# An electron is excited from the n=1 ground state to the n=3 state in a hydrogen atom. Does it take more energy to ionize (completely remove) the electron from n=3 than from the ground state?

Mar 11, 2017

Well, if you are climbing a flight of stairs..........

#### Explanation:

.........does it take you more energy to climb from the bottom of the stairs, than from 3 steps up the flight of stairs? Quite clearly it does.

When we ionize an electron, essentially we raise it from $n = 1$ to $n = \infty$; now clearly this is going to be endothermic:

${\text{Atom, (n=1),"+Delta_1 rarr "Atom}}^{+} \left(n = \infty\right) + {e}^{-}$

But when we raise the electron from $n = 1$ to $n = 3$, then clearly this also will be endothermic:

${\text{Atom, (n=1),"+Delta_2 rarr "Atom}}^{+} \left(n = 3\right) + {e}^{-}$

${\text{Atom, (n=3),"+Delta_3 rarr "Atom}}^{+} \left(n = \infty\right) + {e}^{-}$

Clearly, ${\Delta}_{2} + {\Delta}_{3} = {\Delta}_{1}$ by considerations of conservation of energy. And thus, necessarily, ${\Delta}_{1} > {\Delta}_{3}$ (and the difference is ${\Delta}_{2}$). Are you happy with this treatment?