Week 1 F

# Week 1 F - Par\$cle in a box  a  a  •

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Par\$cle in a box  a  a  •  The par\$cle in a box model (also known as the inﬁnite poten\$al well or the inﬁnite  square well) describes a par\$cle, which is free to move in a small space surrounded  by impenetrable barriers.   •  One dimension only and the poten\$al energy V(x) within the box is equal to 0.  •  This means that the par\$cle is completely trapped inside the box, no forces are  ac\$ng on the it inside the box.  Diﬀerence between classical and  quantum mechanics:  •  In classical system, the par\$cle is no more likely to be found at  one posi\$on than another.   •  However, when the well becomes very narrow (on the scale of  a few nanometers), quantum eﬀects become important.   •  The par\$cle may only occupy certain posi\$ve energy levels and  it can never have zero energy, meaning that the par\$cle can  never "sit s\$ll".   •  Addi\$onally, it is more likely to be found at certain posi\$ons  than at others, depending on its energy level. The par\$cle may  never be detected at certain posi\$ons, known as spa\$al nodes.  Par\$cle in a box  Total solu\$on   •  AMer normaliza\$on   The squared wave func\$ons are the probability  densi\$es, and they show the diﬀerence  between classical and quantum mechanical  behavior.  Classical mechanics predict that the electron  has equal probability of being at any point in  the box.  The wave nature of the electron gives it  extremes of high and low probability at  diﬀerent loca\$on in the box  •  The par\$cle‐in‐a‐box example shows how a wave  func\$on operates in one dimension.  •  The same methods used for the one‐dimensional box  can be expanded to three dimensions for atoms.  •  Mathema\$cally, atomic orbitals are discrete solu\$ons  of the three‐dimensional Schrodinger equa\$ons.  Polar Coordinates  Ψ consists of a radial component R(r) and an angular component A(θ,ϕ).  The hydrogen atom  •  The solu\$on of the Schrodinger equa\$on for the  hydrogen atom is a formidable mathema\$cal problem.  •  The electron in the hydrogen atom sees a spherically  symmetric poten\$al. The poten\$al energy is simply  that of a point charge.  V(r) =  •  The solu\$on is managed by separa\$ng the variables so  that the wave func\$on is represented by the product.  Quantum Numbers from Hydrogen Equations  •  The hydrogen atom solu\$on requires ﬁnding solu\$ons to the  separated equa\$ons which obey the constraints on the  wavefunc\$on.   •  The solu\$on to the radial equa\$on can exist only when a  constant which arises in the solu\$on is restricted to integer  values. This gives the principal quantum number:  Quantum numbers (summary)  Quantum numbers and their proper\$es  •  the quantum number n ( 1, 2, 3…)  –  primarily responsible for determining the overall energy of an atomic orbital.  •  The quantum number l (0, 1, 2, … n‐1)  –  determines the angular momentum and the shape of an orbital;   –  has a small eﬀect on the energy  •  The quantum number ml (‐l ≤ ml ≤ l)  –  determines the orienta\$on of the angular momentum vector in a magne\$c  ﬁeld; or the posi\$on of the orbital in space.  •  The quantum number ms ( +/‐ ½)  –  determines the orienta\$on of the electron’s magne\$c moment in a magne\$c  ﬁeld, either in the direc\$on of the ﬁeld (+1/2) or opposed to it (‐1/2).  •  The inner quantum number j (value of abs(l+s) to abs(l‐s))  •  Total angular momentum, spin‐orbit coupling.  When no magne\$c ﬁeld is present, all ml values (all  three p orbitals and all ﬁve d orbitals) have the same  energy (degenerate).   Together, the quantum numbers n, l, ml deﬁne an  atomic orbital, the quantum number ms describes the  electron spin within the orbital  Hydrogen 1s Radial Probability  The Bohr radius a0 = 52.9 pm   •  It is the value of r at the  maximum of Ψ2 for a  hydrogen 1s orbital  •  It is also the radius of a 1s  orbital according to the Bohr  model  •  The most probable radius is the ground state radius  obtained from the Bohr theory.   •  The Schrodinger equa\$on conﬁrms the ﬁrst Bohr radius as  the most probable radius but goes further to describe in  detail the proﬁle of probability for the electron.  Radial probabili\$es  Boundary surface for the angular part  A(θ,ϕ)  d‐Orbitals  Schrodinger Equa\$on and Bohr’s  model of the hydrogen atom  •  Besides providing informa\$on about the  wavefunc\$ons, solu\$ons of Schrodinger  Equa\$on give orbital energies, E.  •  Shows also the dependence of E on the  principal quantum number n for hydrogen‐like  species and Z the atomic number.  E = ‐kZ2/n2 with k = 1.312 x 103 kJ mol‐1  k = hcR with h=Planck’s constant, c:speed of light and R:  Rydberg constant  For each value of n there is only one energy soluCon ; all atomic  orbitals with the same n are degenerates (valid only for hydrogen‐like  orbitals H, He+, etc..)  NOT TRUE FOR MULTI‐ELECTRON ATOM  Shielding eﬀect / eﬀec\$ve nuclear charge  •  The eﬀec\$ve nuclear charge is the net posi\$ve  charge experienced by an electron in a mul\$‐ electron atom.   •  The term "eﬀec\$ve" is used because the shielding  eﬀect of nega\$vely charged electrons prevents  higher orbital electrons from experiencing the full  nuclear charge by the repelling eﬀect of inner‐ layer electrons.  •  Slater’s rule to es\$mate eﬀec\$ve nuclear charge  experienced by electrons.  Slater’s Rules  to evaluate the shielding eﬀect  Zeﬀ = Z – S  •  Electrons are grouped as follows: (1s), (2s, 2p), (3s, 3p), (3d), (4s,  4p), (4d), (4f), (5s, 5p)  •  Electrons in a group higher than electron considered contribute  nothing to the shielding  •  For an electron in ns or np orbital each of the other electrons in (ns,  np) group contributes S = 0.35; each of the electrons in n‐1 shell  contributes S = 0.85; each of the electrons in shell n‐2 or lower  contributes S = 1.00  •  These values are calculated from the electron probability curves of  the orbitals.  •  For an electron in nd or nf orbital, each of the electrons in the same  group contributes S = 0.35; each of the electrons in a lowergroup  contribute S = 1.00  Example of the Slater’s Rule  •  For K, Z=19. is it 4s1 or 3d1??  •  1s22s22p63s23p64s1  –  Zeﬀ = Z – S = 19 – (8*(0.85) + 10*1) = 2.20.  •   1s22s22p63s23p63d1  –  Zeﬀ = Z – S = 19 – (18*1) = 1.  Periodic Table of Elements  Electron Conﬁgura\$ons of Atoms  •  Electrons ﬁll energy levels star\$ng from the  sublevels with lowest energy.                                       1s                                       2s    2p                                       3s    3p   3d  4s   4p   4d   4f  5s   5p   5d   5f  6s   6p   6d   6f  7s   7p   7d   7f   8s   8p   8d    8f  Aubfau (building up) Principle  •  Is the principle of building up electronic ground state  conﬁgura\$ons used in conjunc\$on with Hund’s rules and  Pauli’s exclusion principle:  –  electrons are placed in orbitals to give the lowest total energy  to the atom (lowest values of n and l ﬁlled ﬁrst). Because the  orbitals within each set (p, d…) have the same energy, the  orders for values of ml and ms are indeterminate.  –  Pauli’s exclusion principle: No two electrons in the same atom  may have the same set of 4 quantum numbers  –  Hund’s rule: In a set of degenerate orbitals, electron may not be  spin paired in an orbital un\$l each orbital in a set contains one  electron; electrons singly occupying orbitals in a degenarate set  have parallel spins  Examples  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  [He]2s22p2     [Ne]3s1  [Ar]4s23d2  [Ar]4s13d5  [Kr]5s24d2  [Kr]5s04d10  [Xe]4f146s25d7  C       Na  Ti  Cr  Zr  Pd  Ir  For more prac\$ce check Table 1.3  Hund’s rule explana\$on  •  Πc : Coulombic energy of  repulsion favors electrons in  diﬀerent orbitals.  •  Πe : Exchange energy, this  energy depends on the  number of possible  exchanges between two  electrons with the same  energy and the same spin.   Example 12C 1s22s22p2  Exchange Energy  •  Exchange energy :  –  number of possible exchanges between two electrons  with the same energy and the same spin.   two possible ways  only one possible way  The higher the number of possible exchanges, the lower the energy  Total Pairing Energy, Π   •  Π = Πc + Πe  –  Πc is posi\$ve and is nearly constant for each pair of  electrons.  –  Πe is nega\$ve and is nearly constant for each pair of  electrons.  •  If there is a diﬀerence in energy between the  orbital levels involved, this diﬀerence in  combinaCon with the total pairing energy Π  determines the ﬁnal conﬁguraCon.  Valence and core electrons  •  Valence electrons = outer electrons.  •  Lower energy quantum levels electrons = core  electrons.  Trends in the Periodic Table  •  Many atomic proper\$es correlate with  electronic structure and so also with their  posi\$on in the periodic table  –  atomic size  –  ion size  –  ioniza\$on energy  –  electron aﬃnity  Atomic size  •  As the nuclear charge increases, the electrons are  pulled in toward the center of the atom, and the  size of any par\$cular orbital decreases.  •  On the other hand, as the nuclear charge  increases, more electrons are added to the atom,  and their mutual repulsion keeps the outer  orbitals large.  •  The interac\$on of the two eﬀects results in  gradual decrease in atomic size across a period.  Shielding eﬀect. Slater Rule.  Varia\$on in Size of Atoms  Rela\$ve Size of Select Ions and Their  Parent Atoms  ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online