What is the shortest wavelength of light that can be emitted by a hydrogen atom that has an initial configuration of 7p1?
Why did the hydrogen atom go to the beach? Because it wanted to show off its shorts-wavelength!
To determine the shortest wavelength of light emitted by a hydrogen atom with an initial configuration of 7p1, we need to calculate the energy difference between the initial state and a lower energy state.
The configuration 7p is referring to the principal quantum number (n = 7) and the orbital angular momentum quantum number (l = 1). The number after the letter 'p' indicates the magnetic quantum number (ml), which can range from -l to +l. In this case, 7p1 corresponds to ml = 1.
1. Determine the energy levels for the hydrogen atom:
The energy levels of a hydrogen atom are given by the equation: En = -13.6 eV / n^2, where n is the principal quantum number.
2. Calculate the energy difference between the initial and lower energy states:
To find the energy difference, we subtract the energy of the lower energy state from the initial state's energy. In this case, we need to find the energy difference between the 7p1 state and a lower state.
The energy of the 7p1 state can be calculated using the formula mentioned above:
Einitial = -13.6 eV / (7^2)
To find the lower energy state, we look at the possible values of ml for the 7p orbital, which are -1, 0, and 1. We need to find the energy associated with ml = -1.
The energy of the lower energy state can also be calculated using the above formula but with the new value of n. Since the principal quantum number remains the same (n = 7), only the value of ml changes.
Elower = -13.6 eV / (7^2)
3. Calculate the energy difference:
To get the energy difference (ΔE), we subtract the energy of the lower state from the initial state:
ΔE = Einitial - Elower
4. Convert the energy difference to wavelength using the formula:
The energy difference can be converted to wavelength using the formula: E = hc / λ, where E is the energy difference, h is Planck's constant (6.626 x 10^-34 J.s), c is the speed of light (2.998 x 10^8 m/s), and λ is the wavelength.
Using the value of ΔE and rearranging the formula, we get:
λ = hc / ΔE
5. Calculate the shortest wavelength:
Plug in the values of Planck's constant and the speed of light, along with the energy difference calculated in step 3, to calculate the wavelength.
Finally, round the answer to the appropriate number of significant figures to get the shortest wavelength of the emitted light.
Remember to always double-check your calculations to ensure accuracy.
Note: The above explanation assumes that the atom transitions from the 7p1 state to a lower energy state by emitting a photon.