P.A. Cox - Inorganic chemistry (793955), страница 4
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The average radius <r> of an orbital is(2)10SECTION A—ATOMIC STRUCTUREwhere a0 is the Bohr radius (59 pm), the average radius of a 1s orbital in hydrogen. Thus electron distributions arepulled in towards the nucleus by the increased electrostatic attraction with higher Z. The energy (see Equation 1) is(3)The factor Z2 arises because the electron-nuclear attraction at a given distance has increased by Z, and the averagedistance has also decreased by Z. Thus the ionization energy of He+ (Z=2) is four times that of H, and that of Li2+(Z=3) nine times.Section A—Atomic structureA3MANY-ELECTRON ATOMSKey NotesThe orbitalapproximationElectron spinPauli exclusionprincipleEffective nuclear chargeScreening andpenetrationHund’s first ruleRelated topicsPutting electrons into orbitals similar to those in the hydrogen atomgives a useful way of approximating the wavefunction of a manyelectron atom.
The electron configuration specifies the occupancy oforbitals, each of which has an associated energy.Electrons have an intrinsic rotation called spin, which may point inonly two possible directions, specified by a quantum number ms. Twoelectrons in the same orbital with opposite spin are paired. Unpairedelectrons give rise to paramagnetism.When the spin quantum number ms is included, no two electrons inan atom may have the same set of quantum numbers. Thus amaximum of two electrons can occupy any orbital.The electrostatic repulsion between electrons weakens their bindingin an atom; this is known as screening or shielding.
The combinedeffect of attraction to the nucleus and repulsion from other electronsis incorporated into an effective nuclear charge.An orbital is screened more effectively if its radial distribution doesnot penetrate those of other electrons. For a given n, s orbitals areleast screened and have the lowest energy; p, d,…orbitals havesuccessively higher energy.When filling orbitals with l>0, the lowest energy state is formed byputting electrons so far as possible in orbitals with different m values,and with parallel spin.Atomic orbitals (A2)Molecular orbitals:homonuclear diatomics (C4)The orbital approximationSchrödinger’s equation cannot be solved exactly for any atom with more than one electron.
Numerical solutions usingcomputers can be performed to a high degree of accuracy, and these show that the equation does work, at least for fairlylight atoms where relativistic effects are negligible (see Topic A5). For most purposes it is an adequate approximation torepresent the wavefunction of each electron by an atomic orbital similar to the solutions for the hydrogen atom.
Thelimitation of the orbital approximation is that electron repulsion is included only approximately and the way inwhich electrons move to avoid each other, known as electron correlation, is neglected.12SECTION A—ATOMIC STRUCTUREA state of an atom is represented by an electron configuration showing which orbitals are occupied by electrons.The ground state of hydrogen is written (1s)1 with one electron in the 1s orbital; two excited states are (2s)1 and (2p)1.For helium with two electrons, the ground state is (1s)2; (1s)1(2s)1 and (1s)1(2p)1 are excited states.The energy required to excite or remove one electron is conveniently represented by an orbital energy, normallywritten with the Greek letter ε.
The same convention is used as in hydrogen (see Topic A2), with zero being taken asthe ionization limit, the energy of an electron removed from the atom. Thus energies of bound orbitals are negative.The ionization energy required to remove an electron from an orbital with energy ε1 is thenwhich is commonly known as Koopmans’ theorem, although it is better called Koopmans’ approximation, as itdepends on the limitations of the orbital approximation.Electron spinIn addition to the quantum numbers n, l and m, which label its orbital, an electron is given an additional quantumnumber relating to an intrinsic property called spin, which is associated with an angular momentum about its own axis,and a magnetic moment.
The rotation of planets about their axes is sometimes used as an analogy, but this can bemisleading as spin is an essentially quantum phenomenon, which cannot be explained by classical physics. The directionof spin of an electron can take one of only two possible values, represented by the quantum number ms, which canhave the values +1/2 and −1/2.
Often these two states are called spin-up and spin-down or denoted by the Greekletters α and β.Electrons in the same orbital with different ms values are said to be paired. Electrons with the same ms value haveparallel spin. Atoms, molecules and solids with unpaired electrons are attracted into a magnetic field, a propertyknow as paramagnetism. The magnetic effects of paired electrons cancel out, and substances with no unpairedelectrons are weakly diamagnetic, being repelled by magnetic fields.Experimental evidence for spin comes from an analysis of atomic line spectra, which show that states with orbitalangular momentum (l>0) are split into two levels by a magnetic interaction known as spin-orbit coupling.
It occursin hydrogen but is very small there; spin-orbit coupling increases with nuclear charge (Z) approximately as Z4 and sobecomes more significant in heavy atoms. Dirac’s equation, which incorporates the effects of relativity into quantumtheory, provides a theoretical interpretation.Pauli exclusion principleElectron configurations are governed by a limitation known as the Pauli exclusion principle:• no two electrons can have the same values for all four quantum numbers n, l, m and ms.An alternative statement is• a maximum of two electrons is possible in any orbital.Thus the three-electron lithium atom cannot have the electron configuration (1s)3; the ground state is (1s)2(2s)1. Whenp, d,…orbitals are occupied it is important to remember that 3, 5,…m values are possible.
A set of p orbitals with any ncan be occupied by a maximum of six electrons, and a set of d orbitals by 10.A3—MANY-ELECTRON ATOMS13Effective nuclear chargeThe electrostatic repulsion between negatively charged electrons has a large influence on the energies of orbitals.
Thusthe ionization energy of a neutral helium atom (two electrons) is 24.58 eV compared with 54.40 eV for that of He+(one electron). The effect of repulsion is described as screening or shielding. The combined effect of attraction tothe nucleus and repulsion from other electrons gives an effective nuclear charge Zeff, which is less than that (Z) ofthe ‘bare’ nucleus. One quantitative definition is from the orbital energy ε using the equation (cf.
Equation 3,Topic A2):where n is the principal quantum number and R the Rydberg constant. For example, applying this equation to He (n=1)gives Zeff=1.34.The difference between the ‘bare’ and the effective nuclear charge is the screening constant σ:For example, σ=0.66 in He, showing that the effect of repulsion from one electron on another has an effect equivalentto reducing the nuclear charge by 0.66 units.Screening and penetrationThe relative screening effect on different orbitals can be understood by looking at their radial probability distributions(see Topic A2, Fig. 2).
Consider a lithium atom with two electrons in the lowest-energy 1s orbital. Which is the lowestenergy orbital available for the third electron? In hydrogen the orbitals 2s and 2p are degenerate, that is, they have thesame energy. But their radial distributions are different. An electron in 2p will nearly always be outside the distributionof the 1s electrons, and will be well screened. The 2s radial distribution has more likelihood of penetrating the 1sdistribution, and screening will not be so effective. Thus in lithium (and in all many-electron atoms) an electron has ahigher effective nuclear charge, and so lower energy, in 2s than in 2p.
The ground-state electron configuration for Li is(1s)2(2s)1, and the alternative (1s)2(2p)1 is an excited state, found by spectroscopy to be 1.9 eV higher.In a similar way with n=3, the 3s orbital has most penetration of any other occupied orbitals, 3d the least. Thus theenergy order in any many-electron atom is 3s<3p<3d.Hund’s first ruleFor a given n and l the screening effect is identical for different m values, and so these orbitals remain degenerate inmany electron atoms. In the ground state of boron (1s)2(2s)2(2p)1 any one of the three m values (−1, 0, +1) for the p electronhas the same energy. But in carbon (1s)2(2s)2(2p)2 the different alternative ways of placing two electrons in the three 2porbitals do not have the same energy, as the electrons may repel each other to different extents.
Putting two electronsin an orbital with the same m incurs more repulsion than having different m values. In the latter case, the exclusion principlemakes no restriction on the spin direction (ms values), but it is found that there is less repulsion if the electrons haveparallel spin (same ms). This is summarized in Hund’s first rule:• when electrons are placed in a set of degenerate orbitals, the ground state has as many electrons as possible indifferent orbitals, and with parallel spin.14SECTION A—ATOMIC STRUCTUREThe mathematical formulation of many-electron wavefunctions accounts for the rule by showing that electrons withparallel spin tend to avoid each other in a way that cannot be explained classically.
The reduction of electron repulsionthat results from this effect is called the exchange energy.Section A—Atomic structureA4THE PERIODIC TABLEKey NotesHistoryBuilding upBlock structureGroup numbers andnameRelated topicsThe periodic table—with elements arranged horizontally in periodsand vertically in groups according to their chemical similarity—wasdeveloped in an empirical way in the 19th century. A more rigorousfoundation came, first with the use of spectroscopy to determineatomic number, and, second with the development of the quantumtheory of atomic structure.The ‘aufbau’ or ‘building up’ principle gives a systematic method fordetermining the electron configurations of atoms and hence thestructure of the periodic table.
Elements in the same group have thesame configuration of outer electrons. The way different orbitals arefilled is controlled by their energies (and hence their differentscreening by other electrons) and by the Pauli exclusion principle.The table divides naturally into s, p, d and f blocks according to the outerelectron configurations, s and p blocks form the main groups, the dblock the transition elements, and the f block the lanthanides andactinides.Modern group numbering runs from 1 to 18, with the f blocks beingsubsumed into group 3.
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