Lecture 25 Quantum Mechanics of Atoms and Atomic Systems

1. Lecture 25Quantum Mechanics of Atoms and Atomic Systems Chapter 28.10  28.14 Outline • Quantum Mechanics and Periodic Table • Atomic Structure and Conductivity

2. Quantum Numbers Three quantum numbers determine the size and shape of the probability cloud of an atomic electron. n  the principal quantum number (0,1,2, … n) l  the orbital quantum number (0,1,2, …n 1) ml  the magnetic quantum number (l, … l) The 4th, spin quantum number (ms) determines the maximum number of electrons allowed on an orbit. ms = +1/2 or 1/2

4. Energy Level Diagram for Hydrogen

5. Electron Shells and Subshells All states with the same principal quantum number n form a shell. K-shell corresponds to n=1. States with a given n and l form a subshell. Subshell s has n = 1, l = 0; p n = 2, l = 1 The maximum number of electrons in a subshell is 2(2l+1). Examples: shell K (n=1) has 1 subshell s (l=0) and the maximum possible number of electrons 2 (ms=1/2). Shell L (n=2) has 2 subshells (s and p, l=0,1). Two electrons can be on s and 6 on p  total of 8 electrons.

7. Find the radius of the second orbit, ionization energy from the 2nd orbit, and transition wavelength between the 2nd and 3rd orbit in a singly ionized helium atom. Problems En =  13.6 Z2/n2 eV Atom: He+, Z = 2 rn = 0.0529 n2/Z nm Orbits: nf = 2, ni = 3 1/= RH Z2(1/nf2  1/ni2) Solution: r2 = 0.0529 22/2 nm = 0.1058 nm E2 = 13.6 22/22 eV = 13.6 eV 1/ = 1.097 107 22 (1/22  1/32) nm   = 164 nm

8. The Exclusion Principle In 1925 Wolfgang Pauli proposed that: No two electrons in an atom can have the same set of the four quantum numbers. This exclusion principle explains the electronic structure of complex atoms. The more electrons the atom has, the more levels is filled. Completely filled shells are called closed.

9. The Periodic Table Halogens are followed in atomic number by noble gases, then by alkali metals. This is suggestive of a periodical law in properties of chemical elements. The Periodic Law Elements (listed in order of atomic number) with similar chemical and physical properties appear at regular intervals. The periodic law was first formulated by Dimitri Mendeleev of Russia in 1869. Predictions: unknown elements Unexpected success: noble gases

10. Chemical Properties of Elements Inert (Noble) Gases: Outer shells are closed the electrons are tightly bound to the atom  chemically passive Hydrogen and Alkali Metals: Single outer electrons, which “see” only +e charge and are loosely bound to the nucleus,  lose the outermost electrons in chemical reactions, have similar behavior, and are chemically active

11. Chemical Properties of Elements Halogen Atoms: Outer shells lack 1 electron  tendency to pick up such an electron through the strong attraction of the poorly shielded nuclear charge. Metals and Nonmetals: Metals have 1 or several electrons outside the closed shells  combine chemically by losing these electrons to nonmetals Nonmetals lack 1 or few electrons in the outermost shell  combine by picking up electrons from metals or by sharing electrons with other nonmetals 3D Periodic Table

12. Energy Bands Closely spaced atoms become one quantum system. The electron energy levels of different atoms form continuous bands, separated by band gaps. Electrons may have any energy within bands, but not within band gaps. Electrons are distributed from the lowest energy levels up in agreement with the exclusion principle. In metals, the highest energy band is partially full. In insulators, it is completely full.

13. Summary The exclusion principle allows to determine the maximum possible number of electrons in each shell and subshell. Quantum mechanics explains chemical properties of the existing variety of elements and large atomic systems.