The distance between `4th` and `3rd` Bohr orbits of `He^(+)` is `:`

Options

A. `2.645xx10^(-10)`m
B. `1.322xx10^(-10)`m
C. `1.851xx10^(-10)`m
D. None of these

Answer

To find the distance between the 4th and 3rd Bohr orbits of the helium ion \( \text{He}^+ \), we can follow these steps:

### Step 1: Understand the Bohr Radius Formula
The radius of the \( n^{th} \) Bohr orbit is given by the formula:
\[ r_n = \frac{0.529 \, n^2}{z} \text{ (in angstroms)} \]
where:
- \( r_n \) is the radius of the \( n^{th} \) orbit,
- \( n \) is the principal quantum number (orbit number),
- \( z \) is the atomic number of the element.

### Step 2: Calculate the Radius of the 3rd Bohr Orbit
For the 3rd orbit (\( n = 3 \)) of \( \text{He}^+ \) (where \( z = 2 \)):
\[ r_3 = \frac{0.529 \times 3^2}{2} = \frac{0.529 \times 9}{2} = \frac{4.761}{2} = 2.3805 \text{ angstroms} \]

### Step 3: Calculate the Radius of the 4th Bohr Orbit
For the 4th orbit (\( n = 4 \)) of \( \text{He}^+ \):
\[ r_4 = \frac{0.529 \times 4^2}{2} = \frac{0.529 \times 16}{2} = \frac{8.464}{2} = 4.232 \text{ angstroms} \]

### Step 4: Calculate the Distance Between the 4th and 3rd Orbits
Now, we find the distance between the 4th and 3rd orbits:
\[ \text{Distance} = r_4 - r_3 = 4.232 - 2.3805 = 1.8515 \text{ angstroms} \]

### Step 5: Convert Angstroms to Meters
To convert angstroms to meters, we use the conversion factor \( 1 \text{ angstrom} = 10^{-10} \text{ meters} \):
\[ \text{Distance in meters} = 1.8515 \times 10^{-10} \text{ meters} \]

### Final Answer
Thus, the distance between the 4th and 3rd Bohr orbits of \( \text{He}^+ \) is approximately:
\[ 1.85 \times 10^{-10} \text{ meters} \]

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