Let be the radius of the Bohr orbit of a ion and be the radius of the Bohr orbit of a ion. If their — Atomic Structure Chemistry Question
Question
Let $r_1$ be the radius of the $4^{th}$ Bohr orbit of a $He^+$ ion and $r_2$ be the radius of the $3^{rd}$ Bohr orbit of a $Be^{3+}$ ion. If their ratio $\frac{r_1}{r_2}$ is expressed as a fraction $\frac{x}{y}$ where $x$ and $y$ are coprime positive integers, find the sum $x+y$.
Answer: 41
💡 Solution & Explanation
The radius of a Bohr orbit is $r \propto \frac{n^2}{Z}$. For $He^+$ ($n=4, Z=2$), $r_1 \propto \frac{4^2}{2} = \frac{16}{2} = 8$. For $Be^{3+}$ ($n=3, Z=4$), $r_2 \propto \frac{3^2}{4} = \frac{9}{4}$. The ratio $\frac{r_1}{r_2} = \frac{8}{9/4} = \frac{32}{9}$. Thus $x=32$ and $y=9$, so $x+y = 41$.
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