An electron in a hydrogen atom absorbs 15.5 x 10-20 J of energy. If the electron originated at energy level 3, to what level was it excited?
To determine the level to which the electron was excited, we need to calculate the change in energy between the initial and final level.
We know that the energy change is given by the formula:
ΔE = E_final - E_initial
Since the electron absorbed energy, the final energy level will be higher than the initial energy level. Therefore, we need to find the energy of the initial and final levels.
The energy of an electron in the hydrogen atom can be calculated using the formula for the energy levels in the hydrogen atom:
E = -13.6 eV / n^2
where E is the energy, n is the principal quantum number, and -13.6 eV is a constant.
First, let's convert the given energy from joules to electron volts (eV). We know that 1 eV is equal to 1.6 x 10^-19 J:
15.5 x 10^-20 J * (1 eV / 1.6 x 10^-19 J) ≈ 0.097 eV
Now, let's find the initial level (E_initial) using the energy formula:
E_initial = -13.6 eV / n_initial^2
We are given that the electron originated at energy level 3, so substituting n_initial = 3:
E_initial = -13.6 eV / (3^2) ≈ -13.6 eV / 9 ≈ -1.51 eV
Now, we can find the final level (E_final) by rearranging the energy formula:
E_final = E_initial + ΔE
E_final = -1.51 eV + 0.097 eV ≈ -1.41 eV
Finally, we can find the principal quantum number (n_final) by rearranging the energy formula:
E_final = -13.6 eV / n_final^2
Substituting E_final = -1.41 eV:
-1.41 eV = -13.6 eV / n_final^2
n_final^2 ≈ 13.6 eV / 1.41 eV ≈ 9.65
Taking the square root of both sides:
n_final ≈ √9.65 ≈ 3.11
Since the principal quantum number must be a whole number, we can approximate n_final as 3.
Therefore, the electron was excited from energy level 3 to energy level 3.