In a hydrogen atom, form what energy level must an electron fall to the n=3 level to give a line at 1.28 micrometers (um) in the infrared region?
1/wavelength = R(1/n^3 - 1/x^2)
R = Rydberg constant.
wavelength = in meters.
Solve for x.
To determine the energy level that the electron must fall from in a hydrogen atom to the n=3 level, we need to apply the equation for energy difference in the hydrogen atom.
The energy difference between two energy levels in a hydrogen atom is given by the formula:
ΔE = -13.6 eV * (1/n^2_final - 1/n^2_initial)
where ΔE is the energy difference, n_final is the final energy level, and n_initial is the initial energy level.
In this case, we know that the electron falls to the n=3 level, which is the final energy level. We need to find the initial energy level.
To find the initial energy level, we can rearrange the formula:
ΔE = -13.6 eV * (1/3^2 - 1/n^2_initial)
ΔE/13.6 eV = 1/9 - 1/n^2_initial
Now, we can substitute the wavelength of the line in the infrared region, which is given as 1.28 micrometers or 1.28 × 10^-6 meters, into the equation for the energy difference:
ΔE = hc/λ
where ΔE is the energy difference, h is Planck's constant (6.626 × 10^-34 J·s), c is the speed of light (3.0 × 10^8 m/s), and λ is the wavelength.
Now, we can calculate the energy difference in joules:
ΔE = (6.626 × 10^-34 J·s * 3.0 × 10^8 m/s) / (1.28 × 10^-6 m)
ΔE ≈ 1.63 × 10^-19 J
Finally, we can substitute this value into the rearranged equation to solve for n_initial:
1.63 × 10^-19 J/13.6 eV = 1/9 - 1/n^2_initial
Let's solve for n_initial:
1.63 × 10^-19 J/13.6 eV ≈ 1/9 - 1/n^2_initial
Convert eV to joules by multiplying by the conversion factor 1.6 × 10^-19 J/eV:
(1.63 × 10^-19 J/13.6 eV) * (1.6 × 10^-19 J/eV) ≈ 1/9 - 1/n^2_initial
0.19 ≈ 1/9 - 1/n^2_initial
Simplify the equation:
1/n^2_initial ≈ 1/9 - 0.19
1/n^2_initial ≈ 0.11
Now solve for n_initial:
n^2_initial ≈ 1 / 0.11
n^2_initial ≈ 9.09
Taking the square root of both sides:
n_initial ≈ √(9.09)
n_initial ≈ 3
Therefore, the electron must fall from the n=9 energy level to the n=3 energy level to produce a line at 1.28 micrometers in the infrared region.