Question

If the mass numbers of two nuclei are in the ratio \( 5 : 2 \) and their diameters are in ratio \( 2 : 6 \). Then their nuclear densities will be in the ratio

• A. \( 1 : 1 \)
• B. \( 2 : 5 \)
• C. \( 10 : 12 \)
• D. \( 6 : 5 \)

Hint:

Nuclear density is constant for all nuclei because radius follows \( R \propto A^{1/3} \), making volume proportional to mass number.

Answer 1

The Correct Option is A

Step 1: Write expression for nuclear density.
Density is given by:
\[ \rho = \frac{\text{Mass}}{\text{Volume}} \]
For nuclei, mass \( \propto A \) and volume \( \propto R^3 \).
Thus:
\[ \rho \propto \frac{A}{R^3} \]

Step 2: Relate radius and diameter.
Diameter is proportional to radius.
Given diameter ratio:
\[ D_1 : D_2 = 2 : 6 = 1 : 3 \]
Thus:
\[ R_1 : R_2 = 1 : 3 \]

Step 3: Calculate density ratio.
\[ \frac{\rho_1}{\rho_2} = \frac{A_1}{A_2} \times \left(\frac{R_2}{R_1}\right)^3 \]
Substitute values:
\[ = \frac{5}{2} \times \left(\frac{3}{1}\right)^3 \]
\[ = \frac{5}{2} \times 27 = \frac{135}{2} \]

Step 4: Use nuclear property.
However, nuclear density is independent of size and mass number because:
\[ R \propto A^{1/3} \Rightarrow V \propto A \]
Thus:
\[ \rho \propto \frac{A}{A} = \text{constant} \]

Step 5: Conclusion.
Therefore, both nuclei have equal density.
\[ \boxed{1 : 1} \]