Question

Four statements are given (A is mass number):
A. The volume of a nucleus is proportional to A.
B. The volume of a nucleus is proportional to A$^{1/3$.}
C. The difference in mass of an atom and its nucleus is called the mass defect.
D. The difference in mass of a nucleus and its constituents is called the mass defect.
Choose the correct answer from the options given below:

• A. A and D are true, but B and C are false
• B. B and D are true, but A and C are false
• C. B and C are true, but A and D are false
• D. A and C are true, but B and D are false

Hint:

Since Volume $\propto$ A and Mass $\propto$ A, the density of a nucleus (Mass/Volume) is constant for all elements!

Answer 1

The Correct Option is A

Step 1: Understanding the Topic:
This question deals with "Nuclei" and fundamental nuclear properties. It focuses on the empirical relationship between the number of nucleons (protons and neutrons) and the physical size of the nucleus, as well as the energetic stability of the nucleus described by the concept of "Mass Defect."
Step 2: Key Formulas and Approach:

Nuclear Radius $R = R_0 \cdot A^{1/3}$.
Nuclear Volume $V = \frac{4}{3} \pi R^3 = \frac{4}{3} \pi (R_0 \cdot A^{1/3})^3$.
Mass Defect $\Delta m = [Z \cdot m_p + (A-Z) \cdot m_n] - M_{nucleus}$.

Step 3: Detailed Explanation:

Statement A & B (Volume): Experimental evidence shows that the radius of a nucleus ($R$) grows with the cube root of the mass number ($A^{1/3}$). Since volume $V$ is proportional to $R^3$, we cube the $A^{1/3}$ term. This results in $V \propto A$. Thus, Statement A is correct and Statement B is incorrect. This implies that nuclear matter has constant density.
Statement C & D (Mass Defect): Mass defect is the "missing mass" that was converted into binding energy when the nucleus was formed.
Statement C: The difference between an atom and its nucleus is essentially the mass of the electrons. This is NOT called mass defect. (False)
Statement D: The mass defect is precisely the difference between the total mass of the individual nucleons (the constituents: protons and neutrons) and the actual combined mass of the resulting nucleus. (True)
Consequently, A and D are the only true statements.
Step 4: Final Answer:
The correct option is (A): A and D are true, but B and C are false.