If `Deltar_(1)` represents the difference in radii of statonary orbitals for n = 3 and n = 4 in a hydrogen atom and `Deltar_(2)` represents the difference in radii of stationary orbits for n = 8 and n = 9, then the value of `(Deltar_(1))/(Deltar_(2))` is

Options

A. 0.71
B. 0.41
C. 2.43
D. 14.82

Answer

To solve the problem, we need to calculate the differences in the radii of stationary orbits for the specified quantum numbers (n = 3 & n = 4 for Δr₁ and n = 8 & n = 9 for Δr₂) using the formula for the radius of the nth orbit in a hydrogen atom.

### Step-by-Step Solution:

1. **Identify the formula for the radius of stationary orbits:**
The radius of the nth orbit in a hydrogen atom is given by:
\[ r_n = 0.529 \frac{n^2}{Z} \text{ Angstroms} \]
For hydrogen, \( Z = 1 \).

2. **Calculate Δr₁ (difference between n = 4 and n = 3):**
\[ \Delta r_1 = r_4 - r_3 \]
- Calculate \( r_4 \):
\[ r_4 = 0.529 \times 4^2 = 0.529 \times 16 = 8.464 \text{ Angstroms} \]
- Calculate \( r_3 \):
\[ r_3 = 0.529 \times 3^2 = 0.529 \times 9 = 4.761 \text{ Angstroms} \]
- Now, find Δr₁:
\[ \Delta r_1 = 8.464 - 4.761 = 3.703 \text{ Angstroms} \]

3. **Calculate Δr₂ (difference between n = 9 and n = 8):**
\[ \Delta r_2 = r_9 - r_8 \]
- Calculate \( r_9 \):
\[ r_9 = 0.529 \times 9^2 = 0.529 \times 81 = 42.849 \text{ Angstroms} \]
- Calculate \( r_8 \):
\[ r_8 = 0.529 \times 8^2 = 0.529 \times 64 = 33.856 \text{ Angstroms} \]
- Now, find Δr₂:
\[ \Delta r_2 = 42.849 - 33.856 = 8.993 \text{ Angstroms} \]

4. **Calculate the ratio \( \frac{\Delta r_1}{\Delta r_2} \):**
\[ \frac{\Delta r_1}{\Delta r_2} = \frac{3.703}{8.993} \]
- Performing the division:
\[ \frac{\Delta r_1}{\Delta r_2} \approx 0.412 \]

### Final Result:
The value of \( \frac{\Delta r_1}{\Delta r_2} \) is approximately **0.412**.