Two nuclei have mass numbers A and B respectively. The density ratio of the nuclei is ______.

The correct option is 1 : 1.

To understand why this is the case, let us analyze the relation between a nucleus's mass number and its physical properties, namely its volume and density.

The mass of a nucleus is approximately proportional to its mass number (total number of protons and neutrons), represented by $A$. Therefore, if we assume the average mass of a nucleon is $m$, the total mass $M$ of a nucleus with mass number $A$ is given by:

$M=m·A$

Experimentally, the radius $R$ of a nucleus is related to its mass number $A$ by the empirical formula:

$R=R_0{A}^{1/3}$

where $R_0$ is a constant (approximately equal to 1.2 x 10^-15 m).

Assuming the nucleus is spherical, its volume $V$ can be calculated as:

$V=\frac{4}{3}\pi R^3=\frac{4}{3}\pi {\left(R_0{A}^{1/3}\right)}^3=\frac{4}{3}\pi R_0^{3}A$

Now, let us find the density $\rho$ of the nucleus, which is defined as mass divided by volume:

$\rho =\frac{M}{V}=\frac{m·A}{\frac{4}{3}\pi R_0^{3}A}$

Notice that the mass number $A$ cancels out from both the numerator and the denominator:

$\rho =\frac{3m}{4\pi R_0^3}$

Since $m$, $\pi$, and $R_0$ are constants, the nuclear density $\rho$ is a constant value that does not depend on the mass number of the nucleus.

Since the density is constant and independent of the mass numbers (A and B) of the two nuclei, both nuclei will have the exact same density. Therefore, their density ratio is:

${\rho }_1:{\rho }_2=1:1$