Question

Two nuclei have mass numbers in the ratio $1:64.$ What is the ratio of their nuclear radii?

Answer 1

Hint: We can easily find the solution of the given question if we know the relation between the mass number and nuclear radius of an atom. We need to find the radius for both the nuclei and then find the required ratio of the nuclei.

Complete step by step solution:
First of all we need to write down the equation used to show the relation between the mass number and radius of the nucleus. It can be written as $R=R_0(A{)}^{{\displaystyle \frac{1}{3}}}$
$R_0=1.2\times {10}^{-15}m$which is known as Fermi constant
Now, let us find the radius of the first nucleus.
Therefore, $R_1=1.2\times {10}^{-15}(1A{)}^{{\displaystyle \frac{1}{3}}}$
Similarly, the radius of the second nucleus will be$R_2=1.2\times {10}^{-15}\times (8A{)}^{{\displaystyle \frac{1}{3}}}$
Now, we need to find the ratio of the radii of the nuclei.
Therefore, ${\displaystyle \frac{R_1}{R_2}}={\displaystyle \frac{1.2\times {10}^{-15}{(1A)}^{{\displaystyle \frac{1}{3}}}}{1.2\times {10}^{-15}{(8A)}^{{\displaystyle \frac{1}{3}}}}}$
$\Rightarrow {\displaystyle \frac{R_1}{R_2}}={\displaystyle \frac{1}{8^{{\displaystyle \frac{1}{3}}}}}={\displaystyle \frac{1}{2}}$
Therefore, we can write the ratios of the radii of the nuclei of two atoms as$1:2$.

Hence, the required ratio of the given question is $1:2$.

Note: When we have to find the ratio of radii when the atomic number of the atom is given, instead of using the relation ${\displaystyle \frac{R_1}{R_2}}={\displaystyle \frac{R_0{(A_1)}^{{\displaystyle \frac{1}{3}}}}{R_0{(A_2)}^{{\displaystyle \frac{1}{3}}}}}$we can also write it as ${\displaystyle \frac{R_1}{R_2}}={\left({\displaystyle \frac{A_1}{A_2}}\right)}^{{\displaystyle \frac{1}{3}}}$. The atomic nucleus of an atom is a small dense region which contains protons and neutrons at the centre of the atom. For the first time nucleus was coined by Geiger and Marsden’s gold foil experiment which they conducted on the instructions of Ernest Rutherford in the year 1909. Also, we need to know that atoms are made up of nuclei which are positively charged. The nucleus is surrounded by the electrons which are negatively charged.