In a hydrogen atom, an electron in its ground state has an energy 𝐸 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 it’s attached to. So if the ionization energy is zero electron volts, we’re told that at the ground state of a hydrogen atom is 13.6 electron volts below that level. We’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 𝑛 value, called the number state, and it’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 we’re solving for the energy of the 𝑛th orbital in the hydrogen atom, that energy equals the ground state energy divided by 𝑛 squared. Since it’s 𝑛 we want to solve for, we can rearrange this equation. And we see that 𝑛 is equal to the square root of 𝐸 sub zero over 𝐸 sub 𝑛.
𝐸 sub zero is our ground state energy level and 𝐸 sub 𝑛 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 𝑛, we find that it’s equal to four. So it’s the 𝑛 equals four energy level that requires this much energy for electron to be ionized.