Question

What is the radius of the second Bohr orbit of \( {He}^+ \) ion?

• A. 158.7 pm
• B. 105.8 pm
• C. 52.9 pm
• D. 211.6 pm

Hint:

The Bohr model simplifies the atomic structure to quantized orbits, particularly useful for single-electron systems like \( {He}^+ \).

Answer 1

The Correct Option is B

Step 1: Use the Bohr radius formula for hydrogen-like atoms. The radius of the \( n \)-th orbit for a hydrogen-like atom is given by: \[ r_n = \frac{n^2 \hbar^2}{Z \mu e^2} \] where \( n \) is the principal quantum number, \( Z \) is the atomic number (2 for helium), \( \mu \) is the reduced mass (approximately the electron mass for light ions), \( e \) is the elementary charge, and \( \hbar \) is the reduced Planck's constant. 
Step 2: Plug in the values for the second orbit (\( n = 2 \)). For the \( {He}^+ \) ion in its second orbit: \[ r_2 = \frac{4 \times 0.529 { Å}}{2} \] \[ r_2 = \frac{2.116 { Å}}{2} = 1.058 { Å} = 105.8 { pm} \]