Question

Which of the following is the ratio of 5\(^\text{th}\) Bohr orbit \( (r_5) \) of He\(^+\) & Li\(^{2+}\)?

• A. \( \frac{2}{3} \)
• B. \( \frac{3}{2} \)
• C. \( \frac{9}{4} \)
• D. \( \frac{4}{9} \)

Hint:

For a hydrogen-like atom, the radius of the Bohr orbit is inversely proportional to the atomic number \( Z \).

Answer 1

The Correct Option is A

The radius of the Bohr orbit for a hydrogen-like atom is given by the formula: \[ r_n = \frac{n^2 h^2}{4 \pi^2 m e^2 Z} \] Where \( n \) is the principal quantum number, \( h \) is Planck's constant, \( m \) is the electron mass, \( e \) is the charge of the electron, and \( Z \) is the atomic number. For the 5\(^\text{th}\) orbit, the radius for an atom is proportional to \( \frac{n^2}{Z} \). Therefore, the ratio of the radius of the 5\(^\text{th}\) Bohr orbit for He\(^+\) (which has \( Z = 2 \)) to Li\(^{2+}\) (which has \( Z = 3 \)) is given by: \[ \frac{r_5(\text{He}^+)}{r_5(\text{Li}^{2+})} = \frac{n^2 / Z_{\text{He}^+}}{n^2 / Z_{\text{Li}^{2+}}} = \frac{Z_{\text{Li}^{2+}}}{Z_{\text{He}^+}} = \frac{3}{2} \] Thus, the ratio is \( \frac{2}{3} \).