Calculation Wavelength of Light From N 4 to N 3

This calculator determines the wavelength of light emitted when an electron transitions from the n=4 energy level to the 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 of light. The wavelength of this emitted light can be calculated using the Rydberg formula, which relates the wavelength to the energy levels involved.

This calculation is important in atomic spectroscopy and helps scientists understand the energy structure of atoms. The result provides insight into the electromagnetic radiation emitted during these electronic transitions.

Rydberg Formula

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

Formula

λ = R × (1/n₂² - 1/n₁²)^-1

Where:

λ = wavelength of emitted light (in nanometers)
R = Rydberg constant (1.0973731568508 × 10⁷ m^-1)
n₁ = initial energy level (4 for this calculation)
n₂ = final energy level (3 for this calculation)

The Rydberg constant is a fundamental physical constant that relates to the wavelengths of spectral lines of many chemical elements. For hydrogen, the Rydberg formula provides precise wavelengths for electronic transitions between quantized energy levels.

Calculation Example

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

Example Calculation

λ = 1.0973731568508 × 10⁷ × (1/3² - 1/4²)^-1

= 1.0973731568508 × 10⁷ × (1/9 - 1/16)^-1

= 1.0973731568508 × 10⁷ × (0.1111... - 0.0625)^-1

= 1.0973731568508 × 10⁷ × (0.048611...)^-1

= 1.0973731568508 × 10⁷ × 20.5614

= 2.2626 × 10^-7 m

= 226.26 nm

This calculation shows that the wavelength of light emitted during this transition is approximately 226.26 nanometers. This is in the ultraviolet region of the electromagnetic spectrum.

Interpreting Results

The wavelength calculated represents the specific color of light emitted when the electron transitions from n=4 to n=3. Different transitions produce different wavelengths, which correspond to different colors in the visible spectrum.

For this specific transition:

The wavelength is 226.26 nm
This is in the ultraviolet range (100-400 nm)
The energy of the photon is inversely proportional to the wavelength

Note

In real hydrogen atoms, other factors like electron spin and relativistic effects can slightly modify these wavelengths. However, the Rydberg formula provides a good approximation for most practical purposes.

FAQ

What is the Rydberg formula used for?

The Rydberg formula is used to calculate the wavelengths of light emitted by atoms when electrons transition between energy levels. It's fundamental in atomic spectroscopy and helps understand the energy structure of atoms.

Why is the wavelength in the ultraviolet range for this transition?

The transition from n=4 to n=3 results in a relatively small energy difference, which corresponds to ultraviolet light (100-400 nm). Larger energy differences would result in visible or infrared light.

Can this formula be used for other atoms besides hydrogen?

Yes, the Rydberg formula can be adapted for other atoms by adjusting the Rydberg constant. However, the formula is most accurate for hydrogen and hydrogen-like atoms.

What units should be used for the wavelength result?

The wavelength is typically expressed in nanometers (nm) for atomic transitions, as this unit provides a convenient scale for visible and ultraviolet light.