfrac{6}{4} \\\ & \therefore \dfrac{{{R}_{1}}}{{{R}_{2}}}=\dfrac{3}{2} \\\ \end{aligned}$

So, the correct answer is “Option A”.

Note:
It is to be noted that ${{R}_{\circ }}$ is a constant whose value for electrons is $1.2\times {{10}^{-15}}m$. It is believed to be the average nucleon size and is known as nuclear unit radius. The value of this radius depends on the nature of the probe particles. From the above expression the basic conclusion we can draw is that the radius depends on the cube root of the mass number of the nuclei.