Video: Initial Energy Level of Electron Ionized from Hydrogen Atom

In a hydrogen atom, an electron in its ground state has an energy ๐ธโ‚€ = โˆ’13.6 eV. 0.850 eV is required to remove an electron from a hydrogen atom. What number state was the atom in before the ionization occurred?

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Video Transcript

In a hydrogen atom, an electron in its ground state has an energy ๐ธ sub zero equals negative 13.6 electron volts. 0.850 electron volts is required to remove an electron from a hydrogen atom. What number state was the atom in before the ionization occurred?

The ground state energy of an electron in a hydrogen atom is equal to the amount of energy below the ionization energy; this is the energy at or above which an electron is free to escape the atom itโ€™s attached to. So if the ionization energy is zero electron volts, weโ€™re told that at the ground state of a hydrogen atom is 13.6 electron volts below that level. Weโ€™re told if an electron which is 0.850 electron volts below the level required to ionize an electron from a hydrogen atom, that energy level corresponds to some quantum number ๐‘› value, called the number state, and itโ€™s that value we want to solve for.

To do that, we can recall the mathematical relationship for hydrogen energy levels, which says that if weโ€™re solving for the energy of the ๐‘›th orbital in the hydrogen atom, that energy equals the ground state energy divided by ๐‘› squared. Since itโ€™s ๐‘› we want to solve for, we can rearrange this equation. And we see that ๐‘› is equal to the square root of ๐ธ sub zero over ๐ธ sub ๐‘›.

๐ธ sub zero is our ground state energy level and ๐ธ sub ๐‘› is the energy given as 0.850 electron volts below the ionization level. When we plug those values in to this expression, when we calculate this value for ๐‘›, we find that itโ€™s equal to four. So itโ€™s the ๐‘› equals four energy level that requires this much energy for electron to be ionized.

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