Atomic Orbitals PDF

Elements, Atoms and The Periodic Table L4: The shape of atoms Dr Daniel J Price The structure of atoms Electrons have wave-like properties! This wave-like property is represented by the ‘wavefunction’, y The wavefunction is a function...

Elements, Atoms and The Periodic Table L4: The shape of atoms Dr Daniel J Price The structure of atoms Electrons have wave-like properties! This wave-like property is represented by the ‘wavefunction’, y The wavefunction is a function y(x,y,z) The wavefunction emerges naturally as a solution to the Schrodinger equation, along with an associated energy E. These wavefunctions turn out to be the ‘atomic orbitals’ Labelling the wavefunctions Electrons in orbitals are fully characterized by Four quantum numbers n, l, ml and ms We use the values of n, l, and ml as labels for the different wavefunctions We have already met n, the principal quantum number In Bohr’s model it defines how far the electron is from the nucleus n = 1, 2, 3, 4, 5...... The Principle quantum number, n Bohr’s atom n determines the size of an orbit n=5 n=6 n=4 Lyman n = 3 series Balmer n=2 series n=1 Principle quantum number and the periods Main The angular momentum quantum number, l l defines the radial spatial distribution of the wavefunction Allowed values of l are limited by the value of n Values l = 0, 1, 2, … (n-1) Allowed values of n and l Traditionally use letters to represent l n = 1, l = 0 l=0 s-orbital n = 2, l = 0 or 1 l=1 p-orbital n = 3, l = 0 or 1 or 2 l=2 d-orbital n = 4, l = 0 or 1 or 2 or 3 l=3 f-orbital … … The first orbital: n = 1, l = 0 (s) Probability amplitude of the first wavefunction y n = 1, l = 0 1s Py2 High 1. The value is positive everywhere 2. It is highest at the nucleus 3. It approaches zero as we move away from the nucleus 4. It is spherically symmetric What is the probability, P, of finding the electron in the finite element dx dy dz ? 1. The value is positive everywhere 2. There is no edge to the atom y 3. The highest chance of finding the electron is at the centre Low How big is the first orbital? How big is an atom? 1s-orbital (n = 1, l = 0) Surface enclosing 99% probability of finding the electron. The orbit is a cloud of electron density, it is helpful to construct a boundary surface Here size depends on our chosen probability; 90%, 99%, 99.9%, 100% is infinitely big! How big is the first orbital? What is the probability of finding Peak maximum the electron at radius r? “Average” distance from nucleus What is the r probability of finding electron Dr in shell, with volume Dr ? Volume enclosed by a shell of “Atomic radius” thickness Dr, is proportional to r2 The second orbital: n = 2, l = 0 (s) y2s n = 2, l = 0 Positive 1. The value is positive and negative 2. There is a sphere where the y = 0 3. It approaches zero as we move away from the nucleus. 4. It is spherically symmetric zero Probability Py2 1. The value is positive everywhere 2. There is a sphere where P = 0 3. There is no edge to the atom negative 4. It is spherically symmetric Probability amplitude The shapes of s-orbitals n = 1, l = 0 n = 2, l = 0 n = 3, l = 0 y1s y2s y3s 1s-orbital 2s-orbital 3s-orbital What about l = 1? (p-orbitals) y2p n = 2, l = 1 z-axis Positive 1. The value is positive and negative 2. 3. There is a plane where the y = 0 It is 0 at the nucleus Probability 4. It approaches zero as we move zero away from the nucleus Py2 1. The value is positive everywhere 2. There is a plane where the y = 0 3. It has two lobes (dumbbell Probability negative shaped) – directed along an axis amplitude The magnetic quantum number, ml ml defines the orientation of the wavefunction Allowed values of ml are limited by the value of l Values ml = -l, -l+1, -l+2, ….., +l Allowed values of l and ml How many orbitals? l = 0, ml = 0 one s-orbital l = 1, ml = -1 or 0 or +1 three p-orbitals l = 2, ml = -2 or -1 or 0 or +1 or +2 five d-orbitals l = 3, ml = -3 or -2 or -1 or 0 or +1 or +2 or +3 seven f-orbitals … … …. ….. The three p-orbitals: l = 1; ml = -1, 0, +1 Q.N. Wavefunction Type of value n=2 Y2p- 2px-orbital y2px l =1 complex numbers z = (a + bi) ml = -1 n=2 l =1 y2p0 positive and negative 2py-orbital ml = 0 y2py n=2 l =1 Y2p+ complex numbers 2pz-orbital y2pz z = (a + bi) ml = +1 The three p-orbitals: l = 1; ml = -1, 0, +1 Most easily visualised with a boundary surface: e.g. enclosing a 99% probability of finding the electron Degenerate orbitals: Same energy, orthogonal functions The shapes of p-orbitals n = 2, l = 1, ml = 0 n = 3, l = 1, ml = 0 n = 4, l = 1, ml = 0 2pz 3pz 4pz z x The shapes of all orbitals n=1 n=2 n=3 n=4 l =0 l =1 l =2 l =3 3d 4f 1s 2p = node P = 0 The five d-orbitals: l = 2; ml = -2, -1, 0, +1, +2 n = 1 l = 0 ml = 0 1s 1 orbital n = 4 l = 0 ml = 0 4s 1 orbital l = 1 ml = -1 n = 2 l = 0 ml = 0 3 orbitals 2s 1 orbital ml = 0 4p l = 1 ml = -1 ml = +1 ml = 0 2p 3 orbitals l = 2 ml = -2 ml = +1 ml = -1 n = 3 l = 0 ml = 0 3s 1 orbital ml = 0 4d 5 orbitals ml = +1 l = 1 ml = -1 3 orbitals ml = +2 ml = 0 3p l = 3 ml = -3 ml = +1 ml = -2 l = 2 ml = -2 ml = -1 ml = -1 ml = 0 4f 7 orbitals ml = 0 3d 5 orbitals ml = +1 ml = +1 ml = +2 ml = +2 ml = +3 Energy levels (Hydrogen only) Schrödinger equation gives y’s, with energies E(y) 5s 5p 5d 5f 4s 4p 4d 4f 3s 3p 3d 2s 2p 1s When only 1 interaction (nucleus – electron), the energy is determined by n only. e.g. E(y2s ) = E(y2p) Shells, sub-shells, blocks and orbitals Looks like 2 electrons per orbital 1 orbital 3 orbitals 5 orbitals 7 orbitals  Pauli exclusion principle Fundamental Property of electrons, they cannot occupy the same space. Only 2 electrons can occupy any one orbital Wolfgang Pauli The spin quantum number, ms Electrons have a spin quantum number, s s is an angular momentum, like l it has an associated ms quantum number Allowed values of ms are limited by the value of s Values ms = -s, -s+1, -s+2, ….., +s For electrons s = ½ therefore ms = -½ or +½ ms defines spin orientation The thing about electrons No two electrons in the same atom can have all 4 quantum numbers same The size, shape and orientation of an orbital are defined by the 3 quantum numbers; n, l and ml The spin orientation of an electron is defined by the 1 quantum number; ms For electrons ms can only take 2 values; ms = +½ (spin up) or ms = -½ (spin down) So only 2 electrons can occupy any one orbital End