Question

Two nuclei have mass numbers in the ratio $1:2$ . What will be the ratio of their nuclear densities?

Answer 1

In order to determine the ratio between nuclear densities of two different nuclei having a particular ratio between their mass numbers, figure out the relationship between the mass number and nuclear density.

Complete answer:
Mass number of a nucleus is the measure of total matter present inside it. A nucleus consists of subatomic particles like protons and neutrons. The sum of the total number of protons and neutrons in a nucleus is equal to the mass number.
Nuclear density is the ratio of total mass present in a nucleus and the volume occupied by the nucleus. The mass number is represented by $A$ . The nucleus is assumed to completely spherical in nature and its volume is given by the formula:
${\text{volume of nucleus}} = \dfrac{4}{3}\pi {R^3}$ , where $R$ stands for the radius of the nuclei.
The relationship between mass number and radius can be written as follows:
$R = {R_0}{A^{\dfrac{1}{3}}}$
On inserting this value of radius into the volume we get,
${\text{volume of nucleus}} = \dfrac{4}{3}\pi {\left( {{R_0}{A^{\dfrac{1}{3}}}} \right)^3} = \dfrac{4}{3}\pi {R_0}^3A$
The nuclear density can be calculated as follows:
${\text{Nuclear density}} = \dfrac{{{\text{mass number}}}}{{volume}}$
${\text{Nuclear density}} = \dfrac{A}{{\dfrac{4}{3}\pi {R_0}^3A}} = \dfrac{1}{{\dfrac{4}{3}\pi {R_0}^3}}$
We observe that the nuclear density is independent of the mass number and remains constant for any nucleus. Therefore, even if the mass numbers get altered, the nuclear densities remain unchanged.
Hence, the nuclear densities of the two nuclei having mass number in the ratio $1:2$ will be in the ratio $1:1$ .

Note:
The mass number can be calculated using the atomic number of a particular element. The atomic number gives the total number of protons or electrons (always equal in a neutral atom) present in an atom. This can be added to the number of neutrons to obtain a mass number.