Chapter 2. Atomic Structure
Chapter 2. Atomic Structure The theory of atomic and molecular structure depend on quantum mechanics to describe atoms and molecules in mathematical terms. Fortunately, it is possible to gain a practical understanding of the principles of atomic and molecular structure with only a moderate amount of mathematics rather than the mathematical sophistication involved in quantum mechanics. This chapter presents the fundamentals needed to explain atomic and molecular structures in qualitative or semiquantitative terms.
Chapter 2. Atomic Structure Finding of the subatomic particle: Balmer s Equation 원자의 subatomic particle 인전자가 E 를방출하거나 흡수할수있음. Generalized by Niels Bohr 하지만이러한이론은전자의 wave nature 때문에 H 외에는잘맞지않음. 3-D Schrodinger equation 의 solution 이바로 atomic orbitals Schroedinger equation 의 realistic solution 을위한조건들을적용한예 : Particle in a box (1-D) 그래서 wave property 를잘나타내는 equation 사용하기로함 : Schroedinger equation Heisenberg s Uncertainty principle Quantum number 로 AO 표현함. Ψ 표현하는두가지방법 : 1) Cartesian, 2) Spherical QN 의제한으로인해 aufbau principle 필요 : 1) Pauli s 2) Hund s If more than 1 e -, shielding effect: Z Ionization energy, electron affinity, covalent/ ionic radii.
2.1.1 The Periodic Table Many Chemists had considered the idea of arranging the elements into a periodic table. But, due to either insufficient data or incomplete classification scheme, it was not done until Mendeleev and Meyer s time. Using similarities in chemical behavior and atomic weight, Mendeleev arranged those families in rows and columns,,, and, he predicted the properties of unknown elements, such as Ga, Sc, Ge, Po.
2.1.1 The Periodic Table In the modern periodic table : : periods (horizontal row of elements) : group/family (vertical column) 3 different ways of designations of groups: : IUPAC, American, European 1) American: main group ІA ⅧA; TMs ⅢB ⅧB ⅡB 2) IUPAC: numbering from 1 through 18 for all group Fig2.1
2.1.2 Discovery of Subatomic Particles and the Bohr Atoms During the 50 years after the Mendeleev s periodic table was proposed, there had been experimental advances and discoveries as shown in Table 2.1.
2.1.2 Discovery of Subatomic Particles and the Bohr Atoms The discovery of atomic spectra showed that each elements emits light of specific energy when excited. : Balmer s equation (1885) energy of visible light emitted by H atom E 1 1 = RH - 2 nh 2 2 n h : integer, with n h >2 R H : Rydberg constant for hydrogen = 1.097 X 10 7 m -1 = 2.179 X 10-18 J ------------------------------------------------------------------------------------ * E is related to the wavelength, frequency, and wave number of the light!! hc E = hυ = = hcυ λ h = Planck constant = 6.626 X 10-34 Js υ = frequency of the light, in s -1 c = speed of light = 2.998 x 10 8 ms -1 λ = wavelength of the light, frequently in nm υ = wavenumber of the light, usually in cm -1
2.1.2 Discovery of Subatomic Particles and the Bohr Atoms Balmer s equation becomes more general by replacing 2 2 by n 2 l. (n l < n h ) Niels Bohr s quantum theory of the atom : - negative e - in atoms move in circular orbitals around positive nucleus. - e - may absorb or emit light of specific E 2 Z e R = π µ (4 πε ) h 2 2 4 0 2 2 E 1 1 = RH - 2 2 n l nh μ = reduced mass of the electron-nucleus combination * m e = mass of the electron m nucleus = mass of the nucleus Z = charge of the nucleus e = electronic charge h = Planck constant n h = quantum number of describing the higher energy state n l = quantum number of describing the lower energy state 4πε 0 = permittivity of a vacuum
2.1.2 Discovery of Subatomic Particles and the Bohr Atoms Electron transition among E levels for the hydrogen atom (Fig. 2.2) E release: as e - drops from n h to n l If correct E is absorbed: e - is raised from n l to n h According to Bohr s model and equation, E is inverse-squarely proportional to n. Thus, at small n large E gap at large n small E gap Exercise 2.1 Fig.2.2
2.1.2 Discovery of Subatomic Particles and the Bohr Atoms However, Bohr s theory works only for H and fails to atoms w/ more e - because of the wave nature of e -. de Broglie equation: all moving particles have wave properties, which can be expressed as shown below h λ = mυ λ = wavelength of the particle h = Planck constant = 6.626 X 10-34 Js m = mass of the particle υ = velocity of the particle e - s wave property is observable due to the very small mass.(1/1836 of the H atom) But, we can not describe the motion of e - w/ the wave property precisely because of Heisenberg s uncertainty principle.
2.1.2 Discovery of Subatomic Particles and the Bohr Atoms Heisenberg s Uncertainty Principle Δx Δp x h 4π Δx = uncertainty in the position of the electron Δp x = uncertainty in the momentum of the electron Thus, there is the inherent uncertainty in the location and momentum of e - (Δx is large) (Δp x is small) e - should be treated as wave (due to its uncertainty in location), not simple particles. We can t describe orbits of e -, but can describe orbitals!!! region that describe the probable location of e - electron density Therefore, one should use an equation which describes wave property well!!!
2.2 The Schrödinger Equation The Schrödinger equation The equation describes the wave properties of e - in terms of its position, mass, total E and potential E. HΨ = EΨ H = the Hamiltonian operator E = energy of the electron Ψ = the e - s wave function Hamiltonian operator (H) includes derivatives that operate on the wave function. The result is a constant (E) times Ψ. Different orbitals have 1) different wave functions, and 2) different E values
2.2 The Schrödinger Equation 2 2 2 2 2 -h δ δ δ Ze H = + + - 8π m δx δy δz 4πε x + y + z 2 2 2 2 2 2 2 0 Kinetic E of the e - potential E of the e - (electrostatic attraction b/w the e - and the nucleus) h = Planck constant m = mass of the particle (e - ) e = charge of the e - (x 2 +y 2 +z 2 ) = r =distance from the nucleus Z = charge of the nucleus 4πε 0 = permittivity of a vacuum
2.2 The Schrödinger Equation If applied to a wave function Ψ,,, 2 2 2 2 -h δ δ δ + + + V( x, y, z) ψ ( x,y,z ) = E ψ ( x,y,z) 2 2 2 2 8π m δx δy δz 2 2 -Ze -Ze V = = 4πε r 4πε x + y + z 0 0 2 2 2 V (potential E): result of electrostatic attractions b/w e - and nucleus negative value 1) e - near nucleus (small r) large attraction 2) e - far from nucleus (large r) small attraction 3) e - at infinite distance (r = ) no attraction every atomic orbital has a unique Ψ no limit to the number of solution of the Schrödinger equation describes the wave properties of a given e - in a particular orbital!!
2.2 The Schrödinger Equation Ψ 2 : the probability of finding an e - at a given point in space conditions for a physically realistic solution for Ψ 1) The wave function Ψ must be single-valued. 2) The wave function Ψ and its first derivatives must be continuous. 3) The wave function Ψ must approach zero as r approaches infinity. 4) The integral 5) The integral * ψaψ Adr =1 all space * ψaψ Bdr = 0 all space
2.2.1 Particle in a Box the 1-D particle in a box shows how these conditions are used. Fig.2.3 Potential E well for the particle in a box potential E V(x) = 0, inside the box V(x) =, outside a box - A particle is completely trapped in the box. - It would require an infinite amount of E to leave the box. Thus, wave equation within a box is,, -h δ 8π m δx 2 2 Ψ(x) 2 2 = Eψ( x),,,since V(x) = 0 1-D box
2.2 The Schrödinger Equation The wave characteristics of our particles can be described by a combination of sine and cosine functions since they have properties associated w/ waves a well-defined wavelength and amplitude. a general solution to describe a possible wave in the box,,, Ψ = Asin rx + Bcos sx * A, B, r, s are constants. * solution for r and s,, r = s = 2mE(2π/h)
2.2 The Schrödinger Equation Ψ must be continuous and equals 0 at x < 0 and x > a, Ψ must go to 0 at x = 0 and x = a 1) because for x = 0, cos sx = 1 Ψ = 0, only if B = 0 *Ψ = Asin rx + Bcos sx Ψ = Asin rx 2) for x = a, Ψ must be 0 sin ra must be 0 it occurs,, only if ra = integral multiple of π (ra = ±nπ or r = ±nπ/a, n = any integer 0) positive & negative values give the same result. so, substitute positive r for the solution r,,, *r = s = 2mE(2π/h) r = nπ/a = 2mE(2π/h) E = n 2 h 2 /8ma 2
2.2 The Schrödinger Equation E = n 2 h 2 /8ma 2 E level predicted by the particle in-a-box model for any particle in a one-dimensional box of length a. E levels are quantized: n = 1, 2, 3,,, For the wave function: r = nπ/a, Ψ = Asin rx Ψ = Asin nπx/a if applying normalizing requirement A = (2/a) * ψ ψ dr =1 Then, total solution is, Ψ = (2/a) sin nπx/a
2.2 The Schrödinger Equation The resulting wave function(ψ) & their squares(ψ 2 ) for the first three states Fig. 2.4 Ψ 2 is a probability densities Fig. 2.4 shows difference b/w classical and quantum mechanical behavior 1) classical: equal probability of being at any point in a box 2) quantum: high and low probability at different location in a box
2.2.2 Quantum Number and Atomic Wave Functions Mathematically, atomic orbitals are discrete solutions of the three dimensional Schrödinger equations.
2.2.2 Quantum Number and Atomic Wave Functions explains two experimental observations (1) doubled emission spectra of alkali-metal (2) split beams of alkali-metal in magnetic field magnetic moment of the electron (spin of the electron, tiny bar magnet) n, l, m l define an atomic orbital, m s describes the electron spin within the orbital.
2.2.2 Quantum Number and Atomic Wave Functions There is i (= -1) in the p and d orbital wave equation. need to convert to real function rather than complex functions use linear combination (sum or differences) of functions ex) for p orbitals having m l = +1 and -1 1) sum + normalizing (by multiplying the const. 1/ 2) Ψ 2px = 1/ 2 (Ψ +1 + Ψ -1 ) = 1/2 (3/π)[R(r)]sinθcosΦ 2) difference + normalizing (by multiplying the const. i/ 2) Ψ 2py = i/ 2 (Ψ +1 - Ψ -1 ) = 1/2 (3/π)[R(r)]sinθsinΦ
2.2.2 Quantum Number and Atomic Wave Functions The same procedure can be used on the d orbitals, and the results are shown in the column headed θφ (θ, φ) in Table 2.3.
2.2.2 Quantum Number and Atomic Wave Functions Ψ can be expressed in terms of Cartesian coordinates (x, y, z) or in terms of spherical coordinates (r, θ, Φ ). x = rsinθcosφ y = rsinθsinφ z = rcosθ This is especially useful because r represents the distance from the nucleus. (Fig 2.5) volume elements: rdθ, rsinθdφ, dr products: r 2 sinθdθdφdr (= dxdydz) volume of the thin shell b/w r and (r + dr) : 4πr 2 dr (= integral over Φ (from 0 π), over θ (from 0 2π)) Fig. 2.5 Describing the electron density as a function of distance from the nucleus
2.2.2 Quantum Number and Atomic Wave Functions Ψ can be factored into a radial component & two angular components. R e - density at diff. distance from the nucleus θ, Φ shape of orbitals (s, p, d) orientations of orbitals Ψ (r, θ, Φ) = R(r)Θ(θ)Φ(φ) = R(r) Y(Θ, Φ) Y 1) Angular functions: θ, Φ shape orientations determined by l determined by m l (1) + (2) 3-D shape in the far-right column in Table 2.3
2.2.2 Quantum Number and Atomic Wave Functions The different shading of the lobes represent diff. signs of the wave function Ψ. but, the same probability.. bonding purpose.. Fig. 2.6
2.2.2 Quantum Number and Atomic Wave Functions Ψ (r, θ, Φ) = R(r)Θ(θ)Φ(φ) = R(r) Y(Θ, Φ) 2) Radial functions(r(r)): determined by n and l : probability of finding e - at a given distance from nucleus Radial probability function: 4πr 2 R 2 at the center of nucleus, 4πr 2 R 2 = 0 ( r = 0) 4πr 2 X R 2 r R 2 (generally) r 4πr 2 R 2 show which orbitals are most likely to be involved in reactions Fig. 2.7
2.2.2 Quantum Number and Atomic Wave Functions 3) Nodal surface: - a surface w/ zero e - density ex) 2s at r = 2a 0 - appears as a result of the wave equation for the e - result from solving the wave function is 0 as it changes sign Ψ = 0, probability of finding e - is 0. Ψ 2 = 0 Ψ = 0
2.2.2 Quantum Number and Atomic Wave Functions Ψ (r, θ, Φ) = R(r) Y(Θ, Φ): for Ψ = 0 either R(r) = 0 or Y(Θ, Φ) = 0 Thus, we can determined nodal surfaces by detecting under what conditions R(r) = 0 or Y = 0. 1) Angular Nodes (Y(Θ, Φ) = 0): planar or conical l angular nodes (conical nodes counts two nodes)
2.2.2 Quantum Number and Atomic Wave Functions 2) Radial Nodes [R(r) = 0] n - l - 1 the lowest E orbitals of each classification (1s, 2p, 3d, 4f,,,) have no radial nodes. Fig. 2.8
2.2.3 The Aufbau Principle in German: aufbau building up when there is more than one e - s, interactions b/w e - s require that the order of filling orbitals be specified 1) to give the lowest total E to the atom lowest n, l are filled first m l & m s s values don t matter 2) Pauli exclusion principle: each e - in an atom have a unique set of quantum number 3) Hund s rule of maximum multiplicity: e - should be placed in orbitals so as to give the max. total spin possible (max. number of parallel spins) to avoid electrostatic repulsion
2.2.3 The Aufbau Principle Coulombic energy of repulsion, Π c : causes a repulsive force b/w e - s in the same orbitals exchange energy, Π e : depends on the number of possible exchanges b/w two e - s w/ the same E & the same spin ex) C: 1s 2 2s 2 2p 2 three arrangements for 2p 2 are possible Πc (e - - e - repulsion) so, higher E than other two two possible ways to arrange e - s Only one way to arrange the e - to give the same diagram (one exchange of e - ) The higher the # possible exchange, the lower the E.
2.2.3 The Aufbau Principle Total pairing E, Π = Π c (+) + Π e (-)
2.2.3 The Aufbau Principle The order of filling of atomic orbitals 1) Klechkowsky s rule lowest available value for the sum n + l (e.g. 4s: n + l = 4 + 0; 3d: n + l = 3 + 2) if same, the smallest value of n 2) the blocked out periodic table Group 1-2: filling s orbitals, l = 0 Group 13-18: filling p orbitals, l = 1 Group 3-12: filling d orbitals, l = 2 lanthanides & actinides: f orbitals, l = 3 Fig2.9
2.2.4 Shielding Effect If there is more than one e -, prediction of E of specific levels are difficult. use the concept of shielding each e - acts as a shield for e - farther from the nucleus reduce the attraction b/w nucleus and distant e - the E changes are somewhat irregular due to the shielding of outer e - by inner e - *see Table 2.7
2.2.4 Shielding Effect the higher E w/ a higher quantum # this is true only for orbitals w/ the lowest n (Fig.2.10) for higher value of n,,, l is also important ex) 4s orbital is lower than 3d orbital thus, 3s, 3p, 4s, 3d, 4p also, 5s before 4d 6s before 5d Fig2.10
2.2.4 Shielding Effect Slater s effective nuclear charge Z* as a measure of the attraction of the nucleus for a particular e -. Z* = Z S nuclear charge shielding constant Rules for determining S 1) write order of e - str. and group them (1s) (2s, 2p) (3s, 3p) (3d) (4s, 4p) (4d) (4f) (5s, 5p) (5d),,,, 2) right side e - s do not shield left side e - s 3) for ns & np valence e - a. same group e - 0.35 (exception: 1s e - 0.30) b. n-1 group e - 0.85 c. n-2 or the lower group e - 1.00 4) for nd and nf a. same group e - 0.35 b. left side e - 1.00
2.2.4 Shielding Effect Examples) Oxygen s 2p, Nickel s 3d and 4s Justification for the Slater s rule comes from e - probability curves for the orbitals. (Fig. 2.7) - for the same n, s & p orbitals have higher probability near the nucleus than d orbitals ex) (3s, 3p) shield 3d w/ 100% effectiveness 1.00 - shielding 3s or 3p by (2s, 2p) shows 85 % effectiveness 0.85 3s & 3p orbitals have regions of significant probability close to the nucleus Fig2.7
2.2.4 Shielding Effect Complicate cases - Cr : [Ar]4s 1 3d 5 [Ar]4s 2 3d 4 Cu : [Ar]4s 1 3d 10 [Ar]4s 2 3d 9 place one extra e - in the 3d & remove one e - from the 4s special stability of half-filled subshell (Fig. 2.11) - Rich s explanation for e - - e - interactions consider E difference b/w 1 e - in 1 orbital 2 e - s in 1 orbital electron pairing E is added due to the electrostatic repulsion
2.2.4 Shielding Effect Fig. 2.12: Schematic E levels for TM - as nuclear change e - more strongly attracted E level decrease more stable - the filling order: from bottom to top * d orbitals changes more rapidly than s, p due to the lack of shielding ex) V: [Ar] 4s 2 3d 3, Cr: [Ar] 4s 1 3d 5, Cu: [Ar]4s 1 3d 10 Fig2.12
2.2.4 Shielding Effect Fig. 2.12(b) formation of positive ions by removing e - among TM, reduce the overall e - repulsion & E of the d orbitals more e - w/ highest n are removed first, as ions are formed no s, occupied d TM cations have no s electrons, only d electrons in their outer levels lanthanide & actinide series show similar patterns.. Fig2.12
2.3 Periodic Properties of Atoms 2.3.1 Ionization Energy = ionization potential (E required to remove an e - from a gaseous atom or ion) A + (g) A (n+1)+ (g) + e - ionization E = ΔU in general, as nuclear charge ionization E experimentally observed breaks in trend at Boron and Oxygen - B: new p e - is farther away from the nucleus than others easier to remove than Be - O: the fourth p e - pairing of 2p e - at the same orbital easier to remove Fig2.13
TM - smaller diff. in IE 2.3.1 Ionization Energy - usually lower value for heavier atoms in the same group due to the shielding effect each new period: much larger decrease in IE change the next major quantum # e - w/ much higher E noble gas: maximum of IE decreases w/ increasing Z e - farther from nucleus in the heavier elements overall, in periodic table,, higher IE from left to right lower IE from top to bottom Fig2.13
2.3.2 Electron Affinity electron affinity = E required to remove e - from a negative ion A - (g) A(g) + e - electron affinity = ΔU (or EA) also described as the zeroth ionization E endothermic (+ ΔU value) except noble gases & alkaline earth metals smaller absolute # than IE - removal of e - from a negative ion is easier than removal from a neutron atom noble gas has the lowest EA - removal of e - past noble gas is easy.
2.3.3 Covalent and Ionic Radii two effects exist as Z : 1) e - are pulled toward nucleus orbital size 2) e - # mutual repulsion orbital size sum Gradual decrease in atomic size across each periods Table 2.8: Nonpolar covalent radii sufficient for general comparison other measure of size of elements difficult to obtain consistent data
2.3.3 Covalent and Ionic Radii ion size: the stable ions of the diff. elements have different charges, e - #, crystal str. difficult to determine the size of ion old method: Pauling s approach isoelectronic ion s ratio of radii was assumed to be equal to the ratio of their effective nuclear charge recent method: Shanon s approach consider many things (e.g. e - density map from X-ray) (Table 2.9) factors influencing ionic size: 1) coordination of ion 2) covalent character of bonding 3) distortions of regular crystal geometries 4) delocalization of e - (metallic or semi-metallic) 5) the size & charge if the cation
2.3.3 Covalent and Ionic Radii for the ions w/ the same # e -, as Z size within a group, as Z size same element, as charge on the cation size