What will be the ratio of radii of `Li^(7)` nucleus to `Fe^(56)` nucleus?

Options

A. `1:3`
B. `1:2`
C. `1:8`
D. `1:6`

Answer

To find the ratio of the radii of the \( Li^{7} \) nucleus to the \( Fe^{56} \) nucleus, we will use the formula for the radius of a nucleus, which is given by:

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

where:
- \( R \) is the radius of the nucleus,
- \( R_0 \) is a constant (approximately \( 1.2 \times 10^{-15} \) meters),
- \( A \) is the mass number of the nucleus.

### Step 1: Identify the mass numbers
For lithium (\( Li^{7} \)), the mass number \( A_{Li} = 7 \).
For iron (\( Fe^{56} \)), the mass number \( A_{Fe} = 56 \).

### Step 2: Write the formula for the radii
Using the formula for the radius of a nucleus, we can express the radii for both nuclei:

\[ R_{Li} = R_0 (7)^{1/3} \]
\[ R_{Fe} = R_0 (56)^{1/3} \]

### Step 3: Find the ratio of the radii
Now, we want to find the ratio of the radius of the lithium nucleus to the radius of the iron nucleus:

\[ \frac{R_{Li}}{R_{Fe}} = \frac{R_0 (7)^{1/3}}{R_0 (56)^{1/3}} \]

The \( R_0 \) cancels out:

\[ \frac{R_{Li}}{R_{Fe}} = \frac{(7)^{1/3}}{(56)^{1/3}} = \left(\frac{7}{56}\right)^{1/3} \]

### Step 4: Simplify the fraction
Now simplify \( \frac{7}{56} \):

\[ \frac{7}{56} = \frac{1}{8} \]

### Step 5: Calculate the cube root
Now we take the cube root:

\[ \left(\frac{1}{8}\right)^{1/3} = \frac{1}{2} \]

### Conclusion
Thus, the ratio of the radii of the \( Li^{7} \) nucleus to the \( Fe^{56} \) nucleus is:

\[ \frac{R_{Li}}{R_{Fe}} = \frac{1}{2} \]

### Final Answer
The ratio of the radii of \( Li^{7} \) nucleus to \( Fe^{56} \) nucleus is \( 1:2 \).

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