For which orbit number of `He^(+)` value of radius equal to `1.058 Å` ?

Explanation

To find the orbit number \( n \) for which the radius of the \( He^+ \) ion is equal to \( 1.058 \, \text{Å} \), we can use the formula for the radius of an electron orbit in a hydrogen-like atom:

\[ R_n = \frac{0.529 \, n^2}{Z} \, \text{Å} \]

where:
- \( R_n \) is the radius of the orbit,
- \( n \) is the principal quantum number (orbit number),
- \( Z \) is the atomic number.

### Step 1: Identify the values
For \( He^+ \):
- The atomic number \( Z = 2 \) (since helium has 2 protons).
- The radius \( R_n = 1.058 \, \text{Å} \).

### Step 2: Substitute the values into the formula
We can set up the equation:

\[ 1.058 = \frac{0.529 \, n^2}{2} \]

### Step 3: Rearrange the equation
Multiply both sides by 2 to eliminate the fraction:

\[ 2 \times 1.058 = 0.529 \, n^2 \]

This simplifies to:

\[ 2.116 = 0.529 \, n^2 \]

### Step 4: Solve for \( n^2 \)
Now, divide both sides by \( 0.529 \):

\[ n^2 = \frac{2.116}{0.529} \]

Calculating the right side gives:

\[ n^2 \approx 4 \]

### Step 5: Find \( n \)
Taking the square root of both sides:

\[ n = \sqrt{4} = 2 \]

### Conclusion
The orbit number for which the radius of \( He^+ \) is \( 1.058 \, \text{Å} \) is \( n = 2 \).

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