An electron in a hydrogen atom could undergo any of the transitions listed below, by emitting light. Which transition would give light of the shortest wavelength?
n=4 to n=2
n=3 to n=1
n=2 to n=1
n=4 to n=1
n=4 to n=1
The one in which the electron travels (moves) the farthest.
n=5 to n=4
To determine the transition that would give light of the shortest wavelength, we need to use the formula for calculating the wavelength of light emitted during a transition in the hydrogen atom.
The formula for calculating the wavelength of light emitted during a transition in the hydrogen atom is known as the Rydberg formula:
1/λ = R (1/n₁² - 1/n₂²)
Where:
- λ represents the wavelength of light emitted.
- R is the Rydberg constant (approximately equal to 1.097 x 10^7 m^-1).
- n₁ and n₂ represent the initial and final energy levels of the electron, respectively.
By substituting the given values into the formula and calculating the wavelengths, we can determine the transition that results in the shortest wavelength.
1) n=4 to n=2:
Using the Rydberg formula:
1/λ = R (1/2² - 1/4²) = R (1/4 - 1/16) = R (3/16)
This transition gives a wavelength of:
λ = 16/3R
2) n=3 to n=1:
Using the Rydberg formula:
1/λ = R (1/1² - 1/3²) = R (1 - 1/9) = R (8/9)
This transition gives a wavelength of:
λ = 9/8R
3) n=2 to n=1:
Using the Rydberg formula:
1/λ = R (1/1² - 1/2²) = R (1 - 1/4) = R (3/4)
This transition gives a wavelength of:
λ = 4/3R
4) n=4 to n=1:
Using the Rydberg formula:
1/λ = R (1/1² - 1/4²) = R (1 - 1/16) = R (15/16)
This transition gives a wavelength of:
λ = 16/15R
Comparing the calculated wavelengths:
λ(n=4 to n=2) = 16/3R
λ(n=3 to n=1) = 9/8R
λ(n=2 to n=1) = 4/3R
λ(n=4 to n=1) = 16/15R
We can see that the transition with the shortest wavelength is n=4 to n=2, which yields a wavelength of 16/3R.
Therefore, the transition that gives light of the shortest wavelength is n=4 to n=2.