What is the approximate radius of a carbon nucleus that has six protons and six neutrons?

Options

A. `1.2xx10^(-15)m`
B. `2.7xx10^(-15)m`
C. `2.2xx10^(-15)m`
D. `2.9xx10^(-15)m`

Answer

To find the approximate radius of a carbon nucleus that has six protons and six neutrons, we can follow these steps:

### Step 1: Determine the number of nucleons
The total number of nucleons (A) in the nucleus is the sum of the number of protons and neutrons. For carbon, which has 6 protons and 6 neutrons:
\[ A = \text{Number of protons} + \text{Number of neutrons} = 6 + 6 = 12 \]

### Step 2: Use the formula for the radius of the nucleus
The radius of a nucleus can be estimated using the formula:
\[ R = R_0 \cdot A^{1/3} \]
where \( R_0 \) is a constant approximately equal to \( 1.2 \times 10^{-15} \) meters, and \( A \) is the number of nucleons.

### Step 3: Substitute the values into the formula
Now we substitute \( A = 12 \) and \( R_0 = 1.2 \times 10^{-15} \) m into the formula:
\[ R = 1.2 \times 10^{-15} \cdot (12)^{1/3} \]

### Step 4: Calculate \( (12)^{1/3} \)
To find \( (12)^{1/3} \):
\[ (12)^{1/3} \approx 2.29 \]

### Step 5: Calculate the radius
Now substitute \( (12)^{1/3} \) back into the radius formula:
\[ R \approx 1.2 \times 10^{-15} \cdot 2.29 \]
\[ R \approx 2.748 \times 10^{-15} \text{ meters} \]

### Step 6: Round the result
Rounding \( 2.748 \times 10^{-15} \) meters gives approximately:
\[ R \approx 2.7 \times 10^{-15} \text{ meters} \]

### Final Answer
The approximate radius of a carbon nucleus is:
\[ R \approx 2.7 \times 10^{-15} \text{ meters} \]

---