Calculate the shortest wavelength of visible light (in nanometers) seen in the spectrum of the hydrogen atom.
nm
What are the principle quantum numbers for the levels in this transition?
It depends somewhat on what you call the visible spectrum. I think most texts use 400 nm to about 700 nm but I have seen some references that list 390 nm to 750 nm as detectable by the human eye. Here is a site that lists the wavelengths of the Balmer Series. Take your pick (and it shows the principle quantum numbers involved).
http://hyperphysics.phy-astr.gsu.edu/hbase/tables/hydspec.html
To calculate the shortest wavelength of visible light in the spectrum of the hydrogen atom, we can make use of the Rydberg formula. The Rydberg formula relates the wavelengths of the spectral lines emitted by hydrogen to the difference in energy between two energy levels.
The general form of the Rydberg formula is:
1/λ = R * (1/n₁² - 1/n₂²)
Where:
- λ is the wavelength of the light emitted
- R is the Rydberg constant (approximately 1.097 × 10^7 m⁻¹)
- n₁ and n₂ are the principal quantum numbers for the energy levels in the transition
In this case, since we are interested in the shortest wavelength of visible light, we need to determine the transition with the highest energy difference. The transition with the highest energy difference occurs when an electron jumps from the highest energy level (n) of the initial state to the lowest energy level (n=1) of the final state.
So, we substitute n₁ = n and n₂ = 1 into the Rydberg formula:
1/λ = R * (1/n² - 1/1²)
Simplifying further:
1/λ = R * (1/n² - 1)
Now, we can calculate the shortest wavelength by plugging in the value of n for the hydrogen atom, which is 2 (since it is the first excited state):
1/λ = R * (1/2² - 1)
1/λ = R * (1/4 - 1)
1/λ = R * (-3/4)
To find λ (wavelength), we can take the reciprocal of both sides:
λ = 1 / (R * (-3/4))
Now, we can substitute the value of the Rydberg constant (R):
λ = 1 / (1.097 × 10^7 m⁻¹ * (-3/4))
Calculating this expression will give you the value of the shortest wavelength of visible light in meters. To convert it to nanometers, you can multiply the value by 10^9 since 1 meter is equal to 10^9 nanometers.
Note: The values used for calculations here are approximate for improved clarity, but the actual values can be found in scientific references and data tables.