Lecture-12 - Chem111 Fall, 2005 Lecture 12: Electron Spin...

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Chem111 Fall, 2005 51 Before we move on to many electron atoms … Electron Spin Æ fourth quantum number (m s ) needed for a complete description of an atomic particle Æ 3 spatial dimensions (x, y, z) or (r, θ , ϕ ) Æ gives us our 3 quantum numbers (n, l, m) Æ 4 th dimension = time Æ intrinsic electron spin Î m s quantum number Dirac showed that electron spin is a natural combination of relativity and QM (also predicted the existence of the positron) m s can have values of +1/2 (“spin up”) or -1/2 (“spin down”) Need all four quantum numbers to completely describe an electron in an atom n, A , m A Æ Ψ Æ spatial part (space and distance) Æ idenfify each orbital m s Æ spin Æ identifies each electron Pauli Exclusion Principle: several ways to state this principle … No two electrons can have exactly the same total wavefunction No two electrons can have the same set of four quantum numbers If (n, l, and m) are the same for two electrons, then m s must be different … one easy consequence no more than 2 e- per orbital – one “up” and one “down” Lecture 12: Electron Spin Pauli Exclusion Many-electron atoms Aufbau rule e- beam N S 50% Stern-Gerlach type of experiment together these identify the total wavefunction for an electron
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Chem111 Fall, 2005 52 Example A 2s orbital can have two electrons. What are their quantum numbers? n = 2 from the "2" in "2s" for an s-orbital, A = 0 if = 0, then m = 0 also there are two electrons, one with spin up (m s = ½) and one spin down (m = -½) {2, 0, 0, ½} and {2, 0, 0, -½} Νοτε: maximum number of orbitals {n, l, m} in a shell = n 2 maximum number of electrons {n, l, m, m s } in a shell= 2n 2 Now you try one … How many electrons are there in the 3 rd shell? Write out all of their quantum numbers. Many Electron Atoms with one electron (H-atom) Æ E depends only on the principal quantum number n This is NOT TRUE when more electrons are added (i.e. apply H atom to larger atoms) H = T + V attraction H = T + V attraction + V repulsion kinetic energy of all electrons original potential energy part = attraction between nuvleus and electrons new potential energy part = repulsion between the electrons one-electron atom – kinetic and attractive potential terms
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Chem111 Fall, 2005 53 Can't solve this problem exactly (ugh!) Æ use approximations So, what do we do? Well, we assume that the hydrogen atom solutions will still work, basically , though we might need to make some relatively small alterations . And then we test our assumption . And it DOES work! (hooray!) End result is that the orbitals within a given shell do not have the same energy any more! Both n and l determine the energy: 1s 2s 2p 3s 3p 3d E 1s 2s 2p 3s 3p 3d 4s E H-atom (one electron) many-electron atoms not degenerate any more! split by the electron-electron repulsion in the many-electron atom!
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Lecture-12 - Chem111 Fall, 2005 Lecture 12: Electron Spin...

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