Two nuclei have their mass numbers in the ratio of \( 1:3 \). The ratio of their nuclear densities would be
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( 1:1 \)
The nuclear radius grows as \( R = R_0A^{1/3} \), so the nuclear volume is proportional to the mass number A, and so is the nuclear mass (about A times the mass m of a nucleon).
\( \displaystyle \rho = \frac{A\,m}{\frac{4}{3}\pi R_0^{3}A} \)
\( \displaystyle \rho = \frac{3m}{4\pi R_0^{3}} \)
A cancels, so the nuclear density is the same for all nuclei (about \( 2.3\times10^{17}\,\mathrm{kg\,m^{-3}} \)), and the ratio is 1 : 1.
Note: A similar question was also asked in CUET (UG) 2023 on 26-05-2023 (Slot 1).
NCERT (2026-27):
Size of the Nucleus – “… It has been found that a nucleus of mass number \( A \) has a radius
\( \displaystyle R = R_0 A^{1/3} \qquad (13.5) \)
where \( R_0 = 1.2\times10^{-15}\,\mathrm{m} \) (\( = 1.2\,\mathrm{fm} \); \( 1\,\mathrm{fm} = 10^{-15}\,\mathrm{m} \)). This means the volume of the nucleus, which is proportional to \( R^{3} \) is proportional to \( A \). Thus the density of nucleus is a constant, independent of \( A \), for all nuclei. Different nuclei are like a drop of liquid of constant density.”