Question

The ratio of radii of second orbit of hydrogen atom to fourth orbit of He$^+$ ion is

• A. 1:4
• B. 2:1
• C. 1:2
• D. 2:3

Hint:

Use the Bohr radius formula \( r_n = \frac{n^2 a_0}{Z} \) for hydrogen-like species to compare orbital sizes.
Don’t forget to square the principal quantum number and account for atomic number \( Z \).

Answer 1

The Correct Option is C

The radius of the $n^{\text{th}}$ orbit in a hydrogen-like atom is given by: \[ r_n = \frac{n^2 a_0}{Z} \] For hydrogen (Z = 1), radius of second orbit: \[ r_{\text{H, 2}} = \frac{2^2 a_0}{1} = 4a_0 \] For He$^+$ ion (Z = 2), radius of fourth orbit: \[ r_{\text{He, 4}} = \frac{4^2 a_0}{2} = \frac{16a_0}{2} = 8a_0 \] Ratio: \[ \frac{r_{\text{H, 2}}}{r_{\text{He, 4}}} = \frac{4a_0}{8a_0} = \frac{1}{2} \]