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Calculate The Wavelength of N 2 to N 1

Reviewed by Calculator Editorial Team

The transition from n=2 to n=1 in the hydrogen atom emits light with a specific wavelength. This calculator computes the wavelength using the Rydberg formula, which is fundamental in atomic physics.

Introduction

When an electron in a hydrogen atom transitions from the n=2 energy level to the n=1 ground state, it emits a photon with a characteristic wavelength. This wavelength can be calculated using the Rydberg formula, which relates the wavelength to the principal quantum numbers of the initial and final states.

The Rydberg formula is particularly important in spectroscopy and quantum mechanics, as it provides a way to predict the wavelengths of light emitted or absorbed by atoms. For hydrogen, the formula simplifies to a well-known constant.

Rydberg Formula

The wavelength (λ) of the emitted photon when an electron transitions from principal quantum number n2 to n1 is given by:

λ = 1 / [R∞ (1/n12 - 1/n22)]

Where:

  • R∞ is the Rydberg constant (1.0973731568539 × 107 m-1)
  • n1 is the principal quantum number of the final state (1 for ground state)
  • n2 is the principal quantum number of the initial state (2 for the transition)

For the specific case of n=2 to n=1 transition, the formula simplifies to:

λ = 1 / [R∞ (1/12 - 1/22)] = 1 / [R∞ (1 - 0.25)] = 1 / [0.75 R∞]

This results in the well-known Lyman-α wavelength of 121.567 nm.

Calculation Steps

  1. Identify the initial and final principal quantum numbers (n2 and n1).
  2. Square both quantum numbers.
  3. Calculate the reciprocal of the difference between the reciprocals of the squared quantum numbers.
  4. Multiply by the Rydberg constant to get the wavenumber.
  5. Take the reciprocal of the wavenumber to obtain the wavelength in meters.
  6. Convert to nanometers if desired.

Note: The Rydberg constant is a fundamental physical constant that appears in the formulas for the spectral lines of many chemical elements.

Worked Example

Let's calculate the wavelength for the n=2 to n=1 transition in hydrogen:

  1. Initial state (n2) = 2
  2. Final state (n1) = 1
  3. 1/n12 = 1/12 = 1
  4. 1/n22 = 1/22 = 0.25
  5. Difference = 1 - 0.25 = 0.75
  6. Wavenumber = R∞ × 0.75 = 1.0973731568539 × 107 × 0.75 ≈ 8.23030367667645 × 106 m-1
  7. Wavelength = 1 / wavenumber ≈ 1 / 8.23030367667645 × 106 ≈ 1.21567 × 10-7 m ≈ 121.567 nm

This confirms the known Lyman-α wavelength of 121.567 nm.

Applications

The calculation of transition wavelengths is crucial in several scientific and practical applications:

  • Spectroscopy: Identifying atomic species through their emission spectra
  • Astronomy: Studying stellar composition and interstellar medium
  • Laser technology: Designing lasers that emit specific wavelengths
  • Quantum computing: Understanding atomic transitions for qubit operations
  • Medical imaging: Using specific wavelengths for diagnostic purposes

FAQ

What is the Rydberg constant?

The Rydberg constant (R∞) is a fundamental physical constant that appears in the formulas for the spectral lines of many chemical elements. Its value is approximately 1.0973731568539 × 107 m-1.

Why is the n=2 to n=1 transition important?

The n=2 to n=1 transition in hydrogen produces the Lyman-α line at 121.567 nm, which is one of the most important spectral lines in astronomy and physics. It's used for studying the interstellar medium and stellar atmospheres.

Can this formula be used for other atoms?

Yes, the Rydberg formula can be generalized for other atoms, but the Rydberg constant is modified by the atomic number and the reduced mass of the electron. For hydrogen, it simplifies to the standard Rydberg constant.