Calculate The Wavelengths of The N 4 to N 3

This calculator helps you determine the wavelength of light emitted when an electron transitions from the n=4 to n=3 energy level in a hydrogen atom. The calculation uses the Rydberg formula, which is fundamental in atomic physics.

Introduction

When an electron in a hydrogen atom moves from a higher energy level (n=4) to a lower energy level (n=3), it emits a photon with a specific wavelength. This transition is part of the Balmer series of spectral lines, which are visible in the visible and near-ultraviolet spectrum.

The wavelength of the emitted light can be calculated using the Rydberg formula, which relates the wavelength to the energy levels involved in the transition. This calculation is essential in atomic spectroscopy and quantum mechanics.

Formula

The wavelength (λ) of the emitted light can be calculated using the Rydberg formula:

λ = 1 / (R × (1/n₁² - 1/n₂²))

Where:

λ is the wavelength in meters
R is the Rydberg constant (1.0973731568160 × 10⁷ m^-1)
n₁ is the lower energy level (3 for n=4 to n=3 transition)
n₂ is the higher energy level (4 for n=4 to n=3 transition)

The result is typically converted to nanometers (nm) for easier interpretation.

Calculation

The calculation involves plugging the energy levels into the Rydberg formula and solving for the wavelength. The result is then converted to nanometers for practical use.

Note: The Rydberg formula assumes the hydrogen atom is in its ground state and does not account for relativistic or quantum electrodynamic effects, which become significant for very high energy levels.

Examples

Let's calculate the wavelength for the n=4 to n=3 transition:

λ = 1 / (1.0973731568160 × 10⁷ × (1/3² - 1/4²))
λ = 1 / (1.0973731568160 × 10⁷ × (0.1111 - 0.0625))
λ = 1 / (1.0973731568160 × 10⁷ × 0.0486)
λ ≈ 1 / 5.348 × 10^-4 ≈ 1.869 × 10^-6 m ≈ 1869 nm

The wavelength for the n=4 to n=3 transition is approximately 1869 nanometers.

FAQ

What is the Rydberg formula?

The Rydberg formula is an empirical equation that describes the wavelengths of light emitted by a hydrogen atom when an electron transitions between energy levels. It was developed by Johannes Rydberg and is fundamental in atomic spectroscopy.

Why is the wavelength in nanometers?

Nanometers are commonly used in spectroscopy because they provide a convenient scale for visible and near-ultraviolet light. The wavelength in meters is converted to nanometers by multiplying by 10⁹.

What is the Balmer series?

The Balmer series is a set of spectral lines in the visible and near-ultraviolet spectrum that result from electron transitions from higher energy levels to the n=2 level in a hydrogen atom. The n=4 to n=3 transition is part of this series.