Question

The energy of second Bohr orbit of hydrogen atom is \(-3.4 \, \text{eV}\). The energy of the fourth Bohr orbit of the He\(^+\) ion will be

• A. \(-3.4 \, \text{eV}\)
• B. \(-13.6 \, \text{eV}\)
• C. \(-6.8 \, \text{eV}\)
• D. \(-0.85 \, \text{eV}\)

Hint:

For hydrogen-like ions, energy levels depend on \( \frac{Z^2}{n^2} \), where \(Z\) is atomic number and \(n\) is orbit number.

Answer 1

The Correct Option is A

Step 1: Use the Bohr model energy formula for hydrogen-like species
\[ E_n = -13.6 \, \text{eV} \times \frac{Z^2}{n^2} \] Step 2: Given energy of 2nd orbit of hydrogen
For hydrogen (\(Z=1\)), \(n=2\): \[ E_2 = -13.6 \times \frac{1^2}{2^2} = -3.4 \, \text{eV} \] Step 3: Use same \(Z^2/n^2\) ratio for He\(^+\)
For He\(^+\) (\(Z=2\)), \(n=4\): \[ E_4 = -13.6 \times \frac{2^2}{4^2} = -13.6 \times \frac{4}{16} = -13.6 \times \frac{1}{4} = -3.4 \, \text{eV} \] Step 4: Final Answer
\[ \boxed{-3.4 \, \text{eV}} \]