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 \(128\), given that the nuclear radius of a nucleus with nucleon number \(16\) is \(3 \times 10^{-15} \, \text{m}\), we can follow these steps:

### Step 1: Understand the relationship for nuclear radius
The nuclear radius \(R\) can be expressed as:
\[ R = R_0 A^{1/3} \]
where:
- \(R_0\) is a constant,
- \(A\) is the nucleon number (mass number).

### Step 2: Set up the equation for nucleon number \(16\)
For the nucleus with nucleon number \(16\):
\[ R_{16} = R_0 (16)^{1/3} \]
We know that \(R_{16} = 3 \times 10^{-15} \, \text{m}\), so we can write:
\[ 3 \times 10^{-15} = R_0 (16)^{1/3} \]

### Step 3: Set up the equation for nucleon number \(128\)
For the nucleus with nucleon number \(128\):
\[ R_{128} = R_0 (128)^{1/3} \]

### Step 4: Divide the two equations
Now, we can divide the equation for \(R_{16}\) by the equation for \(R_{128}\):
\[ \frac{R_{16}}{R_{128}} = \frac{R_0 (16)^{1/3}}{R_0 (128)^{1/3}} \]
This simplifies to:
\[ \frac{3 \times 10^{-15}}{R_{128}} = \frac{(16)^{1/3}}{(128)^{1/3}} \]

### Step 5: Simplify the right side
The right side can be simplified:
\[ \frac{(16)^{1/3}}{(128)^{1/3}} = \left(\frac{16}{128}\right)^{1/3} = \left(\frac{1}{8}\right)^{1/3} = \frac{1}{2} \]
So we have:
\[ \frac{3 \times 10^{-15}}{R_{128}} = \frac{1}{2} \]

### Step 6: Solve for \(R_{128}\)
Now, we can solve for \(R_{128}\):
\[ R_{128} = 3 \times 10^{-15} \times 2 = 6 \times 10^{-15} \, \text{m} \]

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