109 LECTURE 5. PERIODIC TRENDS EXPLAINED BY EFFECTIVE NUCLEAR CHARGE Summary. The periodic table was created as a consequence of the boundary conditions imposed by the quantum mechanical solutions to Schrodinger’s wave equations for multi-electron systems. What we will learn in this lecture is that in defining the properties of atoms and ions in the table, there will turn out to be periodic trends to those properties. Specifically, we will see that as we go across the table from left to right, or down the table from top to bottom, we will see that a systematic increase or decrease in the quantitative measure of a property is observed. This is good for two reasons. First, allows us to cement into place the important ideas that explain the properties. And second, it keeps us from having to to much memorization since we can make nice sweeping generalizations. What are the trends that we will be able to explain? There are a lot of them. But here are six you will see over the next several lectures: • Atomic radius size • Ionic radius size • Ionization energy • Electron affinity • Metallic character • Electronegativity And what are the two big ideas that explain the trends for these six properties? They are the titles of the next two lectures: Rule 1: Effective nuclear charge (ENC) will explain the relative size and interest in electrons for atoms and ions. As will be shown, for example, as ENCØ Size × and as ENC× Size Ø. A similar trend can be defined for how much an ion or atom wants an electron. ENC arguments are the most important argument in explaining the overall trends in the periodic table. Rule 2: Filled and half filled shells have additional stability. As mentioned when drawing the electronic configurations of atoms, whenever a filled shell or half filled shell can be created, it will have a stability out of bounds with that expected by ENC. Filled shell stability, for example, is the root of the idea of the octet rule in explaining chemical bonds. We will also see it impact the fine structure in many of our trends, for example creating exceptions to the Aufbau principle and creating fine structure exceptions to the trends for ionization energy and electron affinity.110 The Multielectron Schrodinger Equation Recall that when we had a single e¯ and a single nucleus, +, there was a simple potential to stick in the Schrodinger Equation 1 e¯ case => V(r) = (-e)(+e) which yields the solution E= hR 4 Π Eo r h² But what happens with more than one e¯? Things quickly get more complicated with a repulsion term in addition to attraction. 2 e- case => V(r) = -2e² - 2e² + e² 4 Π Eo r1 4 Π Eo r2 4 Π Eo r12 × × × attraction of attraction of repulsion of e¯1, to nucleus e¯2, to nucleus e¯1 from e¯2 The key thing to note is that in addition to the simple attractions between protons in nucleus and each e¯, there are REPULSIONS between electrons. This repulsive effect result, called electron shielding, has a profound implication in that it keeps atom sizes from getting smaller and smaller as the number of protons increases. We would live in a very different world if an atom with 100 protons was smaller than an atom with one proton just because attractive forces ruled everything. Sheilding and Effective Nuclear Charge The calculation of V(r) is not some thing we can do to determine the relative extent of repulsion, so we simplify and make a nice freshman chemistry definition of shielding. A singly charged electron has just as much repulsive effective as a singly charged proton. So in the drawing below, the perimeter electron has all of its attraction to the single proton completely canceled and as a result, it has no attraction. It is completely shielded and is on its own.111 Al +13 Protonse¯e¯e¯e¯e¯e¯e¯e¯e¯e¯e¯e¯e¯Not shielded Shielded by 2 e¯ Shielded by 10 e¯ N=1N=2 N=3While this argument is certainly an oversimplification, it is remarkable useful in providing a semi-quantitative measure of shielding. The math equation below determines the effective (or actual) nuclear charge and as mentioned before, ENC explains all of the trends in the periodic table. So be able to do this calculation. ENC= effective nuclear charge= (# of protons in nucleus) – ( # of shielding inner shell electrons) ENC Calculations: An analogy to getting good seats at a concert A better understanding of ENC might come from: creating an analogy to attending a concert. The band is the protons and the people watching are the electrons. Who sees the band best (has the best effective nuclear charge?) The people in front see all of the show. And the people in back rows? They are are shielded and see less of the band (have lower effective nuclear charge.) Let’s imagine a 13 person band and think about ENC for the various rows at the show.Also note in this theatre that there are only 2 seats in the front row, 8 seats in the second row, and 10 seats in the third row (I see an analogy coming.) ENC Calculations when the actual nuclear charge= +13 First row, n=1 with no shielding: ENC= 13-0 =13 Second row, n = 2 with a shielding by the two electrons in the front row: ENC= 13-2 =11 Third row, n = 3 with a shielding by the ten electrons in the front two rows: ENC= 13-10 =3 13 e¯ show up112 s d p 2e¯n=18e¯n=23e¯n=3 13 e¯ The rows of the table are rows of seats N +7 protons e¯e¯e¯e¯e¯e¯e¯Large RadiusF +9 protonse¯e¯e¯e¯e¯e¯e¯e¯e¯Small Radius Note as move to the right, p+ goes up, but e¯ doesn’t shield (people in your row don’t shield you) so going to the right, ENC ×, smaller radius Unless you haven’t been paying attention, you see that I can straighten out the rows in the concert above and I am staring at the periodic table, and specifically looking what ENC is for the electrons attracted to the 13th element, Al. for electrons in the n=3 row, ENC= 13-10=3 *Note as shielding increases the ENC goes down Trends in ENC down and across the table We can see why it is that as we go down the periodic table, electrons are not getting closer. There are more protons, but there are an equivalent number of
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