Question

The radius of stationary state (\(n=2\)) of hydrogen atom is \(x\) pm. The radius of stationary state (\(n=3\)) of \( \text{He}^{+} \) ion (in pm) is

• A. \( \frac{9}{8}x \)
• B. \( \frac{9x}{8} \)
• C. \( \frac{16x}{9} \)
• D. \( \frac{9}{16x} \)

Hint:

Remember the proportionality \( r_n \propto \frac{n^2}{Z} \). You can simply set up a ratio: \( \frac{r_2}{r_1} = \frac{n_2^2 / Z_2}{n_1^2 / Z_1} \).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The radius of the \( n^{\text{th}} \) orbit for a hydrogen-like species is given by the Bohr formula: \[ r_n = r_0 \frac{n^2}{Z} \] where \( r_0 \) is the Bohr radius (constant), \( n \) is the principal quantum number, and \( Z \) is the atomic number.
Step 2: Key Formula or Approach:
1. For Hydrogen atom (\( \text{H} \)): \( Z = 1 \), \( n = 2 \). Given radius \( r_{H,2} = x \). 2. For Helium ion (\( \text{He}^+ \)): \( Z = 2 \), \( n = 3 \). Find radius \( r_{He^+,3} \).
Step 3: Detailed Explanation:
Write the expression for the radius of Hydrogen (\(n=2\)): \[ r_{H,2} = r_0 \frac{2^2}{1} = 4r_0 \] We are given \( r_{H,2} = x \), so: \[ x = 4r_0 \implies r_0 = \frac{x}{4} \] Now, write the expression for the radius of \( \text{He}^+ \) (\(n=3\)): \[ r_{He^+,3} = r_0 \frac{3^2}{2} = r_0 \frac{9}{2} \] Substitute \( r_0 = \frac{x}{4} \) into this equation: \[ r_{He^+,3} = \left(\frac{x}{4}\right) \frac{9}{2} = \frac{9x}{8} \]
Step 4: Final Answer:
The radius is \( \frac{9x}{8} \).