Radius of first orbit of hydrogen is `0.53 Å`. The radius in fourth orbit is:

Options

A. `8.48 Å`
B. `2.12 Å`
C. `4.24 Å`
D. `0.193 Å`

Answer

To find the radius of the fourth orbit of hydrogen, we can use the relationship between the radius of the orbits and the principal quantum number \( n \). The radius of the nth orbit is given by the formula:

\[ r_n \propto n^2 \]

This means that the radius of the nth orbit is directly proportional to the square of the principal quantum number \( n \).

### Step-by-Step Solution:

1. **Identify the radius of the first orbit**:
The radius of the first orbit \( r_1 \) is given as \( 0.53 \, \text{Å} \).

2. **Use the proportionality relationship**:
Since \( r_n \propto n^2 \), we can express the radius of the fourth orbit \( r_4 \) in terms of \( r_1 \):
\[ \frac{r_1}{r_4} = \frac{1}{4^2} = \frac{1}{16} \]

3. **Rearranging the equation**:
From the above relationship, we can rearrange to find \( r_4 \):
\[ r_4 = r_1 \times 16 \]

4. **Substituting the value of \( r_1 \)**:
Now, substitute \( r_1 = 0.53 \, \text{Å} \):
\[ r_4 = 0.53 \, \text{Å} \times 16 \]

5. **Calculating \( r_4 \)**:
Performing the multiplication:
\[ r_4 = 8.48 \, \text{Å} \]

### Final Answer:
The radius of the fourth orbit of hydrogen is \( 8.48 \, \text{Å} \).