The fourth quantum number: Spin

1. The fourth quantum number: Spin • Solving the Schrödinger equation gives 3 quantum number n, l, and ml • A fourth quantum number (spin quantum number) was discovered from experimental observation to account for the emission spectra of atoms (Goudsmit and Uhlenbeck) explaining splitting of lines e.g., doublet yellow line of Na: predict at 590 nm, but observed as 2 lines at 589.59 and 588.99 nm

2. Magnet Pole Face N H-atom beam S Magnet Pole Face

3. Electron Spin Quantum Number The Electron is given one of two spin Quantum numbers: ms = 1/2 or ms =-1/2 ms = 1/2ms = -1/2

4. A spin tells how a particle looks like from different directions Spin = ½ means the particle has to be turned through 2 revolutions to look the same! (particles with spin ½ make up matter in universe) Hawking, a brief history of time Chapter 5 (posted on Moodle)

5. The Pauli Exclusion Principle • In a given atom, no 2 electrons can have the same set of four quantum numbers (n,l,ml,ms) Since electrons in the same orbital have the same values of n, l, and ml; the postulate implies that they should have different mS Only 2 electrons of opposite spin states can be in an orbital n,l,ml

6. Only 2 electrons of opposite spins can be in an orbital n,l,ml An electron in an Orbital implies a certain n, l, ml Example 2pz is a wavefunction with n = 2, l = 1, ml = 0 2 electrons in orbital 2pz (n, l, ml, ms) (2,1,0, 1/2) and (2,l,0, -1/2) More than 2 es: IMPOSSIBLE

7. Polyelectronic Atoms Consider He: 2 protons and 2 es e-Correlation Problem Electron 1 e-e Repulsion 2p in nucleus Electron 2 Attraction • 3 Energy Contributions: • Kinetic energy of the electrons • Potential energy of attraction between nucleus and electron • Potential energy of repulsion between the 2 electrons Schrödinger Equation cannot be solved exactly because the e pathways are unknown---called electron correlation problem

8. Approximation e Z* Z • The electron is treated as if it were moving in a field of charge that is the net result of nuclear attraction and average repulsion of all other es. • The result is that electrons are not bound as strongly to the nucleus • The nucleus is “screened” or “shielded” by the presence of other electrons – This is known as “screening” or “shielding” effect • It is as though the nucleus has a charge Z* that is smaller than its actual charge Z

9. Orbitals of polyelectronic atoms • Hydrogen-like orbitals are obtained, have same general shape as H orbitals but with different sizes and energies • The orbital energy is a function of n and l for polyelectronic atoms E = f (n,l) EnS< Enp< End<Enf s electrons penetrate to the nucleus more than p more than d

10. (a) The radial probability distribution for an electron in a 3s orbital. (b) The radial probability distribution for the 3s, 3p, and 3d orbitals. 3s node node

11. The order of energy for orbitals in the first three levels of polyelectronic atoms.

12. The Periodic Table (independently by Mendeleev and Meyer) Dimitri Mendeleev (1834-1907) In 1869, a Russian chemist named Dmitri Mendeleev came up with a way of organizing the elements that were known at the time. He set them out in order of atomic weight, and then grouped them into rows and columns based on their chemical and physical properties. He predicted the existence and properties of yet unknown elements

13. Mendeleev had no idea what atoms were made of or why they behaved as they did. Nevertheless, he was able to put together the periodic table almost as we know it today--except that some elements were missing, because they were unknown in 1869. Also the current Table is arranged in order of atomic numbers • Based on the gaps in his table, Mendeleev even succeeded in predicting the existence and properties of several new elements. • His basic rule was the following: the elements in any column, or group, of the table are similar to their column-mates. For example, look at the first column on the left, underneath hydrogen (H). The elements in this group are called the alkali metals; they're all soft metals that react violently with water to make hydrogen gas.

14. The Elements Lithium (Li), Sodium (Na), and Potassium (K) all formed oxides in the ratio of two atoms per oxygen atom: R2O The Elements Beryllium (Be), Magnesium (Mg), and Calcium (Ca) all formed oxides in the ratio of one atom per oxygen atom: RO Boron (B) and Aluminum (Al) formed R2O3 Carbon (C) and Silicon (Si) formed RO2 Recognizing the patterns of combining ratios or "valency", Mendeleev created a table organized by placing elements with similar combining ratios in the same group.   He arranged the elements within a group in order of their atomic mass. 

15. As protons are added one by one to the nucleus to build up the elements, electrons are similarly added to the hydrogen-like orbitals. The Aufbau (buildup) Principle 23 Na 11 A, atomic mass Z, atomic number

16. Filling Order Electron Configurations 1s 2s 2p 3s 3p 3d 4s 4p 4d 4f 5s 5p 5d 5f 6s 6p 6d 6f 7s 7p 7d 7f

17. Hund’s Rule • The lowest energy configuration (Ground State Electron Configuration) for an atom is the one having the maximum number of unpaired electrons allowed by the Pauli Exclusion principle in a particular set of degenerate orbitals. Wrong, Defies Pauli Principle Correct-Ground State This is not Ground State