Lecture 32: The Periodic Table (source: What If by Randall Munroe) PHYS 2130: Modern Physics Prof. Ethan Neil (ethan.neil@colorado.edu)
Announcements Homework #9 assigned, due next Wed. at 5:00 PM as usual. Corrections to my last lecture: I told you oxygen was Z=10, but the example I was doing was actually neon (oxygen is Z=8.) Silver has Z=47, not 54 which I had written before - that would be xenon. (My excuse: last time I took chemistry was in high school, too.) 2
From last time, the chemistry rules of how electrons behave in multi-electron atoms: - Electrons fill up orbits in order of increasing energy - Orbital energy determined by principal number n, and spectroscopic label s,p,d,f - Number of orbits per spectroscopic label is 1,3,5,7 (2l+1) - Two electrons are allowed per orbit, spin up and spin down. Only one electron can ever be in the same orbit + spin. 3
Our more complicated example was silver (Ag, Z=47): 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 10 4p 6 5s 1 4d 10 https://ptable.com/#orbital This was what was used for the Stern-Gerlach experiments, remember. Why? Notice the lonely 5s electron 4
Chemistry is all about the outermost electrons in any given atom. Full orbits are very stable (e.g. noble gases don t react with much of anything.) Silver has a bunch of full orbits and a single electron in the 5s state. When we apply a magnetic field to it, it behaves just like a big, heavy single electron! More importantly, it acts like a neutral electron, so our magnetic field just sees the spin and not the charge. 5's m of 5
Spin is just another kind of quantum angular momentum: S ~ p = s(s + 1)~ S z = m s ~ ms between -s and s. For a single electron (or silver atom), we saw two possible values of Sz: +ħ/2 and -ħ/2. So we have total spin s=1/2 - the electron is a spin-1/2 particle. There are particles with every value of spin varying by half-integer increments. For example, Stern-Gerlach experiment with a spin-one object (molecular O2, perhaps) gives three dots: (+,0,-), or ms=1,0,-1. 6
Next question: why are we only allowed to have one electron (of each spin) per orbital? This is another example of particle-like behavior, in a sense. Classical particles take up space: two can t exist at the same place and time. For a quantum particle, we don t know exactly where it is; our knowledge is in terms of probability amplitudes - we can say it exists in a certain quantum state ψ>. The quantum version of saying particles take up space: two electrons are never allowed to exist in the same state at the same time. ( Pauli exclusion principle ) e - states: (n, l,m,ms) Total Energy 3 3p 3s 2p 2s This is why the orbitals fill up from the bottom; electrons fall to lowest energy they can, but they can never coexist in the same state. 1s 7
There is another class of quantum particles, known as bosons; they can overlap in the same state (arbitrarily many!) Bose-Einstein condensate - 2,000 ultracold Rubidium atoms, all in the same state at once! (2001 Nobel prize for Carl Wieman and Eric Cornell, right here at CU.) 8
Finally, the energy levels have this funny ordering. Can Schrödinger explain it? (source: https://chem.libretexts.org/) (note: there is some slight variation in this ordering for individual atoms - which is why silver had only a single 5s electron, instead of a pair and 9 4d electrons.) 9
Let s think about the electron orbitals for sodium (Na, Z=11). Na~1s 2 2s 2 2p 6 3s 1 For hydrogen, the 3s orbit is (on avg.) 9x further out than the 1s orbit (r~n 2 rb.) (source: http://timelineoftheatom.blogspot.com/p/erwin-schrodinger-1926.html) But sodium has more positive nuclear charge, which we know reduces the orbital radius by 1/Z. In the end, the 3s electron in sodium ends up a bit further out than 1s in hydrogen. The difference is due to screening : the 1s, 2s and 2p electrons shield the 3s electron from the nuclear charge. 10
CQ: For the lithium atom (Z=3, 1s 2 2s 1 ), unlike hydrogen, an electron in the 2p state will be less bound than the 2s state. Why? g 2s 1s 2p A. The 2p electron can jump between all three 2p orbitals, increasing its kinetic energy. P o elec elec B. The 2s electron orbit passes closer to the nucleus on average, lowering its energy. C. The 2p electron orbit passes closer to the nucleus on average, raising its energy. 11
n=3 n=2 Hydrogen (1p, 1e) l=0 (s) l=1 (p) l=2 (d) 3s 3p 3d 2s 2p 1s 2 2p 2s 2 Boron (5p, 5e s) NOT TO SCALE! 4s 3s 2s 4p 3p 2p 3d m=-1,0,1-8.3 ev n=1 1s -13.6 ev l=0,m=0 Energy only depends on n ENERGY Splitting of s and p energy levels (shielding) Energy depends on n and l 1s -340.2 ev 12
Summary: higher Z atoms have deep inner bound states (energy 1s ~ Z 2 ), but outer electrons are pushed out due to screening by inner ones. Screening is weaker for s >p >d >f orbitals - higher l spend less time near the nucleus! (source: https://chem.libretexts.org/) 13
Easiest atoms to ionize have single outermost s-shell electron: Li, Na, K these are also the biggest atoms in a given row. (p-wave etc. isn t as good at screening, and full shells are more stable.) 14
Electrons in this column have the same filling of the outer-most electron shell! Common shapes, similar energies > similar chemical properties. l=0 (s-orbitals) l=1 (p-orbitals) l=2 (d-orbitals) Valence (n) l=2 (f-orbitals) 3 Going down the column, atoms become less stable, because there s more screening of the nuclear charge/electron orbits get further out. (So all the radioactive and unstable things live towards the bottom of the table.)
Chemical reactions simply boil down to forming and breaking electron bonds between different atoms. A bond is just a bound state of the electron(s) with lower energy than being around a single atom. There are different types of bonds, which are characterized by how much the electron is shared between the atoms involved: Degree of sharing of electron Ionic Covalent Metallic electron completely electron equally shared electron shared transferred from one between two adjacent between all atoms atom to the other atoms in solid Li + F - H 2 Solid Lead 16
a- Ei rat Ionic bonds occur when an electron is completely removed from one atom and taken by the other. NaCl (salt) is a common example. Na is easy to ionize, and Cl is just one electron away from a full shell. Bond due to Coulomb attraction; minimum-energy distance found by balancing against repulsion if the atoms get too close. Energy Cl - V(r) Repulsion when atoms overlap Separation of ions Na + Coulomb attraction 17
Covalent bonds share one or more electrons in a common orbit around both atoms. Hydrogen gas (H2) is a common example. In terms of QM, we know shared electron = wavefunction extends across both atoms. A bond forms when the shared wavefunction has lower energy than both atoms being isolated. Ea shared g e- to +0 antinbonding e- cloud +0 %+ bonding 18
Quantum Bound States PhET: https://phet.colorado.edu/en/simulation/legacy/bound-states Lower-energy state is the bonding orbital, with higher probability of electron appearing in between the wells. 19
Complete answer comes from solving Schrödinger! But we can argue on general grounds about the shapes of the lowest-energy bonding orbitals. Ψ + puts electron density between protons.. glues together protons. Ψ - no electron density between protons protons repel (not stable) Bonding Orbital Antibonding Orbital 20
Those two sorts of bonds are more important for chemistry. The most interesting kind of bond to a physicist is a metallic bond.. @ # 5. %.. Now we have not two, but lots of atoms involved, all at once. Electrons are shared across all of them. What happens to its available energy levels? 21
22 Increase number of wells, get bands of energy levels close to each other! (Different ways of joining together n=1, n=2, n=3 solutions from each well.)
In solid, `10 22 atoms/cm 3, many!! electrons, and levels countless levels smeared together, individual levels indistinguishable. "bands" of levels. Each level filled with 2 electrons until run out. empty conduction empty band band gap ~ few ev 3 filled with electrons valence band 23 Energy atom level 2 1 more atoms filled with electrons bands (credit to Prof. Finkelstein for this visualization and the next few.) 41