If the radius of a nucleus with a mass number 7 is 2 fermi then the radius of the nucleus with mass number 189 is

Options

A. 27 fm
B. 25 fm
C. 6 fm
D. 3 fm

Answer

To solve the problem, we will use the relationship between the radius of a nucleus and its mass number. The formula that relates the radius \( r \) of a nucleus to its mass number \( A \) is given by:

\[ r = r_0 A^{1/3} \]

where \( r_0 \) is a constant.

### Step-by-Step Solution:

1. **Identify Given Values:**
- For the first nucleus (mass number \( A_1 = 7 \)):
- Radius \( r_1 = 2 \) Fermi
- For the second nucleus (mass number \( A_2 = 189 \)):
- We need to find \( r_2 \).

2. **Write the Radius Formula for Both Nuclei:**
- For the first nucleus:
\[ r_1 = r_0 A_1^{1/3} \]
- For the second nucleus:
\[ r_2 = r_0 A_2^{1/3} \]

3. **Set Up the Ratio of the Radii:**
- Taking the ratio of the two radii:
\[ \frac{r_2}{r_1} = \frac{r_0 A_2^{1/3}}{r_0 A_1^{1/3}} \]
- The \( r_0 \) cancels out:
\[ \frac{r_2}{r_1} = \frac{A_2^{1/3}}{A_1^{1/3}} \]

4. **Substitute Known Values:**
- Substitute \( r_1 = 2 \) Fermi, \( A_1 = 7 \), and \( A_2 = 189 \):
\[ \frac{r_2}{2} = \frac{189^{1/3}}{7^{1/3}} \]

5. **Simplify the Expression:**
- This can be simplified to:
\[ r_2 = 2 \cdot \left( \frac{189^{1/3}}{7^{1/3}} \right) \]
- Since \( \frac{189}{7} = 27 \):
\[ r_2 = 2 \cdot 27^{1/3} \]

6. **Calculate \( 27^{1/3} \):**
- We know that \( 27^{1/3} = 3 \):
\[ r_2 = 2 \cdot 3 = 6 \text{ Fermi} \]

### Final Answer:
The radius of the nucleus with mass number 189 is \( 6 \) Fermi.

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