Calculate The Wavelength of The N 4 N 3 Transition

Calculating the wavelength of spectral lines in hydrogen is fundamental to understanding atomic physics. This guide explains how to calculate the wavelength for the n=4 to n=3 transition using the Rydberg formula, provides a working calculator, and explains the physics behind the result.

Introduction

The n=4 to n=3 transition in hydrogen atoms produces a spectral line in the red part of the visible spectrum. This transition is important in astrophysics, quantum mechanics, and spectroscopy. The wavelength of this transition can be calculated using the Rydberg formula, which relates the wavelength to the principal quantum numbers of the initial and final states.

This guide will walk you through:

The Rydberg formula and its components
How to calculate the wavelength for the n=4 to n=3 transition
A step-by-step worked example
Common questions about hydrogen spectral lines

Rydberg Formula

The wavelength of a spectral line in hydrogen 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 (1.0973731 × 10⁷ m⁻¹)
n₁ is the principal quantum number of the final state (3 for n=4 to n=3 transition)
n₂ is the principal quantum number of the initial state (4 for n=4 to n=3 transition)

The formula shows that the wavelength depends on the difference in energy levels between the initial and final states. For the n=4 to n=3 transition, the calculation simplifies to:

λ = 1 / [1.0973731 × 10⁷ (1/3² - 1/4²)]

Calculation Steps

Identify the initial and final quantum numbers (n₂ = 4, n₁ = 3)
Calculate the difference in energy levels: (1/n₁² - 1/n₂²)
Multiply by the Rydberg constant: R × (1/n₁² - 1/n₂²)
Take the reciprocal to get the wavelength in meters
Convert to nanometers (optional) by multiplying by 10⁹

Note: The Rydberg constant is an empirical constant that fits experimental data. It's not derived from first principles but is fundamental to hydrogen spectroscopy.

Worked Example

Let's calculate the wavelength for the n=4 to n=3 transition step by step.

Identify quantum numbers: n₂ = 4, n₁ = 3
Calculate energy difference:
1/3² - 1/4² = 1/9 - 1/16 ≈ 0.1111 - 0.0625 = 0.0486
Multiply by Rydberg constant:
1.0973731 × 10⁷ × 0.0486 ≈ 5.32 × 10⁵ m⁻¹
Take reciprocal to get wavelength:
λ ≈ 1 / 5.32 × 10⁵ ≈ 1.88 × 10⁻⁷ m
Convert to nanometers:
1.88 × 10⁻⁷ × 10⁹ ≈ 188 nm

The calculated wavelength for the n=4 to n=3 transition is approximately 188 nanometers.

FAQ

What is the Rydberg constant?: The Rydberg constant (R) is a fundamental physical constant that relates to the wavelengths of spectral lines in hydrogen. It's approximately 1.0973731 × 10⁷ m⁻¹.
Why is the wavelength in the red part of the spectrum?: The n=4 to n=3 transition produces red light because the energy difference corresponds to a wavelength in the 620-750 nm range, which is the red part of the visible spectrum.
Can this formula be used for other atoms?: The Rydberg formula is specific to hydrogen and hydrogen-like atoms. For other atoms, more complex quantum mechanical calculations are needed.
What units should I use for the result?: The formula gives wavelength in meters. For spectroscopy, nanometers (nm) are commonly used, so you may want to multiply by 10⁹ to convert.
How accurate is this calculation?: The calculation is theoretically exact for hydrogen. In practice, experimental measurements may differ slightly due to factors like electron mass corrections.