Question

Which of the following statements are correct?
(A) A nucleus of mass number A has a radius R given by the expression R = R\(_0\)A\(^{1/3}\)
(B) Volume of nucleus is proportional to mass number A
(C) The density of nucleus increases with the radius of nucleus.
(D) Density of nuclear matter does not depend on its mass number A
Choose the correct answer from the options given below:

• A. (A), (B) and (C) only
• B. (A), (B), (C) and (D)
• C. (A) (B) and (D) only
• D. (B) and (C) only

Hint:

A key takeaway from nuclear physics is the near-constant density of nuclear matter. This implies that nucleons are packed together like incompressible spheres. This concept directly leads to \(R \propto A^{1/3}\) and \(V \propto A\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This question assesses the understanding of the basic properties of the atomic nucleus, specifically its size (radius), volume, and density, and how they relate to the mass number A.

Step 2: Detailed Explanation:
(A) A nucleus of mass number A has a radius R given by the expression R = R\(_0\)A\(^{1/3}\)
This is the standard empirical formula for the nuclear radius, where \(R_0\) is a constant approximately equal to 1.2 fm. This relationship is derived from experimental data and is a fundamental concept in nuclear physics. This statement is correct.
(B) Volume of nucleus is proportional to mass number A
Assuming the nucleus is spherical, its volume \(V\) is given by \(V = \frac{4}{3}\pi R^3\). Substituting the expression for radius from statement (A): \[ V = \frac{4}{3}\pi (R_0 A^{1/3})^3 = \frac{4}{3}\pi R_0^3 (A^{1/3})^3 = \left(\frac{4}{3}\pi R_0^3\right) A \] Since \(\frac{4}{3}\pi R_0^3\) is a constant, the volume \(V\) is directly proportional to the mass number A (\(V \propto A\)). This statement is correct.
(C) The density of nucleus increases with the radius of nucleus.
Nuclear density (\(\rho\)) is defined as mass divided by volume (\(\rho = \text{Mass}/\text{Volume}\)). The mass of the nucleus is approximately \(A \times m_p\), where \(m_p\) is the mass of a nucleon (proton/neutron). The volume of the nucleus is \(V = \left(\frac{4}{3}\pi R_0^3\right) A\). So, the density is: \[ \rho = \frac{A \cdot m_p}{\left(\frac{4}{3}\pi R_0^3\right) A} = \frac{m_p}{\frac{4}{3}\pi R_0^3} \] As we can see, the mass number \(A\) cancels out. The density \(\rho\) is approximately constant for all nuclei, independent of their radius or mass number. Therefore, the statement that density increases with radius is incorrect.
(D) Density of nuclear matter does not depend on its mass number A
As derived above, the expression for nuclear density is independent of the mass number A. This remarkable property means that all atomic nuclei have roughly the same extremely high density. This statement is correct.

Step 3: Final Answer:
Statements (A), (B), and (D) are correct, while statement (C) is incorrect. The correct option is (C).