Question

Read the following statements:
(A) Volume of the nucleus is directly proportional to the mass number.
(B) Volume of the nucleus is independent of mass number.
(C) Density of the nucleus is directly proportional to the mass number.
(D) Density of the nucleus is directly proportional to the cube root of the mass number.
(E) Density of the nucleus is independent of the mass number.
Choose the correct option from the following options

• (A) and (D) only
• (A) and (E) only
• (B) and (E) only
• (A) and (C) only

Answer 1

The Correct Option is B

To solve this question, we need to analyze each given statement based on our understanding of nuclear physics. 

1. Understanding Statement (A): "Volume of the nucleus is directly proportional to the mass number."
   • The volume of a nucleus \( V \) is proportional to \( A \), the mass number, because the nuclear radius \( R \) is given by \( R = R_0A^{1/3} \), where \( R_0 \) is a constant. Thus, volume \( V \) is proportional to \( R^3 = (R_0^3)A \). Hence, this statement is correct.
2. Understanding Statement (B): "Volume of the nucleus is independent of mass number."
   • As explained above, volume \( V \) is proportional to the mass number \( A \). Therefore, this statement is incorrect.
3. Understanding Statement (C): "Density of the nucleus is directly proportional to the mass number."
   • Density \( \rho \) of a nucleus is defined as mass divided by volume. Since the volume is proportional to the mass number and the nuclear matter density is approximately constant, the density is actually independent of the mass number. Therefore, this statement is incorrect.
4. Understanding Statement (D): "Density of the nucleus is directly proportional to the cube root of the mass number."
   • As previously stated, the density of the nucleus does not depend on the mass number, making this statement incorrect.
5. Understanding Statement (E): "Density of the nucleus is independent of the mass number."
   • The density of a nucleus is generally recognized as approximately constant (around \( \approx 2.3 \times 10^{17} \text{ kg/m}^3 \)) for all nuclei since volume and mass number scale proportionally. Hence, this statement is correct.

By evaluating all statements, we conclude that the correct option is (A) and (E) only. Thus, the correct answer is (A) and (E) only.

Answer 2

\(R=R0A^{\frac{1}{3}}\)
\(⇒V=\frac{4}{3}πR^3=\frac{4}{3}πR_0^3A\)
\(⇒ρ=\frac{M}{V}∝\frac{A}{A}∝A^0\)
So, the correct option is (B): (A) and (E) only.