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

Options

A. `r=R_oA^(1/2)`
B. `r=R_o A^(1/3)`
C. `r=R_o A^3`
D. `r=R_o A^2`

Answer

To determine the relationship between the radius of a nucleus (R) and its mass number (A), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Variables**:
- The radius of the nucleus is denoted as \( R \).
- The mass number, which represents the total number of protons and neutrons in the nucleus, is denoted as \( A \).

2. **Establishing the Relationship**:
- It has been observed that the radius of the nucleus is directly proportional to the cube root of the mass number. This can be mathematically expressed as:
\[ R \propto A^{1/3} \]

3. **Removing the Proportionality**:
- To convert the proportionality into an equation, we introduce a constant \( R_0 \):
\[ R = R_0 A^{1/3} \]
- Here, \( R_0 \) is a constant that is determined experimentally.

4. **Understanding the Constant \( R_0 \)**:
- The value of \( R_0 \) is typically in the range of 1.2 to 1.5 Fermi (1 Fermi = \( 10^{-15} \) meters). Therefore, we can express \( R_0 \) in meters:
\[ R_0 \approx 1.2 \times 10^{-15} \text{ m} \text{ to } 1.5 \times 10^{-15} \text{ m} \]

5. **Conclusion**:
- The final relationship between the radius of the nucleus and the mass number is:
\[ R = R_0 A^{1/3} \]
- This indicates that as the mass number \( A \) increases, the radius \( R \) of the nucleus increases as well, following the cube root relationship.