A line in the infrared region of the hydrogen spectrum has a wavelength of 1.875 micro meters. What is the transition responsible for this wavelength?
To determine the transition responsible for a specific wavelength in the hydrogen spectrum, we can use the Rydberg formula which relates the wavelength of light emitted or absorbed in a spectral line to the energy levels involved in the transition.
The Rydberg formula is given by:
1/λ = Rh (1/n1^2 - 1/n2^2)
where:
- λ is the wavelength of the light in meters,
- Rh is the Rydberg constant for hydrogen (approximately 1.097 × 10^7 m^(-1)),
- n1 is the initial energy level (quantum number),
- n2 is the final energy level (quantum number).
To find the transition responsible for a wavelength of 1.875 micrometers (or 1.875 × 10^(-6) meters):
1/λ = Rh (1/n1^2 - 1/n2^2)
Rearranging the formula and plugging in the given values:
1/(1.875 × 10^(-6)) = 1.097 × 10^7 (1/n1^2 - 1/n2^2)
Simplifying further, we can solve for the values of n1 and n2 by substituting various integers and observing the resulting wavelengths.
By trial and error, we can find that for n1 = 2 and n2 = 3, the calculated wavelength is approximately 1.875 micrometers.
Therefore, the transition responsible for the wavelength of 1.875 micrometers in the hydrogen spectrum is from the n = 3 energy level to the n = 2 energy level.