Calculate The Wavelength of Light Emitted N 4 N 2

When an electron in a hydrogen atom transitions from the n=4 energy level to the n=2 energy level, it emits light with a specific wavelength. This calculator computes that wavelength using the Rydberg formula, which is fundamental in atomic physics.

Introduction

In quantum mechanics, electrons in atoms occupy specific energy levels called principal quantum numbers (n). When an electron transitions from a higher energy level to a lower one, it emits light with a wavelength that can be calculated using the Rydberg formula.

For the transition from n=4 to n=2 in a hydrogen atom, we can calculate the emitted wavelength using the following steps:

Identify the initial (n₁) and final (n₂) principal quantum numbers
Use the Rydberg constant (R_∞) which is 1.0973731568508 × 10⁷ m^-1
Calculate the wavenumber (ν) using the formula
Convert the wavenumber to wavelength using λ = 1/ν

Rydberg Formula

The Rydberg formula for the wavenumber (ν) of light emitted when an electron transitions from level n₁ to n₂ is:

ν = R_∞ × (1/n₂² - 1/n₁²)

Where:

ν = wavenumber (m^-1)
R_∞ = Rydberg constant (1.0973731568508 × 10⁷ m^-1)
n₁ = initial principal quantum number (4 for our calculation)
n₂ = final principal quantum number (2 for our calculation)

The wavelength (λ) is then calculated as the reciprocal of the wavenumber:

λ = 1/ν

Calculation Example

Let's calculate the wavelength for the transition from n=4 to n=2:

Identify n₁ = 4 and n₂ = 2
Use R_∞ = 1.0973731568508 × 10⁷ m^-1
Calculate the wavenumber:

ν = 1.0973731568508 × 10⁷ × (1/2² - 1/4²)

ν = 1.0973731568508 × 10⁷ × (0.25 - 0.0625)

ν = 1.0973731568508 × 10⁷ × 0.1875

ν = 2.069038345276 × 10⁶ m^-1
Calculate the wavelength:

λ = 1/ν = 1/2.069038345276 × 10⁶ m^-1

λ = 4.833 × 10^-7 m or 483.3 nm

The wavelength of light emitted during this transition is approximately 483.3 nanometers.

Interpreting Results

The calculated wavelength of 483.3 nm falls in the visible light spectrum (400-700 nm), which explains why this transition is often used in spectroscopy demonstrations.

This calculation is particularly important in:

Atomic spectroscopy
Laser technology
Understanding atomic energy levels
Quantum mechanics education

Note: The Rydberg formula is an approximation that works best for hydrogen-like atoms. For more complex atoms, quantum electrodynamics must be considered.

FAQ

What is the Rydberg constant?: The Rydberg constant (R_∞) is a fundamental physical constant that relates to the wavelengths of spectral lines of many chemical elements. It's approximately 1.0973731568508 × 10⁷ m^-1.
Why is this wavelength in the visible spectrum?: The 483.3 nm wavelength falls within the visible light range (400-700 nm), which is why this transition produces visible light that can be observed in spectroscopy experiments.
Can this formula be used for other atoms?: The Rydberg formula is most accurate for hydrogen atoms. For other atoms, more complex quantum mechanical calculations are needed due to electron-electron interactions and nuclear effects.
What units should I use for the result?: The wavelength is typically expressed in nanometers (nm) for visible light, but can also be shown in meters or other appropriate units depending on the context.