Chemistry

For hydrogen-like ions, the electron's orbital speed depends on the nuclear charge $Z$ and the principal quantum number $n$—higher charge pulls the electron faster, while larger orbits slow it down. You'll use the Bohr model formula $v_n = \frac{Z e^2}{2\epsilon_0 h n}$ (or equivalently $v_n = \frac{Z c \alpha}{n}$ where $\alpha$ is the fine-structure constant) to find the actual speed, then compare it to $c$ to judge whether relativistic effects matter. The key insight is that as $Z$ increases or $n$ decreases, $v/c$ grows—and when that ratio approaches 0.1 or higher, relativity becomes important.

For hydrogen-like ions, the electron's orbital speed depends on the nuclear charge $Z$ and the principal quantum number $n$—higher charge pulls the electron faster, while larger orbits slow it down. You'll use the Bohr model formula $v_n = \frac{Z e^2}{2\epsilon_0 h n}$ (or equivalently $v_n = \frac{Z c \alpha}{n}$ where $\alpha$ is the fine-structure constant) to find the actual speed, then compare it to $c$ to judge whether relativistic effects matter. The key insight is that as $Z$ increases or $n$ decreases, $v/c$ grows—and when that ratio approaches 0.1 or higher, relativity becomes important.