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

### Step 1: Understand Nuclear Density
Nuclear density is defined as the mass of the nucleus divided by its volume. The formula for density (\( \rho \)) is given by:

\[ \rho = \frac{\text{mass}}{\text{volume}} \]

### Step 2: Define Mass and Volume of Nuclei
Let the mass numbers of the two nuclei be \( A \) and \( 3A \). The mass of a nucleus can be approximated as:

\[ \text{mass} = A \times m_p \]

where \( m_p \) is the mass of a proton (approximately \( 1.67 \times 10^{-27} \) kg).

The volume \( V \) of a nucleus can be modeled as a sphere, which is given by:

\[ V = \frac{4}{3} \pi R^3 \]

The radius \( R \) of a nucleus can be approximated using the formula:

\[ R = R_0 A^{1/3} \]

where \( R_0 \) is a constant (approximately \( 1.1 \times 10^{-15} \) m).

### Step 3: Calculate the Volume for Each Nucleus
For nucleus with mass number \( A \):

\[ R_A = R_0 A^{1/3} \]
\[ V_A = \frac{4}{3} \pi (R_0 A^{1/3})^3 = \frac{4}{3} \pi R_0^3 A \]

For nucleus with mass number \( 3A \):

\[ R_{3A} = R_0 (3A)^{1/3} = R_0 \cdot 3^{1/3} A^{1/3} \]
\[ V_{3A} = \frac{4}{3} \pi (R_0 \cdot 3^{1/3} A^{1/3})^3 = \frac{4}{3} \pi R_0^3 (3A) \]

### Step 4: Calculate the Densities
Now we can calculate the densities for both nuclei.

For nucleus with mass number \( A \):

\[ \rho_A = \frac{A \cdot m_p}{\frac{4}{3} \pi R_0^3 A} = \frac{3 m_p}{4 \pi R_0^3} \]

For nucleus with mass number \( 3A \):

\[ \rho_{3A} = \frac{3A \cdot m_p}{\frac{4}{3} \pi R_0^3 (3A)} = \frac{3 m_p}{4 \pi R_0^3} \]

### Step 5: Compare Densities
From the calculations, we see that:

\[ \rho_A = \rho_{3A} = \frac{3 m_p}{4 \pi R_0^3} \]

### Conclusion
The ratio of the nuclear densities of the two nuclei is:

\[ \frac{\rho_A}{\rho_{3A}} = 1 \]

Thus, the ratio of their nuclear densities is **1:1**.