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Calculate The Wavelength of Light Emitted N 3 N 2

Reviewed by Calculator Editorial Team

When an electron in a hydrogen atom transitions from the n=3 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

The emission of light from atoms occurs when electrons transition between energy levels. For hydrogen atoms, the Rydberg formula allows us to calculate the wavelength of emitted light based on the initial and final energy levels.

In this case, we're calculating the wavelength for the transition from n=3 to n=2, which is a visible light emission in the red part of the spectrum.

Rydberg Formula

Formula

The wavelength (λ) of light emitted when an electron transitions from energy level ni to nf is given by:

λ = 1 / [R∞ (1/nf2 - 1/ni2)]

Where:

  • R∞ = Rydberg constant (1.0973731568508 × 107 m-1)
  • ni = initial energy level (3 for this calculation)
  • nf = final energy level (2 for this calculation)

The Rydberg formula is derived from the Bohr model of the atom and provides a precise way to calculate emission wavelengths for hydrogen-like atoms.

Worked Example

Let's calculate the wavelength for the n=3 to n=2 transition:

  1. Identify the initial and final energy levels: ni = 3, nf = 2
  2. Plug the values into the formula:

    λ = 1 / [1.0973731568508 × 107 (1/22 - 1/32)]

  3. Calculate the denominator:

    1/22 = 0.25

    1/32 ≈ 0.1111

    Difference = 0.25 - 0.1111 ≈ 0.1389

  4. Multiply by Rydberg constant:

    1.0973731568508 × 107 × 0.1389 ≈ 1.518 × 106 m-1

  5. Take the reciprocal to get wavelength:

    λ ≈ 1 / 1.518 × 106 ≈ 6.59 × 10-7 m or 659 nm

This calculation shows the wavelength is approximately 659 nanometers, which is in the red part of the visible spectrum.

Interpreting Results

The calculated wavelength provides several important insights:

  • The emitted light is in the visible spectrum (659 nm is red)
  • This transition is part of the Balmer series of hydrogen emission lines
  • The wavelength is characteristic of the hydrogen atom's energy structure

Note

In real-world conditions, the actual wavelength may be slightly different due to environmental factors and the atom's quantum state.

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. For hydrogen, it's approximately 1.097 × 107 m-1.
Why is this wavelength in the red part of the spectrum?
The transition from n=3 to n=2 falls within the Balmer series of hydrogen emission lines, which produces visible light in the red to violet range.
Can this formula be used for other atoms?
Yes, the Rydberg formula can be adapted for other hydrogen-like atoms (atoms with a single electron) by adjusting the Rydberg constant.
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 length units.