If the radius of a nucleus with mass number 125 is 1.5 fermi then radius of nucleus with mass number 64 is

Options

A. 0.48 fm
B. 1.92 fm
C. 1.2 fm
D. 1.2 fm

Answer

To solve the problem of finding the radius of a nucleus with mass number 64, given that the radius of a nucleus with mass number 125 is 1.5 fermi, we can use the relationship between the radius of a nucleus and its mass number.

### Step-by-Step Solution:

1. **Understand the Relationship**: The radius \( r \) of a nucleus is directly proportional to the cube root of its mass number \( A \). This can be expressed mathematically as:
\[ r \propto A^{1/3} \]

2. **Set Up the Known Values**: We know:
- For the first nucleus (mass number \( A_1 = 125 \)):
- Radius \( r_1 = 1.5 \) fermi
- For the second nucleus (mass number \( A_2 = 64 \)):
- We need to find \( r_2 \).

3. **Write the Proportionality Equations**:
- For the first nucleus:
\[ r_1 = k \cdot A_1^{1/3} \]
- For the second nucleus:
\[ r_2 = k \cdot A_2^{1/3} \]
Here, \( k \) is a constant of proportionality.

4. **Divide the Two Equations**:
\[ \frac{r_2}{r_1} = \frac{A_2^{1/3}}{A_1^{1/3}} \]

5. **Substitute the Known Values**:
\[ \frac{r_2}{1.5} = \frac{64^{1/3}}{125^{1/3}} \]

6. **Calculate the Cube Roots**:
- \( 64^{1/3} = 4 \) (since \( 4^3 = 64 \))
- \( 125^{1/3} = 5 \) (since \( 5^3 = 125 \))

7. **Substitute the Cube Roots into the Equation**:
\[ \frac{r_2}{1.5} = \frac{4}{5} \]

8. **Solve for \( r_2 \)**:
\[ r_2 = 1.5 \cdot \frac{4}{5} \]
\[ r_2 = 1.5 \cdot 0.8 = 1.2 \text{ fermi} \]

### Final Answer:
The radius of the nucleus with mass number 64 is **1.2 fermi**.