Choose the correct statements from the following based on the properties of atomic nucleus.
I. Radius of a nucleus of mass no. A depends on its mass number as \( R \propto A^3 \)
II. Radius of nucleus and mass number are related as \( R \propto A^{\frac{1}{3}} \)
III. Density of nuclei is always a constant
IV. Density of nuclei depends on its mass number A
V. Density of nuclei depends on its size, volume
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → II and III only
The volume of a nucleus is proportional to the number of nucleons in it, since the nucleons are packed closely like drops of a liquid.
\( \displaystyle \frac{4}{3}\pi R^{3} \propto A \;\Rightarrow\; R = R_0\,A^{1/3}, \quad R_0 \approx 1.2\,\mathrm{fm} \)
The nuclear density is then mass divided by volume, with \( m \) the mass of a nucleon:
\( \displaystyle \rho = \frac{A\,m}{\frac{4}{3}\pi R_0^{3} A} = \frac{3m}{4\pi R_0^{3}} \)
\( A \) cancels, so the density is the same for all nuclei.
I. Radius of a nucleus of mass no. A depends on its mass number as \( R \propto A^3 \) — Incorrect. The radius grows only as the cube root of A.
II. Radius of nucleus and mass number are related as \( R \propto A^{1/3} \) — Correct. This follows from volume \( \propto A \).
III. Density of nuclei is always a constant — Correct. \( A \) cancels in the expression for \( \rho \) (about \( 2.3\times10^{17}\,\mathrm{kg\,m^{-3}} \)).
IV. Density of nuclei depends on its mass number A — Incorrect. The density is independent of A.
V. Density of nuclei depends on its size, volume — Incorrect. A bigger nucleus has proportionally more mass, so its density is unchanged.
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 R0 = 1.2 × 10–15 m (=1.2 fm; 1 fm = 10–15 m). This means the volume of the nucleus, which is proportional to R3 is proportional to A. Thus the density of nucleus is a constant, independent of A, for all nuclei.”