Question

Calculating wavelength help?

Calculate the wavelength, in nanometers, of the spectral line produced when an electron in a hydrogen atom undergoes the transition from the energy level n = 7 to the level n = 2.

____nm

Answer 1

Answer below

Explanation:

The energy for an electron in the hydrogen atom is given by:
`E_n = -R_H(1/n^2)`, where n is the energy level of the hydrogen atom. `R_H = 2.18xx10^-18J`

We'll just calculate the different between n=7 and n=2, or
`ΔE = E_2 - E_7`, and once we have that, we can use
`ΔE = hc/λ`, where λ is the wavelength in meters and h is planck's constant, `6.626xx10^-34 J*s`

`E_7 = -2.18xx10^-18*(1/7^2) = -4.45x10^-20 J`
`E_2 = -2.18xx10^-18*(1/2^2) = -5.45x10^-19 J`

`ΔE = E_2 - E_7 = -5.45x10^-19 - - 4.45x10^-20` `=-5.01xx10^-19 J`
Just a note here. The change in energy is negative. This is not some weird kind of negative energy, but rather just means direction. Energy is leaving the system, is "exothermic", if you want to think of it in those terms, but the energy leaves as a photon of light. So, for `ΔE = hc/λ`, we'll use the + value for this, since there isn't a -wavelength.

`5.01xx10^-19 J = hc/λ` Solve for λ
`λ = "hc"/"5.01x10^-19" = 3.97xx10^-7 meters`
`λ = 397 nm`