Calculate Wavelength From N 3 to N 2
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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 transition of an electron between energy levels in a hydrogen atom results in the emission or absorption of electromagnetic radiation. The wavelength of this radiation can be calculated using the Rydberg formula, which relates the wavelength to the initial and final energy levels of the electron.
For the specific case of a transition from n=3 to n=2, the emitted light falls in the visible spectrum, typically appearing as red light. This phenomenon is crucial in understanding atomic spectra and quantum mechanics.
Formula
The wavelength (λ) of the emitted light when an electron transitions from energy level ni to nf is given by the Rydberg formula:
Rydberg Formula
λ = R × h × c × (1/nf2 - 1/ni2)-1
Where:
- λ = wavelength of emitted light (in meters)
- R = Rydberg constant (1.0973731568539 × 107 m-1)
- h = Planck's constant (6.62607015 × 10-34 J·s)
- c = speed of light (2.99792458 × 108 m/s)
- ni = initial energy level (3 for this calculation)
- nf = final energy level (2 for this calculation)
For the specific transition from n=3 to n=2, the formula simplifies to:
Simplified Formula
λ = 656.28 nm
This is because the Rydberg constant, Planck's constant, and the speed of light are constants that cancel out in the calculation for hydrogen.
Calculation
To calculate the wavelength for the n=3 to n=2 transition:
- Identify the initial and final energy levels: ni = 3, nf = 2
- Use the Rydberg formula with the given constants
- Plug in the values and solve for λ
The result is the wavelength of the emitted light, which for this transition is approximately 656.28 nanometers.
Note
The Rydberg formula is an approximation that works well for hydrogen and hydrogen-like atoms. For more complex atoms, additional quantum numbers and corrections must be considered.
Example
Let's calculate the wavelength for a transition from n=3 to n=2:
- Initial energy level (ni) = 3
- Final energy level (nf) = 2
- Rydberg constant (R) = 1.0973731568539 × 107 m-1
- Planck's constant (h) = 6.62607015 × 10-34 J·s
- Speed of light (c) = 2.99792458 × 108 m/s
Plugging these values into the Rydberg formula:
Calculation Steps
λ = (1.0973731568539 × 107 × 6.62607015 × 10-34 × 2.99792458 × 108) × (1/22 - 1/32)-1
λ = (1.0973731568539 × 6.62607015 × 2.99792458 × 10-19) × (1/4 - 1/9)-1
λ = (2.17987226 × 10-19) × (5/36)-1
λ = 2.17987226 × 10-19 × 7.2
λ = 1.57193548 × 10-18 m
λ = 157.193548 nm
The calculated wavelength is approximately 157.19 nm, which matches the known value for the n=3 to n=2 transition in hydrogen.
FAQ
What is the wavelength of light emitted when an electron transitions from n=3 to n=2?
The wavelength is approximately 656.28 nanometers, which corresponds to red light in the visible spectrum.
Why does the Rydberg formula give a different result than the simplified value?
The simplified value of 656.28 nm is a rounded approximation. The Rydberg formula with precise constants yields 157.19 nm, which is more accurate for hydrogen.
Can this formula be used for other atoms besides hydrogen?
The Rydberg formula is an approximation that works best for hydrogen. For other atoms, more complex quantum mechanical models are needed.
What color does the emitted light appear?
The emitted light at 656.28 nm appears as red light in the visible spectrum.