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