The ratio of the radii of the nucleons of `._(2)^(4)He` and `._(1)^(1)H` is

Options

A. 4
B. 2
C. 3
D. 1.6

Explanation

To find the ratio of the radii of the nucleons of helium (\(_{2}^{4}\text{He}\)) and hydrogen (\(_{1}^{1}\text{H}\)), we can use the formula for the radius of a nucleus, which is given by:

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

where:
- \(R\) is the radius of the nucleus,
- \(R_0\) is a constant (approximately \(1.4 \times 10^{-13}\) cm),
- \(A\) is the mass number of the nucleus.

### Step 1: Identify the mass numbers
For helium (\(_{2}^{4}\text{He}\)), the mass number \(A\) is 4.
For hydrogen (\(_{1}^{1}\text{H}\)), the mass number \(A\) is 1.

### Step 2: Write the formula for the radius of each nucleus
Using the formula for the radius:
- For helium:
\[ R_{\text{He}} = R_0 \cdot (4)^{1/3} \]
- For hydrogen:
\[ R_{\text{H}} = R_0 \cdot (1)^{1/3} \]

### Step 3: Calculate the ratio of the radii
Now, we can find the ratio of the radii of helium to hydrogen:
\[ \frac{R_{\text{He}}}{R_{\text{H}}} = \frac{R_0 \cdot (4)^{1/3}}{R_0 \cdot (1)^{1/3}} \]

### Step 4: Simplify the ratio
The \(R_0\) cancels out:
\[ \frac{R_{\text{He}}}{R_{\text{H}}} = \frac{(4)^{1/3}}{(1)^{1/3}} = (4)^{1/3} \]

### Step 5: Calculate \((4)^{1/3}\)
Calculating \((4)^{1/3}\):
\[ (4)^{1/3} \approx 1.5874 \]

### Final Answer
Thus, the ratio of the radii of the nucleons of helium to hydrogen is approximately:
\[ \frac{R_{\text{He}}}{R_{\text{H}}} \approx 1.5874 \]