With the aid of tunable lasers, Rydberg atoms of sodium have...

With the aid of tunable lasers, Rydberg atoms of sodium have been produced with $n \approx 100$. The resulting atomic diameter would correspond in hydrogen to $n \approx 600$. (a) What would be the diameter of a hydrogen atom whose electron is in the $n \approx 600$ orbit? $(b)$ What would be the speed of the electron in that orbit? $(c)$ How does the result in $(b)$ compare with the speed in the $n \approx 1$ orbit?

Key Concepts

Rydberg Atoms

Rydberg atoms refer to atoms in which one or more electrons have been excited to very high quantum numbers. These states are characterized by their large size, exaggerated properties, and strong response to external fields. Understanding Rydberg atoms is essential when discussing highly excited states and their unique behavior compared to ground state atoms.

Principal Quantum Number

The principal quantum number, n, determines the energy level and average distance of an electron from the nucleus in quantum systems like the hydrogen atom. Its value directly affects orbital size, with the radius scaling approximately as n². This concept is crucial for describing the spatial extent of the atom and how properties like atomic diameter change as n increases.

Bohr Model of the Hydrogen Atom

The Bohr model provides an early quantum-mechanical description of the hydrogen atom, where electrons occupy discrete circular orbits with quantized energy levels. It introduces the concept of the Bohr radius—the fundamental unit of length in the atom—and explains how orbital radius and electron energy depend on the principal quantum number. This model underlies the estimation of atomic diameters and electron speeds in different orbits.

Scaling Laws for Atomic Radii and Electron Speed

In the Bohr model, the radius of an electron's orbit scales with the square of the principal quantum number (n²), meaning that for larger n, the atom becomes much larger. Conversely, the electron's speed in its orbit scales inversely with n, decreasing as the electron occupies higher energy levels. These scaling laws are essential for understanding how properties such as the diameter of the atom and the electron’s velocity change with n.

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