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

Question

Accepted Answer

The question asks about the ratio of nuclear densities of two nuclei whose mass numbers are in the ratio of 1:3. The correct answer is Option 4, which states that the ratio of their nuclear densities is 1:1. Let's explore why this is the case and why the other options are incorrect.

### Explanation for Option 4 (Correct Answer):

1. **Nuclear Density Concept**: Nuclear density is defined as the mass of the nucleus divided by its volume. The volume of a nucleus is proportional to the cube of its radius, and the radius of a nucleus is proportional to the cube root of its mass number (A). Therefore, the volume $ V $ of a nucleus is proportional to $ A^{1/3} $.

2. **Density Calculation**: The density $ \rho $ of a nucleus can be expressed as:
   $$
   \rho = \frac{\text{Mass of the nucleus}}{\text{Volume of the nucleus}} = \frac{A}{V} \propto \frac{A}{A^{1/3}} = A^{2/3}
   $$
   However, since the mass number $ A $ is directly proportional to the mass of the nucleus and the volume is proportional to $ A^{1/3} $, the nuclear density simplifies to a constant value, independent of $ A $.

3. **Conclusion**: Since nuclear density is constant for all nuclei, regardless of their mass number, the ratio of their nuclear densities is 1:1. This supports Option 4 as the correct answer.

### Explanation for Why Other Options are Incorrect:

1. **Option 1 (1:3)**: This option suggests that the nuclear density is directly proportional to the mass number. However, as explained, nuclear density is independent of the mass number, so this option is incorrect.

2. **Option 2 (3:1)**: This option implies that the nuclear density is inversely proportional to the mass number, which is not the case. Nuclear density remains constant regardless of the mass number, making this option incorrect.

3. **Option 3 $(3)^{\frac{1}{3}}:1$**: This option suggests a relationship based on the cube root of the mass number ratio. While the radius of a nucleus is proportional to the cube root of the mass number, the density itself does not follow this relationship. Therefore, this option is incorrect.

In summary, the nuclear density of a nucleus is a constant value that does not depend on the mass number, leading to the conclusion that the ratio of nuclear densities for any two nuclei is always 1:1. This confirms that Option 4 is the correct answer.