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 will use Bohr's model of the hydrogen atom, which gives us the radius of stationary orbits. The formula for the radius of the nth orbit in a hydrogen atom is given by:

\[ r_n = a_0 n^2 \]

where \( a_0 \) is the Bohr radius and \( n \) is the principal quantum number.

### Step 1: Calculate \( \Delta r_1 \) for \( n = 3 \) and \( n = 4 \)

1. **Calculate \( r_3 \)**:
\[ r_3 = a_0 \cdot 3^2 = a_0 \cdot 9 \]

2. **Calculate \( r_4 \)**:
\[ r_4 = a_0 \cdot 4^2 = a_0 \cdot 16 \]

3. **Calculate \( \Delta r_1 \)**:
\[ \Delta r_1 = r_4 - r_3 = (16 a_0 - 9 a_0) = 7 a_0 \]

### Step 2: Calculate \( \Delta r_2 \) for \( n = 8 \) and \( n = 9 \)

1. **Calculate \( r_8 \)**:
\[ r_8 = a_0 \cdot 8^2 = a_0 \cdot 64 \]

2. **Calculate \( r_9 \)**:
\[ r_9 = a_0 \cdot 9^2 = a_0 \cdot 81 \]

3. **Calculate \( \Delta r_2 \)**:
\[ \Delta r_2 = r_9 - r_8 = (81 a_0 - 64 a_0) = 17 a_0 \]

### Step 3: Calculate \( \frac{\Delta r_1}{\Delta r_2} \)

Now, we can find the ratio of the differences in radii:

\[ \frac{\Delta r_1}{\Delta r_2} = \frac{7 a_0}{17 a_0} = \frac{7}{17} \]

### Step 4: Approximate the value

Calculating the decimal value:

\[ \frac{7}{17} \approx 0.4118 \approx 0.41 \]

Thus, the final answer is:

\[ \frac{\Delta r_1}{\Delta r_2} \approx 0.41 \]

### Final Answer:
The value of \( \frac{\Delta r_1}{\Delta r_2} \) is approximately \( 0.41 \).

---