Question

If the mass numbers of two nuclei are in the ratio 3: 2, then the ratio of their nuclear densities is :

• A. 1:1
• B. $2^{1/3}: 3^{1/3}$
• C. 2:3
• D. $3^{1/3}:2^{1/3}$
• E. 3:2

Answer 1

The Correct Option is A

Given: The mass numbers of two nuclei are in the ratio 3:2.

Concept: Nuclear density is given by:

$ρ = \frac{\text{Mass}}{\text{Volume}}$ 

Since the mass of a nucleus is approximately proportional to its mass number $A$, and the radius of a nucleus is given by $R = R_0 A^{1/3}$,

Volume $V ∝ R^3 ∝ (A^{1/3})^3 = A$

So, density $ρ ∝ \frac{A}{A} = \text{constant}$

Thus, nuclear density is independent of mass number, meaning:

Ratio of nuclear densities = 1:1

Correct Answer: Option A: 1:1

Answer 2

Step 1: The nuclear density is given by the formula: \[ \rho = \frac{\text{Mass of nucleus}}{\text{Volume of nucleus}} = \frac{A}{\frac{4}{3}\pi R^3} \] 

Step 2: Radius of a nucleus is given by: \[ R = R_0 A^{1/3} \Rightarrow R^3 = R_0^3 A \]

Substitute into the volume: \[ \text{Volume} = \frac{4}{3} \pi R^3 = \frac{4}{3} \pi R_0^3 A \]

Therefore, nuclear density becomes: \[ \rho = \frac{A}{\frac{4}{3} \pi R_0^3 A} = \text{constant} \]

Conclusion: Nuclear density is independent of mass number \( A \), so the ratio of their nuclear densities is: \[ 1 : 1\]

Final Answer: 1:1