CHEM 231 Inorganic Chemistry Lecture Notes PDF

CHEM 231 Inorganic Chemistry I, 3.0 Credits Dr. Mohamed Ahmed Ass. Prof. @ Chemistry Department, UAEU 1.5 An introduction to wave mechanics 1.6 Atomic orbitals 1.7 Many-electron atoms 2  Wave Nature of Matter ❑Einstein’s Photoelect...

CHEM 231 Inorganic Chemistry I, 3.0 Credits Dr. Mohamed Ahmed Ass. Prof. @ Chemistry Department, UAEU 1.5 An introduction to wave mechanics 1.6 Atomic orbitals 1.7 Many-electron atoms 2  Wave Nature of Matter ❑Einstein’s Photoelectric effect illustrated the particle like properties (photons) associated with light waves. ❑Waves have particle like properties and particles have wave like properties ❑Matter has a wavelength associated with it, which was first described by de Broglie ❑Using Einstein’s and Planck’s equations, de Broglie showed the wave-particle duality: h Wavelength = mv Particle momentum ❑ The momentum, mv, is a particle property, whereas  is a wave property. ❑ h is Planck’s constant 6.626 x 10-34 Js 3 Heisenberg’s Uncertainty Principle 4 Schrödinger Wave Equation d  d  d  8 m 2 2 2 2 2 + 2 + 2 + 2 ( E − V ) = 0 d x d y d z h 5  Schrodinger Wave Equation ❑It is easier to solve the Schrodinger equation in terms of polar coordinates: Cartesian(x,y,z) = radial(r) angular (θ, ϕ) = R(r)A(θ, ϕ) ❑In this coordinate system the wavefunction (solution to the Schrodinger equation) is expressed as a product of a radial function R(r) and an angular function A(θ, ϕ). ❑The solutions of the Schrodinger equation for an atom are the atomic orbitals and are described in terms of the quantum numbers n, l, ml. 6  Quantum Numbers 1. Principal Quantum Number, n. This is the same as Bohr’s n. As n becomes larger, the atom becomes larger, and the electron is further from the nucleus. Energy increases as n increases. (related to the radial part of Sch. Eq.) 2. Azimuthal (Orbital) Quantum Number, l. The values of l can be 0,1,2,….,(n - 1) and it determines the shape of the atomic orbitals. We usually use letters for l (s, p, d and f for l = 0, 1, 2, and 3). Usually we refer to the s, p, d and f-orbitals. When more than 1 electron is present, energy increases with increasing l. 3. Magnetic Quantum Number, ml. This quantum number depends on l. The magnetic quantum number has integral values between -l and +l. Magnetic quantum numbers give the 3D orientation of each orbital. 7  Shells, Subshells & Orbitals ❑An Energy (or principal) shell corresponds to a particular value of n and can hold 2n2 electrons. For example: n=1 → 2(1)2 = 2 electrons n=3 → 2(3)2 = 18 electrons This is equivalent to the Bohr value of n ❑An orbital is a region of space that can hold two electrons. ❑For a given value of n there are n2 orbitals. For example: n=1 → (1)2 = 1 orbital n=3 → (3)2 = 9 orbitals ❑Orbitals sometimes exist in degenerate sets called subshells: that is, several orbitals of the same energy. The subshells correspond to the values of l. ❑The subshells are labeled with the letters s, p, d, f according to value of l, l = 0, → s, l = 1, → p, l = 2, → d, l = 3, → f, l = 4, → g (rarely discussed) & alphabetically thereafter 8  Shell, Subshells & Orbitals ❑For a given value of n, there are n subshells For example: n=1 → 1 subshells (l= 0) (s) n=3 → 3 subshells (l= 0,1,2) (s,p,d) ❑The number of orbitals in a subshell is equal to 2l+1 For example: a p-subshell (l =1) has → 2(1)+1 = 3 p-orbitals a d-subshell (l=2) has → 2(2)+1 = 5 d-orbitals ❑Each orbital in a subshell corresponds to a particular value of ml ❑ Subshells are labeled as nl. For example: - 1s (n=1, l=0), - 2s (n=2, l=0) 2p (n=2, l=1), - 3s (n=3, l=0) 3p (n=3, l=1) 3d (n=3, l=2), - 4s (n=4, l=0) 4p (n=4, l=1) 4d (n=4, l=2) 4f (n=4, l=3) 9  Principle Qu. No. Orbital Qu. No. Magnetic Qu. No. Quantum Numbers in one shot 10  Radial and Angular Functions ❑As discussed earlier, the solutions to the Schrodinger equation in spherical polar coordinates can be written as a product of separate radial and angular functions: (x,y,z) = R(r)A(θ, ϕ) ❑The radial function gives variation of the wavefunction with distance (r) from the nucleus. ❑The angular functions give rise to the characteristic shape of the atomic orbitals. 11  Radial Part of Wave Function R(r) 1s atomic orbital 2s atomic orbital 2p, 3p, 4p and 3d orbitals Radial Functions for H atom 2p, 3p, 4p and 3d orbitals 12  Points we learn form the radial function ✓ s atomic orbitals have a finite value of R(r) at the nucleus; ✓ For all orbitals other than s, R(r) = 0 at the nucleus; ✓ For the 1s orbital, R(r) is always positive; for the first orbital of other types (i.e. 2p, 3d, 4f ), R(r) is positive everywhere except at the origin; ✓ For the second orbital of a given type (i.e. 2s, 3p, 4d, 5f ), R(r) may be positive or negative and the point at which R(r) = 0 (not including the origin) is called a radial node; ✓ For the third orbital of a given type (i.e. 3s, 4p, 5d, 6f ), R(r) has two sign changes, i.e. it possesses two radial nodes. ✓ Radial nodes occur (at fixed radius) as the principal quantum number increases. ❑The number of radial nodes for an orbital is given by, # of radial nodes = n-l-1 ns orbitals have (n-1) radial nodes. np orbitals have (n-2) radial nodes. nd orbitals have (n-3) radial nodes. nf orbitals have (n-4) radial nodes. 13  Radial Distribution Functions (RDF) ❑The plots of radial functions show the variation in the radial component of ψ with distance (r) from the nucleus. ❑More useful is the variation in the probability of finding an electron with distance from the nucleus. The probability of finding an electron is related to ψ 2. ❑The radial distribution function (RDF) gives the probability of finding an electron at any distance (r) from the nucleus. ❑Radial distribution function (RDF) = 4πr2R(r)2 14  Radial Distribution Functions (RDF) for H atom RDF for 1s, 2s and RDF for 3s, 3p and 3s atomic orbitals 3d atomic orbitals ❑ Note that all the RDFs have a zero value at the nucleus (as r=0 at the nucleus) ❑ The RDF is a probability function and always positive. ❑ The RDF = 0 at radial nodes (as R(r)=0). ❑ Note that orbitals may have more than one maximum in the RDF. Where the electron has the higher probability of being found. 15  Angular Part of Wave Function A(θ,Ф) ❑ This part is independent of n (principal quantum no.). ❑For s-orbitals, A(θ,Ф) is independent of the angles θ,Ф. s-orbital is spherically symmetric. ❑px, py, and pz are degenerate set of p-orbitals. 16  s-Orbitals p-Orbitals For an s-subshell l = 0, For a p-subshell l = 1 so, there is 2l+1 = 1 orbital. so. there are 2l+1 = 3 orbitals 17  d-Orbitals For a d-subshell l = 2 so there are 2l+1 = 5 orbitals 18  f-Orbitals For an f-subshell l = 3 so there are 2l+1 = 7 orbitals fy3 - 3x2y fxyz f5yz2 - yr2 fz3 - 3zr2 f5xz2 - xr2 fzx2 - zy2 fx3 - 3xy2 19  Angular Nodes (nodal plans) ❑Nodes are the points in space around a nucleus where the probability of finding an electron is zero. ❑It is clear that, s-orbitals have no angular nodes, p- orbitals have 1 angular node, d-orbitals have 2 angular nodes, …. ❑ Angular nodes are typically flat plane (at fixed angles, nodal plan) In general, the number of angular nodes for a given orbital is equal to the quantum number l: # angular nodes = l 20  Energy of Subshells (s,p,d and f) ❑In the H atom, all subshells corresponding to the same value of n are degenerate ❑For atoms with more than 1 electron, the energy of a subshell depends on both n and l. ❑For a given value of n, energy increases with l. ❑This is related to penetration/shielding. 21  Penetration/Shielding ❑ RDF as a function of r. ❑ The 2s and 2p RDFs penetrate the 1s function. ❑ The 2s penetrates more than the 2p. ❑ Hence an e in the 2s experiences a greater effective nuclear charge Zeff than an e in the 2p. ❑ Subsequently the 2s is of lower energy than 2p for the many e atom. ❑While for Li (Z=3), 1s orbital if fully occupied. Where to add the 3ed electron? Figure 1.13 Radial distribution functions, 4πr2 R(r)2, for the 1s, 2s and 2p atomic orbitals of the hydrogen atom. ❑ The 3rd electron will occupies either 2s or 2p, both suffer the fully occupied 1s orbital (shielding). ❑ Penetration/Shielding: electron arrangement in the atomic orbitals to produce low energy system (more stable system). 22  Slater’s Rules ❑ One of the empirical rules to estimate the Effective Nuclear Charge Zeff. ❑ Effective Nuclear Charge: The charge ‘experienced’ by an electron in a many electron atom. Zeff = Z-S Zeff = Effective Nuclear Charge, Z = Nuclear Charge, and S = Shielding/Screening Constant To find the value of S: 1. Write electron configuration as: (1s) (2s,2p) (3s,3p) (3d) (4s,4p) (4d) (4f) … 2. Electrons to right of ē of interest contribute 0 (higher energy). 3. Each ē in same group contributes 0.35 4. For ē in ns or np: - Each ē in (n-1) group contributes 0.85 - Each ē in (n-2) or lower contributes 1.0 1. For ē in nd or nf: - Each ē in lower group contributes 1.0 23  Slater’s Rules-Example 1 Q. Use Slater’s rules to predict the most favored electron configuration for Ca from the following: I. 1s2 2s2 2p6 3s2 3p6 4s2 II. 1s2 2s2 2p6 3s2 3p6 3d2 I. (1s)2 (2s,2p)8 (3s,3p)8 (4s)2 II. (1s)2 (2s,2p)8 (3s,3p)8 (3d)2 S = (0.35x1) + (8x0.85) + (10x1.0) S = (0.35x1) + (18x1.0) S = 17.15 S = 18.35 Z = 20 Z = 20 ➔ Zeff = Z – S = 20 – 17.15 = 2.85 ➔ Zeff = Z – S = 20 – 18.35 = 1.65 We predict that the 4s2 configuration is more stable than the 3d2 configuration as the Zeff is greater for the 4s electrons than for 3d. 24  Slater’s Rules- Example 2 Q. Calculate the effective nuclear charge experienced by a 4s electron in V and compare to the effective nuclear charge experienced by a 3d electron in V. Which electron do you predict will be lost on formation of V+? V: (1s)2 (2s,2p)8 (3s,3p)8 (3d)3 (4s)2 For 4s ē : S = 0.35 + (11x0.85) + (10x1.0) = 19.7 Test yourself ➔ Zeff = 23 – 19.7 = 3.3 For 3d ē: S = 2x0.35 + (18x1.0) = 18.7 Zeff = 23 – 18.7 = 4.3 The 4s electron sees a lesser effective nuclear charge and is expected to be lost most readily. 25