If r is radius of first orbit , the radius of nth orbit of the H atom will be

Options

A. `rn^(2)`
B. rn
C. rin
D. `r^(2)n^(2)`

Answer

To find the radius of the nth orbit of a hydrogen atom, we can use the formula for the radius of the nth orbit in a hydrogen-like atom. The formula is given by:

\[ r_n = \frac{0.529 \, n^2}{Z} \, \text{angstroms} \]

where:
- \( r_n \) is the radius of the nth orbit,
- \( n \) is the principal quantum number (orbit number),
- \( Z \) is the atomic number of the atom (for hydrogen, \( Z = 1 \)),
- 0.529 is a constant that represents the radius of the first orbit in angstroms.

### Step-by-Step Solution:

1. **Identify the radius of the first orbit**:
The radius of the first orbit (n=1) for hydrogen is given as:
\[ r_1 = \frac{0.529 \times 1^2}{1} = 0.529 \, \text{angstroms} \]
We denote this radius as \( R \).

2. **Apply the formula for the nth orbit**:
Now, we can find the radius of the nth orbit using the formula:
\[ r_n = \frac{0.529 \, n^2}{1} = 0.529 \, n^2 \, \text{angstroms} \]

3. **Express \( r_n \) in terms of \( R \)**:
Since we have defined \( R = 0.529 \, \text{angstroms} \), we can express \( r_n \) as:
\[ r_n = R \cdot n^2 \]

4. **Final expression**:
Thus, the radius of the nth orbit of the hydrogen atom can be expressed as:
\[ r_n = R \cdot n^2 \]

### Conclusion:
The radius of the nth orbit of the hydrogen atom is \( R \cdot n^2 \), where \( R \) is the radius of the first orbit.