Calculate The Wavelength of The N 4 N 1 Transition

This calculator computes the wavelength of the n=4 to n=1 transition in hydrogen using the Rydberg formula. The result is expressed in nanometers (nm) and includes the energy difference between the two states.

Introduction

The n=4 to n=1 transition in hydrogen is a fundamental atomic emission process that produces light in the ultraviolet spectrum. This transition occurs when an electron in the fourth energy level drops to the ground state (n=1).

The wavelength of emitted light can be calculated using the Rydberg formula, which relates the wavelength to the principal quantum numbers of the initial and final states. This calculation is essential in atomic physics and spectroscopy.

Rydberg Formula

The Rydberg formula for the wavelength of emitted light is:

λ = R × (1/n₁² - 1/n₂²)⁻¹
Where:
λ = wavelength (in meters)
R = Rydberg constant (1.0973731568160 × 10⁷ m⁻¹)
n₁ = initial principal quantum number
n₂ = final principal quantum number

For the n=4 to n=1 transition, n₁ = 4 and n₂ = 1.

Calculation Steps

Identify the initial and final quantum numbers (n₁ = 4, n₂ = 1)
Plug these values into the Rydberg formula
Calculate the term (1/n₁² - 1/n₂²)
Take the reciprocal of the result
Multiply by the Rydberg constant to get wavelength in meters
Convert meters to nanometers (1 m = 10⁹ nm)

Worked Example

Let's calculate the wavelength for the n=4 to n=1 transition:

λ = 1.0973731568160 × 10⁷ × (1/4² - 1/1²)⁻¹
= 1.0973731568160 × 10⁷ × (1/16 - 1/1)⁻¹
= 1.0973731568160 × 10⁷ × (0.0625 - 1)⁻¹
= 1.0973731568160 × 10⁷ × (-0.9375)⁻¹
= 1.0973731568160 × 10⁷ × (-1.0666666666666667)
= -1.1716 × 10⁶ m
(Negative sign indicates emitted photon energy)

The absolute value of the wavelength is 117.16 nm, which is in the ultraviolet range.

Interpreting Results

The calculated wavelength of 117.16 nm corresponds to ultraviolet light. This emission line is part of the hydrogen spectrum and is commonly used in spectroscopic analysis.

The negative sign in the calculation indicates that energy is released as a photon, which is consistent with the transition from a higher to a lower energy state.

Frequently Asked Questions

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. Its value is approximately 1.0973731568160 × 10⁷ m⁻¹.
Why is the wavelength negative in the calculation?: The negative sign indicates that energy is released as a photon during the transition. The absolute value of the wavelength is what's physically meaningful.
Can this formula be used for other elements?: The Rydberg formula is specifically for hydrogen and hydrogen-like ions. For other elements, more complex atomic models are needed.
What units should I use for the result?: The calculator provides results in nanometers (nm), which is a common unit for visible and ultraviolet wavelengths.
How accurate is this calculation?: The calculation uses the exact value of the Rydberg constant and follows the standard Rydberg formula, providing precise results for hydrogen transitions.