Topic 1. Atomic Structure and Periodic Properties

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1 Topic Atomic Structure and Periodic Properties Atomic Structure 1-2 History Rutherford s experiments Bohr model > Interpretation of hydrogen atom spectra Wave - particle duality Wave mechanics Heisenberg s uncertainty principle Electron density and orbitals The Schrödinger equation and its solutions Electron spin, the Pauli principle, Hund s rule Aufbau principle Effective nuclear charge, shielding and penetration Structure of the periodic table

2 History 1-3 Democritus ( BC): atoms Lucretius: (94-55 BC): atoms assembled Aristoteles ( ) BC: matter and essence Robert Boyle: ( ) revived Democritus ideas John Dalton s experiments (1808): showed that matter consist of elementary particles (=atoms), which, combined in fixed relative portions, form molecules John Dalton 1-4 We might as well attempt to introduce a new planet into the solar system, or to annihilate one already in existence, as to create or destroy a particle of hydrogen. John Dalton (A New System of Chemical Philosophy, 1808)

3 Rutherford s experiment (1906) 1-5 Atoms must consist of a small, but very massive, positively charged nucleus in order to explain the observed scattering of α-particles on gold atoms. The Bohr Model (1913) 1-6 First model that could account for the spectra of atomic hydrogen centrifugal force = coulombic attraction - m e + m n Energy = kinetic energy + potential energy

4 Bohr introduced quantization 1-7 Atomic spectra of hydrogen is not continuous but consists of discrete lines > Bohr suggested that the electron can adopt only certain distances r (orbits) where k is a constant (Bohr radius = 52.9 pm, also a 0 ), and n is any integer = QUANTUM NUMBER of the orbit Each allowed orbit corresponds to a different energy level: E n 4 2 me Z k' = = ε hn n 0 More mathematical background from centrifugal force = coulombic attraction quantizing of the angular momentum: Energy = kinetic energy + potential energy 4 0 πε rmv mvr nh = 2π = Ze E = mv 2 4πε 0 Ze r 2 With these three equations, the radius, the energy and the velocity of the electron of the H atom with quantum number n (and nuclear charge Z) can be calculated: m = x kg e = x C h = x Js ε 0 = x Fm 1 1 ev = x J 1 J = x cm 1

5 1-9 Energy for hydrogenlike atoms: R = Rydberg constant ( ev) Z = nuclear charge Energy levels 1-10 Each orbit corresponds to a specific energy level The allowed energies are often displayed in a energy level diagram: Note: The energy levels are negative numbers and indicate the energy of an electron in the corresponding orbit: This is the energy required to remove this electron from the orbit (=Ionization energy). The principle quantum number n determines this energy value.

6 Emission and Absorption Spectra 1-11 Atoms are excited either via electrical discharge (A) or with a white light source After passing through a prism the absorbed energy appears as discrete lines (A) Excited sample Increasing wavelength Emission spectrum (B) Sample Prism Increasing wavelength Absorption spectrum Prism Emission Spectrum of Hydrogen 1-12 Transition between energy levels R H = ev or cm 1

7 Quiz: Ionization Energy 1-13 What line in the hydrogen spectrum (Balmer series) corresponds to the ionization energy of the electron with principal quantum number 2?? A B Conclusions 1-14 Ionization energy is proportional to Z 2 (nuclear charge) Radius of hydrogen atom in ground state (n=1) is 52.9 pm (= Bohr radius) The radius is inversely proportional to Z The excited state radius is proportional to n 2 In the ground state the electron has a velocity of v = cms 1 Bohr s model allows accurate prediction of the hydrogen atom spectrum BUT fails to describe atoms or ions with more than ONE electron => New theory required, which is not based on classical mechanics

8 Electron: Particle or Wave? 1-15 Depending on the experimental conditions, electrons appear either as particles, or show properties usually associated with waves Electrons are diffracted by crystalline materials, just as observed with X-rays De Broglie relationship: h: Planck s constant m e v: momentum of electron The photoelectric effect revealed a linear relationship between the kinetic energy of the photon and frequency: 1 2 E = mev = h( ν ν0) E = hν From Orbits to Orbitals: The Uncertainty Principle Werner Heisenberg (1927): It is not possible to determine simultaneously the position and momentum of an electron with good precision: But: the probability of finding an electron at a particular point can be calculated > this probability distribution is called an ORBITAL rather than an orbit.

9 Representation of Orbitals 1-17 The Schrödinger Equation 1-18 The probability distribution and energy levels for electrons in atoms and molecules can be calculated using the Schrödinger equation: HΨ = EΨ H: Hamilton Operator (Energy) E: Energy of solution Ψ Ψ: wavefunction Each solution Ψ of the equation corresponds to a different electron probability distribution with a distinct energy E The probability of finding an electron at some point is proportional to ΨΨ* (Ψ* is the complex conjugate of Ψ)

10 The Wavefunction 1-19 Atomic wavefunctions are expressed in polar coordinates: > value of Ψ at any given point in space is specified by r, θ and φ The wave function Ψ can be written as: Ψ( r, θφ, ) = R() r Y(, θφ) Quantum Numbers 1-20 Each of the solutions of the Schrödinger equation is labeled by a set of quantum numbers n, l, and m l = principle quantum number = angular momentum quantum number = magnetic quantum number Hydrogen atom: n determines the energy and size of the orbital > can be any integer l determines the shape of the orbital > any integer between (n 1) and 0 m l determines the orientation of the orbital > any integer between + l and l

11 1-21 The names s, p, d, and f is historical and connected with the appearance of the atomic spectroscopy lines: The Radial Distribution Function 1-22 The radial part R(r) of a wavefunction ψ(r) is related to the probability [ψ(r)ψ(r)* or ψ 2 ] of finding an electron at a specific point with distance r from the nucleus The probability of finding an electron at a given distance r from the nucleus is described by the radial probability density function:

12 Radial Distribution Function 1-23 ψ 1s = 1 1 r/ a0 e π a or using a 0 as unit of length (a 0 =1): ψ 1s = 1 e π r Note: the maximum probability is found at r = a 0 2s and 3s Orbitals 1-24 The s orbitals with n > 1 have radial nodes (change of function sign): ψ 2s 1 = ( 2 re ) 4 2π r / 2 ψ 3s 1 = ( 27 18r+ 2r ) e 81 3π 2 r / 3

13 Hydrogen p Orbitals 1-25 Unlike s orbitals, p orbitals are not spherically symmetric > they are directional and have an angular node: p x, p z and p y Orbitals 1-26 According to their orientation in space, the three p orbitals are denoted p x, p y and p z However, the solutions for m l = +1 and 1 with exponential imaginary functions. > these can be converted to real functions using their linear combinations: 1 ψ2p = ( ψ2p + ψ x 2p ) ψ2p = ( ψ2p ψ y 2p ) 2 1 1

14 Orbital Shading and Wavefunction Sign 1-27 Wavefunctions are signed quantities The sign is important when we consider bonding in molecules The wavefunctions sign is often indicated graphically by shading: Note: The orbital represents a probability function [ψ(r)ψ(r)* or ψ 2 ], which sign is always positive. Only ψ(r) or is a signed quantity! The Five d Orbitals 1-28 As for the p orbitals, only one d orbital (3d z 2) corresponds directly to the value of m l (= 0). The wavefunction solutions with m l = ±1 and ±2 are exponential imaginary functions. A real wavefunction is again obtained by linear combinations of these solutions.

15 Hydrogen Atom Wavefunctions: Angular Factors Hydrogen Atom Wavefunctions: Radial Factors

16 Multielectron Atoms 1-31 The Schrödinger equation can be solved accurately only for H, He +, Li 2+ etc. > He corresponds to the classic three body problem in physics and can be solved only through approximation (Hartree-Fock method or self consistent field (SCF) method Results from these calculations: Compared to the H atom, the orbitals are somewhat contracted due to the increased nuclear charge Within a given major energy level (quantum number n), the energy increases with s < p < d < f But: At higher energy levels staggering may occur: Electron Spin 1-32 Beam of hydrogen atoms (ground state, n =1, l = 0, m l = 0) is split when sent through a magnetic field N Atomic H source S > introduction of a fourth quantum number, m s explains experiment Note: the spin quantum number comes from relativistic effects (not included in the Schrödinger equation)

17 The Pauli Principle 1-33 In a given atom, no two electrons may have all four quantum numbers n, l, m l, and m s identical > each orbital can contain maximal two electrons with Diamagnetic Atom: total spin S = 0 (all electrons are paired) Paramagnetic Atom: total spin S 0 (one or more unpaired electrons) The Aufbau Principle (Filling Principle) 1-34 energy Based on the Pauli principle, the distribution of electrons among the atom orbitals can be determined (= electron configuration) The electrons fill up the available energy levels (orbitals), starting with the lowest available level

18 Orbital Energies 1-35 The energy of orbitals in a multielectron atom is not identical with the corresponding orbitals of the hydrogen atom > the electrons experience an effective nuclear charge (Z eff ) that is different from 1.0. Each electron is attracted by the nucleus and repelled by all the other electrons Z+ attraction Zeff = Z σ attraction Shielding s electrons shield the third 2s electron from the full effect of the nuclear charge The shielding constant is dependent on which orbitals are occupied: Radial distribution function for Li

19 Orbital Penetration 1-37 The 4s orbital shows a considerable electron probability with maxima close to the nucleus > it is therefore more penetrating than the 3d orbital, even though the principle quantum number is larger P 2 r Variation of Orbital Energy with Z 1-38 Orbitals energies do not always vary very smoothly with atomic numbers > depends on what orbitals are occupied Note: They vary with ion charge, the plot applies only to neutral atoms!

20 Hund s Rule 1-39 There is often more than one way of arranging electrons within a set of degenerate (=energetically equivalent) orbitals > they correspond often to different energies Hund s rule of maximum multiplicity: The ground state of an atom will be the one having the greatest multiplicity Electron configuration for atoms 1-40 Some atoms do not have the electron configuration you would expect based on the Aufbau principle: Half filled and filled subshells (here the five 3d orbitals) provide additional stabilization [Ar] 4s 1 3d 10 is the lower energy configuration compared to 4s 2 3d 9

21 Electron Configuration for Ions 1-41 Ease of removal of an electron (=Ionization energy) often does not mirror the filling order embodied in the Aufbau principle Mn [Ar] 4s 2 3d 5 Mn 2+ [Ar] 4s 0 3d 5 and not [Ar] 4s 2 3d 3 Different electron configurations correspond to different species, and have different properties (color, magnetic behavior, etc.) Note: All transition metal atoms loose their ns electrons before their (n 1)d electrons! Periodic Trends in Atomic Properties 1-42 Effective nuclear charge Ionization energies Electron affinities Covalent and ionic radii Bond strength Electronegativity Orbital energies Promotion energies Common oxidation states Relativistic effects

22 Effective Nuclear Charge 1-43 The valence shell electrons feel a nuclear charge much less due to shielding effects of the core electrons > Many atomic properties can be rationalized by understanding how the effective nuclear charge varies throughout the periodic table Slater s rules allow estimation of Z eff by simple empirical rules Slater s Rules 1-44 [1s] [2s2p] [3s3p] [3d] [4s4p] [4d] [4f] [5s5p] [5d] [5f] All electrons to the right contribute 0 to shielding All electrons in the same group (n) contribute 0.35 For s and p electrons: all electrons in the (n 1) shell contribute 0.85 each All other electrons to the left contribute 1.0 For [nd] or [nf] electrons: all electrons to the left contribute 1.0

23 1-45 Periodic Variation of Z eff Ionization Energies 1-46 Energy required to remove an electron from an atom or ion 1st ionization energy or potential (IE or IP) 2nd ionization energy or potential Depends on the effective nuclear charge experienced by the electron and the average distance from the nucleus > with increasing Z eff increases IP > with increasing distance decreases IP Note: Distance increases as principle quantum number increases (n)

24 Periodic Trends in Ionization Energies 1-47 half filled filled shell IPs do not uniformly increase from left to right > changing orbital and spin pairing breaks trend IPs do not always decrease going down a group > transition series and actinides upset this trend Electron Affinity 1-48 Energy change associated with addition of an electron to an atom or ion 1st electron affinity (EA) Favorable process for most elements Influenced by Z eff and atom size (principle quantum number) Note: Positive sign per definition: F + e > F EA = + 328kJmol 1 H = 328kJmol 1 Trends in EAs parallel those of IPs, but for one lower Z value Exceptions: EA of F is lower than of Cl EA of N is lower than of P EA of O is lower than of S > smaller size of F (and N or O) causes greater electron-electron repulsion!

25 Trends in Electron Affinities 1-49 Covalent Radii 1-50 Covalent radius of an atom A = half the distance between a diatomic molecule A A They are approximately additive: For a non-polar molecule A B the sum of the radii of A and B should be equal to the A B bond length Note: For polar molecules this approximation does not work as well!

26 Ionic Radii 1-51 The sizes of ions follow similar trends BUT: Changes in charge have a very big impact on size Bond Strength 1-52 The bond energy is the energy needed to break a chemical bond Strong bond: > 800 kjmol 1 Average bond: 500 kjmol 1 Weak bond: < 200 kjmol 1 Bond strength often depend upon the size of the elements that are bonded together > bond strength often decreases on going down a group Example: HF 568 kjmol 1, HCl 432 kjmol 1, HBr 366 kjmol 1, HI 298 kjmol 1

27 Electronegativity (EN) 1-53 Pauling (1930): Electronegativity = the ability of an atom to attract electron density towards itself in a molecule Not amenable to direct experimental measurement But: very useful concept which allows to predict, whether a given combination of elements is likely to result a molecule with polar bonds Various quantifications of EN: Pauling: based on bond-strength Alfred-Rochow: based on size and effective nuclear charge Mulliken: based on IPs and EAs Allen s spectroscopic values: based on orbital energies Pauling Scale of ENs 1-54 Based on differences in bond strength: > For a polar molecule A B the strength of the A B bond is greater than the average of the strength of A A and B B (due to an ionic contribution to the bonding) > This difference in bond strength,, was related to the difference in electronegativity using the expression: = ( χ χ ) 2 A B ( in kjmol 1 ) χ E is the electronegativity of element E

28 Mulliken Electronegativity 1-55 Defined as the average of the ionization energy and electron affinity: χ = IE + EA 2 > Can easily calculated from tabulated values Periodic Trends in Electronegativity 1-56

29 Orbital Energies 1-57 Note: Separation between s and p orbitals increases when going from left to right Promotion Energy 1-58 Many atoms form more bonds than they are expected based on the number of unpaired electrons in their ground states Carbon Atom: 1s 2 2s 2 2p 2 > has only 2 unpaired electrons, but forms 4 covalent bonds Prior to bonding interaction, one of the 2s electrons is promoted to the empty 2p orbital (=promotion energy): 1s 2 2s 1 2p 3 > 4 unpaired electrons can form 4 bonds Note: Promotion energies increase when going down the B or C group

30 Relativistic Effects 1-59 Important for heavy metal elements: Einstein s special relativity theory: Objects moving close to the speed of light increase in mass Due to the high nuclear charge of heavy elements electrons close to the nucleus (s orbitals!) have a big velocity > mass of electron increases > effective size of orbital decreases (relativistic orbital contraction) > energy of of electron is lowered The contraction of the s orbitals (and somewhat also the p orbitals) leads to an expansion of the d and f orbitals due to increased shielding of the nuclear charge

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