A : Nuclear density is almost same for all nuclei .
R: The radius (r ) of a nucleus depends only on the mass number (A) as `r prop A^(1//3)`.

Options

A. if both assertion & reason are true and the reason is correct explanation of the assertion then mark (1)
B. if both assertion & reason are true and the reason is not correct explanation of the assertion then mark (2)
C. if assertion is true statement but reason is false then mark (3)
D. if both assertion & reason are false then mark (4)

Answer

To assess the statements provided in the question, we need to analyze both the assertion (A) and the reason (R) step by step.

### Step 1: Understand the Assertion (A)
The assertion states that "Nuclear density is almost the same for all nuclei."

- **Density Formula**: Density (ρ) is defined as mass (m) divided by volume (V):
\[ \rho = \frac{m}{V} \]

- **Nuclear Mass**: The mass of a nucleus is approximately proportional to its mass number (A), since each nucleon (proton or neutron) contributes roughly the same mass.

- **Nuclear Volume**: The volume of a nucleus can be approximated using the formula for the volume of a sphere, given by:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the nucleus.

### Step 2: Understand the Reason (R)
The reason states that "The radius (r) of a nucleus depends only on the mass number (A) as \( r \propto A^{1/3} \)."

- **Radius Formula**: The empirical formula for the radius of a nucleus is given by:
\[ r = r_0 A^{1/3} \]
where \( r_0 \) is a constant (approximately \( 1.1 \times 10^{-15} \) m).

### Step 3: Analyze the Relationship
- Since the radius \( r \) is proportional to \( A^{1/3} \), we can express the volume \( V \) of the nucleus as:
\[ V \propto r^3 \propto (A^{1/3})^3 = A \]
- Thus, the volume is directly proportional to the mass number \( A \).

### Step 4: Calculate Density
- Substituting the expressions for mass and volume into the density formula:
\[ \rho = \frac{m}{V} \propto \frac{A}{A} = \text{constant} \]
- This shows that the density remains constant for different nuclei, confirming the assertion.

### Conclusion
Both the assertion (A) and the reason (R) are true, and the reason correctly explains the assertion. Therefore, the answer is that both statements are true, and the reason is the correct explanation of the assertion.

### Final Answer
Both A and R are true, and R is the correct explanation of A.

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