If the nuclear of radius of `O^8` is `2.4` fermi, then radius of `Al^27` would be

Options

A. 6.0 fermi
B. 5.6 fermi
C. 3.6 fermi
D. 3.0 fermi

Answer

To find the radius of the nucleus of \( Al^{27} \) given that the radius of the nucleus of \( O^{8} \) is \( 2.4 \) fermi, we can use the formula for the radius of a nucleus:

\[ R = R_0 A^{1/3} \]

where:
- \( R \) is the radius of the nucleus,
- \( R_0 \) is a constant (approximately \( 1.2 \) fermi),
- \( A \) is the mass number of the nucleus.

### Step 1: Calculate the radius of \( O^{8} \)

Given:
- Mass number \( A = 8 \)
- Radius \( R = 2.4 \) fermi

Using the formula:
\[ R = R_0 A^{1/3} \]
We can express \( R_0 \) as:
\[ R_0 = \frac{R}{A^{1/3}} = \frac{2.4}{8^{1/3}} \]

### Step 2: Calculate \( 8^{1/3} \)

Calculating the cube root:
\[ 8^{1/3} = 2 \]

### Step 3: Substitute back to find \( R_0 \)

Now substitute \( 8^{1/3} \) back into the equation for \( R_0 \):
\[ R_0 = \frac{2.4}{2} = 1.2 \text{ fermi} \]

### Step 4: Calculate the radius of \( Al^{27} \)

Now we will calculate the radius of \( Al^{27} \) using the mass number \( A = 27 \):
\[ R = R_0 A^{1/3} = 1.2 \cdot 27^{1/3} \]

### Step 5: Calculate \( 27^{1/3} \)

Calculating the cube root:
\[ 27^{1/3} = 3 \]

### Step 6: Substitute back to find the radius of \( Al^{27} \)

Now substitute \( 27^{1/3} \) back into the equation for \( R \):
\[ R = 1.2 \cdot 3 = 3.6 \text{ fermi} \]

### Final Answer:
The radius of the nucleus of \( Al^{27} \) is \( 3.6 \) fermi.

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