The most probable radius (in pm) for finding the electron in `He^(+)` is

Options

A. `0.0`
B. `52.9`
C. `26.5`
D. `105.8`

Explanation

To find the most probable radius for the electron in the helium ion \( He^{+} \), we can use the formula derived from Bohr's model of the atom. Here’s a step-by-step solution:

### Step 1: Understand the Formula
The radius \( r \) of the electron in a hydrogen-like atom is given by the formula:
\[ r = \frac{0.529 \, n^2}{Z} \, \text{Å} \]
where:
- \( n \) is the principal quantum number (the shell number),
- \( Z \) is the atomic number of the element.

### Step 2: Convert the Formula to Picometers
To express the radius in picometers (pm), we convert the angstroms to picometers. Since \( 1 \, \text{Å} = 100 \, \text{pm} \), the formula becomes:
\[ r = \frac{52.9 \, n^2}{Z} \, \text{pm} \]

### Step 3: Identify the Values for \( He^{+} \)
For the helium ion \( He^{+} \):
- The atomic number \( Z = 2 \) (since helium has 2 protons).
- The principal quantum number \( n = 1 \) (as it is in the first shell).

### Step 4: Substitute the Values into the Formula
Substituting \( n \) and \( Z \) into the formula:
\[ r = \frac{52.9 \times 1^2}{2} \, \text{pm} \]

### Step 5: Calculate the Radius
Now, calculate the radius:
\[ r = \frac{52.9}{2} \, \text{pm} = 26.45 \, \text{pm} \]

### Step 6: Conclusion
The most probable radius for finding the electron in \( He^{+} \) is:
\[ \boxed{26.45 \, \text{pm}} \]