Calculate Wavelength When Hydrogen Drops From N-3 to N-1

When an electron in a hydrogen atom transitions from a higher energy level to a lower one, it emits light with a specific wavelength. This calculator determines the wavelength of light emitted when hydrogen transitions from n=3 to n=1 using the Rydberg formula.

Introduction

Hydrogen is the simplest atom, consisting of a single proton and electron. When the electron moves between energy levels (quantum states), it emits or absorbs photons of specific wavelengths. The Rydberg formula allows us to calculate these wavelengths.

For a transition from level n₂ to n₁, where n₂ > n₁, the wavelength (λ) is given by:

Rydberg Formula

λ = R × (1/n₁² - 1/n₂²)^-1

Where:

λ = wavelength (in meters)
R = Rydberg constant (1.0973731 × 10⁷ m^-1)
n₁ = lower energy level
n₂ = higher energy level

For the transition from n=3 to n=1, we substitute these values into the formula to find the emitted wavelength.

Rydberg Formula

The Rydberg formula is a mathematical relationship that describes the wavelengths of light emitted by atoms. It was developed by the Swedish physicist Johannes Rydberg in the late 19th century.

The formula accounts for:

The energy difference between quantum states
The Rydberg constant, which is specific to hydrogen
The quantum numbers of the initial and final states

For hydrogen transitions, the formula simplifies to the one shown above, where the Rydberg constant is approximately 1.0973731 × 10⁷ m^-1.

Calculation Process

To calculate the wavelength for a hydrogen transition:

Identify the initial (n₂) and final (n₁) quantum numbers
Square both quantum numbers
Calculate the reciprocal of the difference between the reciprocals of the squared quantum numbers
Multiply by the Rydberg constant
Convert the result to nanometers (optional)

For the n=3 to n=1 transition, this process yields the characteristic red light emitted by hydrogen.

Worked Example

Let's calculate the wavelength for the transition from n=3 to n=1:

n₁ = 1, n₂ = 3
n₁² = 1, n₂² = 9
1/n₁² - 1/n₂² = 1 - 1/9 = 8/9
Reciprocal = 9/8
λ = (1.0973731 × 10⁷ m^-1) × (9/8) = 1.1721314 × 10⁷ m^-1
Convert to nanometers: 1/1.1721314 × 10⁷ m^-1 = 82.71 nm

The result is approximately 82.71 nanometers, which corresponds to red light in the visible spectrum.

FAQ

What is the Rydberg constant?: The Rydberg constant (R) is a fundamental physical constant that appears in the Rydberg formula. For hydrogen, its value is approximately 1.0973731 × 10⁷ m^-1.
Why does hydrogen emit light at specific wavelengths?: Hydrogen emits light at specific wavelengths because the electron can only occupy certain discrete energy levels. Transitions between these levels result in photons of specific energies and wavelengths.
What happens if the transition is from n=2 to n=1?: The wavelength for the n=2 to n=1 transition is 121.567 nm, which is in the ultraviolet range. This transition produces the Lyman-alpha line, which is important in astrophysics.
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 models are needed.
What units should I use for the result?: The formula gives wavelength in meters. For more intuitive values, you can convert to nanometers (nm) by multiplying by 10⁹.