Which of the following curves may represent the radius of orbit `(r_n)` in a H-atoms as a function of principal quantum number(n)

Options

A.
B.
C.
D. None of these

Explanation

To determine which curve represents the radius of orbit \( r_n \) in a hydrogen atom as a function of the principal quantum number \( n \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula**: According to Bohr's model of the hydrogen atom, the radius of the nth orbit is given by the formula:
\[ r_n = \frac{0.529 \, n^2}{Z} \text{ angstroms} \]
For hydrogen, \( Z = 1 \), so the formula simplifies to:
\[ r_n = 0.529 \, n^2 \text{ angstroms} \]

2. **Identify the Relationship**: The relationship between the radius \( r_n \) and the principal quantum number \( n \) is quadratic, meaning that \( r_n \) is proportional to \( n^2 \). This indicates that if we plot \( r_n \) on the y-axis and \( n \) on the x-axis, the graph will be a parabola.

3. **Analyze the Options**:
- **Option 1**: If \( r_n \) is plotted against \( n \), it will not be a straight line but a curve (parabola), which is incorrect.
- **Option 2**: If \( r_n \) is plotted against \( n^2 \), we can express it as:
\[ r_n = 0.529 \times n^2 \]
Here, if we let \( n^2 \) be on the x-axis and \( r_n \) on the y-axis, this is a linear relationship of the form \( y = mx \) where \( m = 0.529 \). This is a straight line, which is correct.
- **Option 3**: This option suggests a graph that starts as a straight line and then curves, which is incorrect because the relationship is purely linear when plotted against \( n^2 \).
- **Option 4**: This option states "none of these," which is also incorrect since we have identified Option 2 as correct.

4. **Conclusion**: The correct option that represents the radius of orbit \( r_n \) in hydrogen atoms as a function of the principal quantum number \( n \) is **Option 2**.