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Wave Mechanics • Better description of atoms is obtained using wave mechanics. Particles (Bohr) Electrons Waves (de Broglie) • In 1927, Schrödinger suggested that electrons can be described by wave functions Understanding Waves • Traveling waves i.e. waves in ocean • Standing waves i.e. a guitar string Standing Waves boundaries 1 node Petrucci Fig. 8-18 2 nodes node: A point which undergoes no displacement at all Standing Waves Fig. 8-18 What are the possible values of l? L L = length boundaries l=2L l=L 2L l= n n=1,2,3,etc 1 node 2L l= 3 2 nodes In General, Permitted values of l are said to be quantized! Wave functions • Erwin Schrödinger (1927) – suggested electrons could be described by an equation – called wave function (y) - psi • y is a standing wave within the boundary of the system • Simplest system - electron in a 1D box! The Particle in a Box!! The Particle in the Box n ( x) = 2 nπx sin L L L = length of box x = displacement along x axis n = quantum number Fig. 9-19, pg. 321 The Particle in a Box I • particle must be inside a box of length L and the particle moves in x direction • similar picture to before… Fig. 8-20, pg. 301 The Particle in a Box II What mathematical function to use for y? y is a function of x Functions for waves… sine or cosine ??? sine x=0; sin(x) = 0 Fig. 8-20, pg. 301 The Particle in a Box III General solution 2 nx n (x) = sin L L L = length of box x = displacement along x axis n = quantum number Fig. 8-20, pg. 301 Particle in a Box IV • What about energy? Particle has kinetic energy: 1 2 m 2u2 p2 E K = mu = = 2 2m 2m Subatomic particles have wavelike properties: h h l= = p mu Substituting for p: de Broglie Eqn! h2 EK = 2 2ml Particle in a Box V Recall from earlier that for standing waves… In General, 2L l= n Permitted values of l are said to be quantized! n=1,2,3,etc Therefore, h2 EK = 2 2ml n 2h 2 EK = 2 8mL n=1,2,3… Energy is quantized!!! Particle in a Box VI n 2h 2 EK = 2 8mL Note: lowest E n=1,2,3… 0 Consequence: the particle cannot be at rest (consistent with Heisenberg principle) Lowest E is when n=1 Called zero point energy Particle in a Box VII Where can we find the electron? Heisenberg said “both its position and momentum cannot be known exactly” In 1926, Born proposed: We should think about the probability of finding an electron in a particular position. probability 2 Max Born Probability 2 nx n (x) = sin L L 2 2 nx (x) = sin L L 2 n Fig. 8-20 Fig 9-20, pg. 322 Summary • Particle in 1D box – Can be represented by wave function () with quantum numbers (n; n=1,2,3…) – # of nodes = n-1 – Energy is quantized – Probability of finding an electron in a particular position is proportional to 2 IN AN ATOM THE ELECTRON CAN MOVE IN 3-DIMENSIONS Particle in 3D Box n 2h 2 EK = 8mL2 1D box 2 2 2 n n h nx y EK = 2 2 2z 8m L x L y L z 2 3D box In 3-dimensions we need 3 quantum numbers! Wave functions for Atoms • Describing is very complex… we will not go into detail… we will look at the solutions. • Atom is 3D; also +ve charge (protons) must be considered (I.e. electrostatic attraction of protons and electrons) Schrödinger Equation: ˆ H = E Atomic # or h 2 2 2 2 Ze2 2 2 2 2 = E 8 me x y z r Hydrogen Atom Must solve… Ze 2 2 2 2 = E 8 me x y z r h 2 2 2 2 2 For the hydrogen atom, the solution to the Schrödinger equation gives the wave functions for the electrons in this atom. These wavefunctions () are called orbitals. • Wave functions are analyzed in terms of 3 variables required to define a point with respect to the nucleus. • Cartesian coordinates (x,y,z) or spherical polar coordinates (r, q, f) can be used. theta Fig. 8-22 (pg. 303) phi In the spherical polar coordinate system, orbitals can be expressed in terms of two functions: r,q , f = R(r )Y (q , f ) R(r) is the radial wave function Y(q,f) is the angular wave function Each orbital has three quantum numbers to define it. The particular set of quantum numbers gives a certain functional form to R(r) and q,f. Quantum Numbers Principal Quantum Number n = 1,2,3,4… Orbital Angular Momentum Quantum Number l = 0,1,2,3,…n-1 Magnetic Quantum Number ml = -l, -l + 1, -l +2,…0, 1, 2,…l - 1, l Electron spin • A fourth quantum number is used for electron spin ms = +1/2 or –1/2 • ms does not depend on any of the other quantum numbers. • An electron generates a magnetic field due to its spin. Example For an orbital with n = 3 and ml = -1, what is (are) the possible value(s) of l? • All orbitals with the same value of n are in the same principal electronic shell: n = 1 (first principal shell) n = 2 (second principal shell) etc. • All orbitals with the same values of n and l are in the same subshell. n=1 n=2 n=3 etc. l=0 l = 0,1 l = 0,1,2 • Number of subshells = n • Names given to subshells: l=0 l=1 l=2 l=3 l =4 etc. s p d f g Number of orbitals in a subshell = the number of allowed values of ml = 2l + 1 l=0 ml = 0 one s orbital l=1 ml = -1,0,1 three p orbitals l=2 ml = -2,-1,0,1,2 five d orbitals • • To designate the s orbital in the first principal shell: 1s To designate the p orbitals in the second principal shell: 2p – Fig 9-22 Pg. 326 Orbitals of the Hydrogen Atom I • Orbitals are wave functions • The square of the wave function, 2 gives the threedimensional probability distribution. • We commonly draw orbitals as these probability distributions. • Orbitals have nodes and exhibit phase behavior just like other waves. • Recall, that (r,q,f = R(r)Yq,f) Orbitals of the Hydrogen Atom II For the 1s orbital • Angular part: Yq,f) is the same for all s orbitals 1/ 2 • Radial part: R(r) 1 Y ( s) = 4 Z R(1s) = 2 ao σ = 2Zr/na0 3/ 2 e / 2 The 1s orbital of Hydrogen Fig. 9-23 (pg. 329) Orbitals of the Hydrogen Atom II For the 2s orbital • Angular part: Yq,f) is the same for all s orbitals 1/ 2 • Radial part: R(r) 1 Y ( s) = 4 1 Z R(2s) = 2 2 ao 3/ 2 (2 ) / 2 Number of radial nodes: 0 1 2 Fig. 9-23 Pg. 329 Orbitals of the Hydrogen Atom Probabilities of finding an electron versus distance from the nucleus, we see that s orbitals possess n−1 nodes, or regions where there is 0 probability of finding an electron. p Orbitals of Hydrogen • The 2p orbital has no radial nodes at finite values of r • The p orbitals vanish at r = 0 • The angular part of the wave function is a function of q and f Number of radial nodes = n-l-1 Number of angular nodes = l The three p Orbitals Fig. 8-28, pg 311 d Orbitals dxy dxz dyz dx2-y2 dz2