Calculate Wavelength Photon N 5 to N 3

Introduction

When an electron in a hydrogen atom moves from a higher energy level to a lower one, it emits a photon with a specific wavelength. This phenomenon is described by the Rydberg formula, which relates the wavelength of emitted light to the quantum numbers of the energy levels involved.

The transition from n=5 to n=3 is a common example in atomic spectroscopy. Understanding these transitions helps in analyzing emission spectra and studying atomic structure.

Rydberg Formula

The Rydberg formula for the wavelength of emitted light is:

The Rydberg constant is a fundamental physical constant that appears in the formulas for many spectral lines of many chemical elements.

Worked Example

Let's calculate the wavelength for a transition from n=5 to n=3:

1. Identify the quantum numbers: n₁ = 5, n₂ = 3
2. Plug values into the formula:

   λ = 1.0973731568508 × 10⁷ × (1/5² - 1/3²)^-1

   = 1.0973731568508 × 10⁷ × (1/25 - 1/9)^-1

   = 1.0973731568508 × 10⁷ × (0.04 - 0.1111...)^-1

   = 1.0973731568508 × 10⁷ × (-0.0711...)^-1

   = 1.0973731568508 × 10⁷ × -14.0625

   = -1.5517 × 10⁶ m^-1

3. Convert to nanometers: 1/λ = 1.5517 × 10⁶ m^-1 = 1.5517 × 10^-7 m = 155.17 nm

The negative sign indicates the wavelength is in the infrared region of the electromagnetic spectrum.

Interpreting Results

The calculated wavelength of 155.17 nm is in the infrared range, which is typical for transitions between higher energy levels in hydrogen atoms. This wavelength corresponds to a specific color in the electromagnetic spectrum that can be observed using infrared spectroscopy equipment.

Understanding these wavelengths helps scientists analyze the composition of stars and other celestial bodies, as well as study chemical reactions at the atomic level.