If the radius of a nucleus `""^(256) X` is 8 fermi , then the radius of `""^(4) He` nucleus will be

Options

A. 16 fermi
B. 2 fermi
C. 32 fermi
D. 4 fermi

Explanation

To find the radius of the helium nucleus \( ^4He \) given the radius of the nucleus \( ^{256}X \) is 8 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 \( 1.2 \) to \( 1.3 \) fermi),
- \( A \) is the mass number of the nucleus.

### Step 1: Write down the known values for the nucleus \( ^{256}X \)

For the nucleus \( ^{256}X \):
- Mass number \( A_1 = 256 \)
- Radius \( R_1 = 8 \) fermi

Using the formula, we can express this as:

\[ R_1 = R_0 A_1^{1/3} \]

Substituting the known values:

\[ 8 = R_0 (256)^{1/3} \]

### Step 2: Calculate \( (256)^{1/3} \)

Calculating \( (256)^{1/3} \):

\[ 256 = 2^8 \implies (256)^{1/3} = 2^{8/3} = 2^{2.67} \approx 6.3496 \]

### Step 3: Solve for \( R_0 \)

Now, substituting back into the equation:

\[ 8 = R_0 \cdot 6.3496 \]

To find \( R_0 \):

\[ R_0 = \frac{8}{6.3496} \approx 1.26 \text{ fermi} \]

### Step 4: Find the radius of the helium nucleus \( ^4He \)

Now, for the helium nucleus \( ^4He \):
- Mass number \( A_2 = 4 \)

Using the radius formula again:

\[ R_2 = R_0 A_2^{1/3} \]

Substituting \( R_0 \) and \( A_2 \):

\[ R_2 = 1.26 \cdot (4)^{1/3} \]

Calculating \( (4)^{1/3} \):

\[ (4)^{1/3} = 2^{2/3} \approx 1.5874 \]

Now substituting back:

\[ R_2 = 1.26 \cdot 1.5874 \approx 2.00 \text{ fermi} \]

### Final Answer

The radius of the helium nucleus \( ^4He \) is approximately **2 fermi**.

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