How are the radius of nucleus `r` and mass number` (A)` related to each other?

Options

A. `r=R_@A^(1/2)`
B. `r=R_@A^(1/3)`
C. `r=R_@A^3`
D. `r=R_@A^2`

Answer

To determine the relationship between the radius of a nucleus (r) and its mass number (A), we can use the formula that describes this relationship.

### Step-by-Step Solution:

1. **Understanding the Variables**:
- Let \( r \) be the radius of the nucleus.
- Let \( A \) be the mass number, which is the total number of protons and neutrons in the nucleus.

2. **Nuclear Radius Formula**:
- The radius of a nucleus can be expressed with the formula:
\[ r = r_0 A^{1/3} \]
- Here, \( r_0 \) is a constant that represents the radius of a nucleus for a mass number of 1, typically around \( 1.2 \, \text{fm} \) (femtometers).

3. **Interpreting the Formula**:
- From the formula, we can see that the radius \( r \) is directly proportional to the cube root of the mass number \( A \).
- This means that as the mass number increases, the radius of the nucleus also increases, but at a rate that is the cube root of the mass number.

4. **Conclusion**:
- The relationship between the radius of the nucleus and the mass number can be summarized as:
\[ r \propto A^{1/3} \]
- This indicates that if you know the mass number of a nucleus, you can calculate its radius using the above formula.