Atomic Quantum Numbers

1. R = - E H n , l , m 2 n l Atomic Quantum Numbers Principle quantum number n = 1,2,3,.... The energy of an orbital depends only on n: RH = 2.1810-18 J Rydberg constant: The size of an orbital increases with n.

2. Atomic Quantum Numbers Azimuthal quantum number l = 0,1,2,..., n-1 l determines the shape of an orbital. The value of l is designated by a letter: l = 0 , 1 , 2 , 3 , 4 ,... s p d f g Magnetic quantum number ml = -l, -l+1, ... , l-1, l ml determines the orientation of an orbital.

3. Electron Shells • An electron shell is a group of orbitals with the same energy (same n). • A subshell contains orbitals with the same shape and energy (same n and l).

4. Shells and Subshells n l mlorbitalenergy 1 0 0 1s -RH 2 0 0 2s -RH/4 2 1 -1,0,1 2p -RH/4 3 0 0 3s -RH/9 3 1 -1,0,1 3p -RH/9 3 2 -2,-1,0,1,2 3d -RH/9

5. Orbital Representations Electron density plots show the probability of locating the electron.

6. Orbital Representations • A contour diagram is a surface that encloses most (say 90%) of the probability density. • The s orbitals are spherical:

7. Orbital Representations Electron density plots show the probability of locating the electron.

8. 2s 1s 3s Orbital Representations • A contour diagram is a surface that encloses most (say 90%) of the probability density. • The s orbitals are spherical:

9. p Orbitals • p orbitals (l = 1) have two lobes lying along the x, y, or z axis. • Rather than ml = -1,0,1, the orbitals are labelled px, py, and pz.

10. d and f Orbitals These orbitals have complicated shapes, but the electron density at the nucleus is always zero.

11. N Magnetic Field Direction S Magnetic Moment Electron Spin • In 1928, it was discovered that an electron has an intrinsic angular momentum, or spin. • In a magnetic field, the rotation axis has only two possible orientations. Spin quantum number: ms = ½

12. S Magnetic Field Direction N Magnetic Moment Electron Spin • In 1928, it was discovered that an electron has an intrinsic angular momentum, or spin. • In a magnetic field, the rotation axis has only two possible orientations. Spin quantum number: ms = -½

13. e e +Z 2 Z = - E R n 2 H n The Helium Atom Consider a two-electron atom with nuclear charge Z. He: Z = 2 Neglecting electron repulsion, each electron is in a hydrogen-like orbital with energy:

14. Effective Nuclear Charge • We account for electron repulsion by assum- ing that the electrons shield each other from the nuclear charge. • The net nuclear charge experienced by an electron is the effective nuclear charge, Zeff. • If S is the average number of screening electrons: Zeff = Z - S

15. 0 E 3s 2p 5p 2p 2s 5s 4s 4d 4p 3p 3d 4f 1s Electron Energies Due to screening, different subshells have different energies, increasing in the order: s < p < d < f

16. The Exclusion Principle • How many electrons can fit in, or “occupy” an orbital? • The Pauli Exclusion principle states: No two electrons in an atom can have the same four quantum numbers. • The ground state of helium has two electrons in the 1s orbital, but with opposite spins. n l ml ms electron 1 1 0 0 +½ electron 2 1 0 0 -½

17. [He] 2s 1s 2s Many-Electron Atoms • The Aufbau principle: Electrons are assigned, one at a time, to hydrogen orbitals with lowest possible energy. • An orbital diagram shows the number of electrons in each occupied orbital. or Li: • The electron configuration shows only the num- ber of electrons in each occupied subshell. Li: 1s2 2s1 or [He] 2s1

18. 1s 2s 2p 1s 2s 2p Hund’s Rule Hund’s rule: The lowest energy state has the most unpaired electrons. Carbon: Higher energy Lower energy Summary:

19. Electron Configurations and the Periodic Table • The electron configuration of an atom can be estimated from the Periodic table. • The actual configuration must be determined by experiment.

20. 4s 3d Electron Configurations and the Periodic Table Write electron configurations for: 1s2 2s2 2p6 3s2 3p1 13Al: [Ne] 3s2 3p1 [Ar] 4s2 3d6 [Ar] 26Fe: 50Sn: [Kr] 5s2 4d10 5p2 [Xe] 6s2 4f14 5d10 82Pb+2: 92U: [Rn] 7s2 6d1 5f3