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Bohr model only works for one electron atom, or ions. i.e. H, He +, Li 2+, …. It can’t account for electron-electron interactions properly The wave “like” properties of electron need to be explored to do job properly. Electrons as Waves Louis de Broglie : All matter has a corresponding wave character, with a wavelength determined by its momentum p (= mv) = h/p =h/mv Example : Electron moving at 0.1000 C: = (6.626*10 -34 Js)/(9.109*10 -28 kg)*(0.1000*2.998*10 8 m/s) = 2.423*10 -14 m = 242.3 pm

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Example: A 0.100 kg baseball moving at 150. km/hr: = (6.626*10 -34 Js)/(0.1 kg)*(41.7m/s) =1.59*10 -34 m v = (150000 m)/(3600 s) = 41.7 m/s Correspondence Principle Macroscopic bodies don’t feel the effect of quantum mechanics due to their large masses and slow motion Ex) The wavelength of the base ball is insignificant on the scale of the base ball Microscopic bodies do feel the effect of quantum mechanics strongly due to their small masses and fast motion Ex) The wavelength of the electron ball is very large on the scale of the size of the electron i.e. 10 -30 m

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Wavefunctions Standing waves Ex) String on a guitar Only a few wave forms are suitable, which is determined by length = 2L, L, 2L/3 …. Wavefunction: (x) depends on number of lobes n =1, 2, 3, … (x) = sin (x) = sin(2 ) (x) = sin(3 ) Therefore (n,x) is s series of solutions, each with a different energy E(n) = x/2L x

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A Wave in Orbit A circular path imposes a length on the wave form, allowing for only an whole number of nodes.

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A Little Math F(y) - is function that depends on y F(G(x)) - is the function, F, that acts on G(x), where G(x) s a function of x. F(y) = cy - the function F acting on y to give back y multiplied by some constant c Ex) F(y) = 3 y requires that c = 3 Ex) F(y) = 3 y, and y = G(x) = 4x 2 +2 F(G(x)) = 3 (4x 2 +2) = 12x 2 +2 = 3 G(x) where c = 3 F(G(x)) = cG(x) - The function F acting on G(x) gives back G(x) multiplied by some number c

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H( (n))- means that H is a function that acts on (n) - H is a function that calculates the total energy using, (n) - The result is the Energy, E(n), which depends on n, and the original wavefunction (n) H( (n) ) = E(n) (n) r = 3-D coordinates of electron (n) - is the wavefunction corresponding to the electron The Schrödinger Equation - H contains electron-electron, electron-nuclear iteration, and kinetic energy terms.

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H( (n, l,m, s) ) = E(n) (n, l, m, s) The wavefunction depends on four quantum number, each associated, with a different property on the electron. n – Principle Quantum Number l – Angular momentum Quantum Number Determines which shell the electron is in and the energy of the electron, E(n) n = 1, 2, 3, 4, … E(n) = -R y Z/n 2 Subshells exist for each shell differing in the angular momentum value. l = 0, 1 …n-1 m- Magnetic Quantum Number Related to the orientation in space that of the orbital. m = - l …+ l s - Spin Quantum Number Related to symmetry of wavefunction s = 1/2, -1/2 L = h l /2

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Wavefunctions of H n = 1m = 0 Lets for the moment ignore spin l = 0 States of m are labeled as: l = 0 S l = 1 P l = 2 D l = 3 F Therefore this state is: 1s 0 = 1s n = 2 l = 0 m = 0 n = 2 l = 1 m= 1, 0, -1 2s 2p 1, 2p 0, 2p -1 2p x, 2p y, 2p z

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Wavefunction of H n = 3 l = 0 m = 0 n = 3 l = 1 m = 1,0, -1 n = 3 l = 2 m= 2,1,0, -1,-2 3s 3p 1, 3p 0, 3p -1 3d 2, 3d 1, 3d 0, 3d -1, 3d -2 3d(xy), 3d(xz), 3d(yz), 3d(x 2 -y 2 ), 3dz2

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Wavefunction of H n = 4 l = 0 m = 0 n = 4 l = 1 m = 1,0, -1 n = 4 l = 2 m = 2,1,0, -1,-2 4s 4p 1, 4p 0, 4p -1 4d 2, 4d 1, 4d 0, 4d -1, 4d -2 n = 4 l = 3 m= 3,2,1,0, -1,-2,-3 4f 3, 4f 2, 4f 1, 4f 0, 4f -1, 4f -2, 4f -3

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Heisenberg Uncertainty Principle p Observation p ? Measurement effects state of system. There is a limit imposed on the degree of certainty to which you can know the position (r) and momentum (p) of a particle r r1r1 r2r2 r ? p r ≥ h/4 p – error in p r – error in r

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Exercise An electron is traveling between 0.11000 C and 0.11500 C. What is the smallest error in the position you can expect? What is the error in position if it were a proton? Need error in momentum p? We know that v = 0.00500 C Therefore p = m v p = (9.109*10 -31 kg)*(0.00500 * 2.998*10 8 m/s) p = 1.36*10 -24 kg m/s For an electron

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Recall that p r ≥ h/4 r ≥ h/(4 p r ≥ (6.626*10 -34 Js)/[(4*3.14159)*(1.36*10 -24 kg m/s)] r ≥ 3.88*10 -11 m r ≥ (6.626*10 -34 Js)/[(4*3.14159)*(2.51*10 -21 kg m/s)] For an proton p = (1.674 × 10 -27 kg)*(0.00500 * 2.998*10 8 m/s) p = 2.51*10 -21 kg m/s p = m v r ≥ 2.10*10 -14 m

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Probability Distribution A particle position and momentum cannot be known exactly Therefore a particle is characterized by a probability distribution function The probability distribution is determined by the wavefunction: P α r 2 (n,l,m,s;r) (x) P(x)

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Hydrogen Orbitals The hydrogen orbital are determined from the wavefunctions Ex) 1s ( 1, 0, 0 r ) 2s ( 2, 0, 0; r ) S orbitals - are spherical, i.e. they are identical in all directions The probability distribution can be graphed as a function of the radius as P(r) = r 2 2

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radial probability density plot Notice the shell structure as n increases Notice the nodes in the wavefunction For 1s 0 nodes For 2s 1 nodes For 3s 2 nodes

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P Orbitals The p orbitals are constructed from the hydrogen wavefunctions with n > 1, and l = 1. (2,1,1) and (2,1,-1), are complex values, and are combined to make them real valued. The resulting functions are aligned along the x and y axis. The remaining function (2,1,0) is aligned along z axis. i.e. (2,1,0), (2,1,-1)

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P Orbitals These dumbbell shaped orbital are referred to as the p (polar) orbitals, which are labeled according to their orientation, 2p x, 2p y, 2p z The number of nodes increases with n as n-1, i.e 1 for 2p Note that when the nodal plane is crossed the orbital changes sign The orbitals are plotted as the boundary enclosing total of 90% probability PxPx PyPy PzPz

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D Orbitals The d orbitals are constructed from the hydrogen wavefunctions with n > 2, and l = 2. (3,2,2) and (3,2,-2) a well as (3,2,1) and (3,2,-1), are combined to make them real valued functions. The four corresponding distribution functions are have four lobes in the xy, xz and yz plane. (in between the axes) i.e. (3,2,2) (3,2,1), (3,2,0), (3,2,-1), (3,2,-2),

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D Orbitals A fourth orbital exists in the xy plane aligned on the axes, the other fits between the axes. The remaining fifth orbital, d z2, resembles a P z orbital with a donut like shape in the xy plane (z 2 -x 2 and z 2 -y 2 are superimposed) Sign changes when nodal plane (cone) is crossed There are 2 nodes 3d

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F Orbitals Constructed from seven H wavefunctions to make them real valued Composed of 8 lobes There are 3 nodes for 4f

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The Orbitals of the Hydrogen Atom 0 nodes 1 node 2 nodes Radial nodes 1 planar node 2 planar nodes

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Concepts Properties of waves (wavelength, frequency, amplitude, speed) Electromagnetic spectrum, speed of light Planck’s equation and Planck’s constant Wave-particle duality (for light, electrons, etc.) Atomic line spectra and relevant calculations Ground vs. excited states Heisenberg uncertainty principle Bohr and Schrödinger models of the atom Quantum numbers (n, l, ml) Shells (n), subshells (s,p,d,f) and orbitals Different kinds of atomic orbitals (s, p, d, f) and nodes

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