The radius of the first orbit of H-atom is 0.53 Å. Find the radius of the fifth orbit.

Options

A. `121.5Å`
B. `111Å`
C. `13.25Å`
D. `333Å`

Answer

To find the radius of the fifth orbit of a hydrogen atom, we can use the formula for the radius of the nth orbit in a hydrogen-like atom:

\[ R_n = R_1 \cdot \frac{n^2}{Z} \]

Where:
- \( R_n \) is the radius of the nth orbit,
- \( R_1 \) is the radius of the first orbit,
- \( n \) is the principal quantum number (the orbit number),
- \( Z \) is the atomic number of the element.

### Step-by-Step Solution:

1. **Identify the given values:**
- The radius of the first orbit \( R_1 = 0.53 \, \text{Å} \).
- The principal quantum number for the fifth orbit \( n = 5 \).
- The atomic number for hydrogen \( Z = 1 \).

2. **Substitute the values into the formula:**
\[ R_5 = R_1 \cdot \frac{n^2}{Z} \]
\[ R_5 = 0.53 \, \text{Å} \cdot \frac{5^2}{1} \]

3. **Calculate \( n^2 \):**
\[ n^2 = 5^2 = 25 \]

4. **Now substitute \( n^2 \) back into the equation:**
\[ R_5 = 0.53 \, \text{Å} \cdot 25 \]

5. **Perform the multiplication:**
\[ R_5 = 0.53 \cdot 25 = 13.25 \, \text{Å} \]

6. **Conclusion:**
The radius of the fifth orbit of the hydrogen atom is \( 13.25 \, \text{Å} \).

### Final Answer:
The radius of the fifth orbit is \( 13.25 \, \text{Å} \).

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