The nuclear radius of a nucelus with nucleon number `16` is `3 xx 10^(-15) m`. Then, the nuclear radius of a nucleus with nucleon number `128` is .

Options

A. ` 3 xx 10^(-15) m`
B. ` 1.5 xx 10^(-15) m`
C. ` 6 xx 10^(-15) m`
D. ` 4.5 xx 10^(-15) m`

Explanation

To find the nuclear radius of a nucleus with nucleon number 128, we can use the formula for nuclear radius, which is given by:

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

where:
- \( R \) is the nuclear radius,
- \( R_0 \) is a constant,
- \( A \) is the nucleon number.

### Step 1: Identify the given values
We know:
- For the nucleus with nucleon number \( A_1 = 16 \), the radius \( R_1 = 3 \times 10^{-15} \, \text{m} \).
- We need to find the radius \( R_2 \) for the nucleus with nucleon number \( A_2 = 128 \).

### Step 2: Write the formula for both nuclei
Using the formula for both cases:
1. For \( A_1 = 16 \):
\[ R_1 = R_0 (16)^{1/3} \]
\[ 3 \times 10^{-15} = R_0 (16)^{1/3} \]

2. For \( A_2 = 128 \):
\[ R_2 = R_0 (128)^{1/3} \]

### Step 3: Divide the two equations
We can divide the second equation by the first to eliminate \( R_0 \):
\[ \frac{R_2}{R_1} = \frac{R_0 (128)^{1/3}}{R_0 (16)^{1/3}} \]
This simplifies to:
\[ \frac{R_2}{3 \times 10^{-15}} = \frac{(128)^{1/3}}{(16)^{1/3}} \]

### Step 4: Simplify the right-hand side
We can simplify the right-hand side:
\[ \frac{R_2}{3 \times 10^{-15}} = \left(\frac{128}{16}\right)^{1/3} = (8)^{1/3} \]
Since \( 8 = 2^3 \), we have:
\[ (8)^{1/3} = 2 \]

### Step 5: Solve for \( R_2 \)
Now we can solve for \( R_2 \):
\[ R_2 = 2 \times (3 \times 10^{-15}) = 6 \times 10^{-15} \, \text{m} \]

### Final Answer
Thus, the nuclear radius of the nucleus with nucleon number 128 is:
\[ R_2 = 6 \times 10^{-15} \, \text{m} \]