isabelle ruedisueli 4e wrote:Hi! On page 47 of the Course Reader, it says that as n approaches infinity, E approaches 0, and that (negative means bound e has lower energy than free e).
Can someone explain what "negative means bound e has lower energy than free e" means? What is a bound electron versus a free electron?
Thanks so much :)
A bound electron is an electron that is "bound" to the nucleus, as in it is part of an orbital and remains around the atom. A free electron is an electron that is off by itself; it isn't a "part" of any particular atom. For example, consider the photoelectric effect example that was talked about before. The atoms in the metal that is being bombarded with light all have bound electrons and the electrons that are ejected are free and are of a higher energy than the bound electrons because a photon of light had to be absorbed in order to eject the electron.
The phrase "negative means bound e has lower energy than free e" is also referencing the different energy levels that Dr. Lavelle was diagramming and talking about in lecture.
Here's a crude diagram for your reference: (the vertical line on the left represents energy)
____________________________ n=4
____________________________ n=3

____________________________ n=2



____________________________ n=1
( + ) < atomic nucleus
Anyway, when the atom is hit with a photon, the electron moves from one energy state to a higher one, lets say from n=1 to n=2, for example, because it absorbed the energy of the photon. This means that n=2 is a higher energy state than n=1 and n=3 is a higher energy state than n=2, etc. Therefore, it logically follows that the energy of a free electron, which is basically at n=infinity, would have a higher energy than the bound electron.
To explain the phrase, "negative means bound e has lower energy than free e" you'll have to consider the equation E=hR/n^2, where h is Planck's constant, R is the Ryberg constant, and n is the energy state.
This equation gives the energies of the different n states in the hydrogen atom, which is the atom we're talking about.
***It is important to note the negative sign on the term because it indicates that the energy's sign is negative, which tells us that this equation is talking about the energy of the electron in terms of energy lost, e.g. the energy emitted in the form of a photon, not in terms of energy added from the ground state, e.g. energy absorbed.*** Because the reference point is n=infinity and E=0 and not the ground state of the atom, any value of n, except n=infinity will be negative, and if n=infinity represents a truly free electron, and n=some smaller value like n=1,2,3,4,5,6... represents different feasible energy states where the electron is bound, then the energy of those states where the electron is bound will always be less than the energy of the free electron.
Does this make sense? I know its a bit long and convoluted...
To put it another, less mathematical way, you can think of a free electron as an that electron is very far away from the nucleus of an atom, like when n is some massive number or infinity. If an electron is very far from the nucleus then it doesn't really interact with the nucleus at all. This means that the absorption of a photon isn't necessary to eject the electron from the nucleus because if there is no attraction between the nucleus and the electron, then no energy is required to pull it away from the nucleus. To free a bound electron, an electron around an atom, you must put in energy. If energy must be put in to free an electron, and no energy is required to remove a free electron, then that means that the free electron already has that energy, meaning that it has a higher energy when compared to a bound electron.
Hope this helps!