Question

The wavelength of a particular electron transition for He\(^+\) is 100 nm. The wavelength (in \(\unicode{x212B}\)) of H atom for the same transition is

• A. 1000
• B. 100
• C. 4000
• D. 2000

Hint:

For hydrogen-like species, the transition wavelength varies inversely with the square of atomic number \( Z \): \(\lambda \propto \frac{1}{Z^2}\).

Answer 1

The Correct Option is C

Step 1: Understand the relationship of wavelength with nuclear charge
For hydrogen-like atoms, the wavelength of emitted radiation is inversely proportional to the square of the atomic number (\( Z^2 \)): \[ \lambda \propto \frac{1}{Z^2} \] Step 2: Given values
For He\(^+\): \( Z = 2 \), \( \lambda = 100 \text{ nm} = 1000 \, \unicode{x212B} \)
For H: \( Z = 1 \), we need to find \( \lambda_H \) Step 3: Use ratio based on inverse square law
\[ \frac{\lambda_H}{\lambda_{He^+}} = \left(\frac{Z_{He^+}}{Z_H}\right)^2 = \left(\frac{2}{1}\right)^2 = 4 \] \[ \lambda_H = 4 \times 1000 = 4000 \, \unicode{x212B} \] Step 4: Final Answer
\[ \boxed{4000 \, \unicode{x212B}} \]