Two nuclei have their mass numbers in the ratio of 1:3. The ratio of their nuclear densities would be

Options

A. `(3)^(1//3):1`
B. `1:1`
C. `1:3`
D. `3:1`

Explanation

To solve the problem of finding the ratio of nuclear densities of two nuclei with mass numbers in the ratio of 1:3, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Mass Numbers**:
Let the mass number of the first nucleus be \( A_1 \) and the mass number of the second nucleus be \( A_2 \). Given that the ratio of their mass numbers is \( A_1 : A_2 = 1 : 3 \), we can express this as:
\[ A_1 = 1 \quad \text{and} \quad A_2 = 3 \]

2. **Use the Formula for Nuclear Radius**:
The nuclear radius \( R \) is given by the formula:
\[ R = R_0 A^{1/3} \]
where \( R_0 \) is a constant. Therefore, we can express the radii of the two nuclei as:
\[ R_1 = R_0 A_1^{1/3} \quad \text{and} \quad R_2 = R_0 A_2^{1/3} \]

3. **Calculate the Ratio of Radii**:
Substituting the values of \( A_1 \) and \( A_2 \):
\[ R_1 = R_0 (1)^{1/3} = R_0 \]
\[ R_2 = R_0 (3)^{1/3} = R_0 \cdot 3^{1/3} \]
Thus, the ratio of the radii is:
\[ \frac{R_1}{R_2} = \frac{R_0}{R_0 \cdot 3^{1/3}} = \frac{1}{3^{1/3}} \]

4. **Use the Formula for Nuclear Density**:
The nuclear density \( \rho \) is given by:
\[ \rho = \frac{A}{\frac{4}{3} \pi R^3} \]
Therefore, for the two nuclei:
\[ \rho_1 = \frac{A_1}{\frac{4}{3} \pi R_1^3} \quad \text{and} \quad \rho_2 = \frac{A_2}{\frac{4}{3} \pi R_2^3} \]

5. **Calculate the Ratio of Densities**:
The ratio of the densities is:
\[ \frac{\rho_1}{\rho_2} = \frac{\frac{A_1}{\frac{4}{3} \pi R_1^3}}{\frac{A_2}{\frac{4}{3} \pi R_2^3}} = \frac{A_1 R_2^3}{A_2 R_1^3} \]
Substituting \( A_1 = 1 \) and \( A_2 = 3 \):
\[ \frac{\rho_1}{\rho_2} = \frac{1 \cdot (R_0 \cdot 3^{1/3})^3}{3 \cdot (R_0)^3} \]
Simplifying this gives:
\[ = \frac{(R_0^3 \cdot 3)}{3 \cdot R_0^3} = \frac{3}{3} = 1 \]

6. **Conclusion**:
The ratio of the nuclear densities \( \rho_1 : \rho_2 \) is:
\[ \rho_1 : \rho_2 = 1 : 1 \]

### Final Answer:
The ratio of their nuclear densities is \( 1 : 1 \).