Energy levels found by Schrodinger where iden6cal to those found by Bohr!
RJ En = − 2 n Instead of orbits the electron moves in regions of space called orbitals In the Bohr model the orbit was characterized by one quantum number n which could be any posi6ve integer. 1
In the Schrodinger theory there are 3 quantum numbers Symbol name values allowed significance n principal 1,2,3,… energy l secondary 0, 1, 2, n-‐1 shape m magne6c –l to +l orienta6on integer increments Secondary (azimuthal) quantum number (determines angular momentum of electron) is o9en designated by a le:er l 0 1 2 3 4 leKer s p d f g 2
Note the rela6onship between the quantum numbers n=1; l=0 ; m=0 ; 1s
1 orbital
n=2 ; l=0 ; m=0 ; 2s l=1 ; m=-‐1,0,+1 ; 2p
4 orbitals
n=3 ; l=0 ; m=0 ; 3s 9 orbitals l=1 ; m=-‐1,0,+1 ; 3p l=2; m=-‐2,-‐1,0,+1,+2 ; 3d n=4 ; l=0; m=0 ; 4s l=1; m=-‐1,0,+1 ; 4p 16 orbitals l=2 ; m=-‐2,-‐1,0,+1,+2 ; 4d l=3; m=-‐3,-‐2,-‐1,0,+1,+2,+3 ; 4f 3
Note the rela6onship between the quantum numbers n=1; 1 orbital n=2
Orbitals with same n cons@tute a shell
4 orbitals
n=3
9 orbitals
n=4
16 orbitals
Number of orbitals within a shell is called the degeneracy of the shell
The energy in the H atom depends only on the n quantum number 4
Loca@on of electron in atom is uncertain up to a point The square of an atomic orbital tells you the probability/volume of finding the electron in the space around the atomic nucleus If we could see the electron (which we can’t! ) and if, for example, the electron were a fire-‐fly (which it isn’t) it would appear to light up in a different region of space around the nucleus.
6
Dot diagram
Contour diagram 1s orbital
2s orbital Radial node
90% contours 7
3s orbital Dot diagram
Radial nodes
Contour diagram
90 % contour
All s orbitals are spherically symmetric
As n increases the orbitals get larger
The number of radial nodes is n-‐1 A node is a region of space where the probability of finding an electron is zero
Electron is most likely found outside of last node
Dot diagram
Contour diagram 2p orbital
Dot diagram Radial node
Contour diagram
90% contour
3p orbital
p orbitals
There are three p orbitals called px, py, pz Correspond to l=1 and m=-‐1, 0, +1 Contour diagrams for the orbitals are dumbbell shape Node at the nucleus As n gets larger the p orbitals get larger
Dot diagram
Contour diagram
3d orbital
90% contour
d orbitals
There are 5 d orbitals
Three of the d orbitals lie in a plane bisec6ng the x, y, & z axes
Two of the d orbitals lie in a plane aligned along the x,y & z axe
Contour diagrams for four of the d orbitals have 4 lobes each
Contour diagram for one d orbital has two lobes and a collar 14
If n=4 what values of I are permiKed ? l can go from 0 to n-‐1
s, p, d, f
Which of the following are not permiKed ? 3s
2p
2d
3f
6s
5d
Electron Spin Line spectra of many electron atoms show each line as a closely spaced pair of lines Stern and Gerlach designed an experiment to determine why A beam of Ag atoms was passed through a slit and into a magne@c field and the atoms were then detected Two spots were found: one with the electrons spinning in one direc@on and one with the electrons spinning in the opposite direc@on
Electron spin
There is a fourth quantum number associated with the electron spin The electron behaves like a bar magnet Some electrons have the north pole poin6ng up : Spin up : ms = +1/2 Some electrons have the north pole poin6ng down Spin down : ms=-‐1/2 18
Electron spin quantum number
1 ms = + 2
1 ms = − 2
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Many electron atoms Orbitals in many electron atoms closely resemble those in H The angular dependence is the same s, p, d,… Because nuclear charge is larger the radial dependence somewhat different Orbitals have the same shape but electrons are closer to the nucleus
Principal quantum number n s6ll describes the shell
Energies of orbitals within a shell are different s < p < d < f
H atom
Many electron atom 4p 3d
4s
4s 4p 4d 4f 3s 3p 3d
2s 2p
1s
Energy
Energy
3p 3s 2p 2s
1s
Recall constraints on quantum numbers
n = 1, 2, 3 … ∞ l = 0, 1, … , n-‐1 -‐l ≤ ml ≤ l integer increments ms=+1/2 or -‐1/2 Certain rules for pu[ng electrons in orbitals
Electron configura6ons Three principles Aufau or building up principle Add electrons so that energy of atom is minimized Pauli principle Two electrons may occupy the same orbital so long as they have opposite spins Hunds rule When electrons fill a degenerate set of orbitals they are placed singly in each orbital with parallel spins before pairing them in a single orbital
Detailed electron configura6on of N
Energy
2p 2s
1s 25
Detailed electron configura6on of O
Energy
2p 2s
1s 26
Detailed electron configura6on of F
Energy
2p 2s
1s 27
Detailed electron configura6on of Ne
Energy
2p 2s
1s 28
Li Be B C N Ne Na
Orbital energies
1s < 2s < 2p < 3s< 3p < 4s < 3d < 4p
