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Particle in a Box

Particle in a Box. The wavefunctions for the particle are identical to the displacements of a stretched string as it vibrates. where n=1,2,3,…. n is the quantum number It defines a state. . Particle In a Box. Now we know that the allowable energies are :. Where n=1,2,3,….

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Particle in a Box

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  1. Particle in a Box • The wavefunctions for the particle are identical to the displacements of a stretched string as it vibrates. where n=1,2,3,… • n is the quantum number • It defines a state 

  2. Particle In a Box • Now we know that the allowable energies are : Where n=1,2,3,… • This tells us that: • The energy levels for heavier particles are less than those of lighter particles. • As the length b/w the walls decreases, the ‘distance’ b/w energy levels increases. • The energy levels are Quantized. 

  3. Zero Point Energy • A particle in a container CANNOT have zero energy • A container could be an atom, a box, etc. • The lowest energy (when n=1) is: Zero Point Energy • This is in agreement with the Uncertainty Principle: • ∆p and ∆x are never zero, therefore the particle is always moving

  4. Wavefunctions and Probability Densities • Examine the 2 lowest energy functions n=1 and n=2 • We see from the shading that when n=1, 2 is at a maximum @ the center of the box. • When n=2, we see that2 is at a maximum on either side of the center of the box

  5. Wavefunction Summary • The probability density for a particle at a location is proportional to the square of the wavefunction at the point • The wavefunction is found by solving the SchrÖdinger equation for the particle. • When the equation is solved to the appropriate boundary conditions, it is found that the particle can only posses certain discrete energies. • The wavefunctions and their associated energies placed electrons in defined orbitals whose size and shape is determined by the quantum numbers of the particle.

  6. Types of Orbitals s orbital p orbital d orbital

  7. Orbitals • No more than 2 e- assigned to an orbital • Orbitals grouped in s, p, d (and f) subshells s orbitals p orbitals d orbitals

  8. s orbitals p orbitals d orbitals s orbitals p orbitals d orbitals No. orbs. 1 3 5 No. e- 2 6 10

  9. Subshells & Shells • Subshells grouped in shells. • Each shell has a number called thePRINCIPAL QUANTUM NUMBER, n • The principal quantum number of the shell is the number of the period or row of the periodic table where that shell begins.

  10. n = 1 n = 2 n = 3 n = 4 Subshells & Shells

  11. QUANTUM NUMBERS The shape, size, and energy of each orbital is a function of 3 quantum numbers: n(major; Principal) shell Determines the energy and size of the orbital • The bigger the number, the higher the energy and the larger the orbital radius l (angular or Azimuthal) subshell l = n – 1 Determines the shape of the orbital ml(magnetic) designates an orbital within a subshell ml = l, l - 1, …, -l

  12. QUANTUM NUMBERS Symbol Values Description n (major) 1, 2, 3, .. Orbital size and energy where E = -R(1/n2) l (angular) 0, 1, 2, .. n-1 Subshell (Orbital Shape) ml (magnetic) -l..0..+l Orbital orientation # of orbitals in subshell = 2l + 1

  13. s Orbitals— Always Spherical Dot picture of electron cloud in 1s orbital. Surface density 4πr2y versus distance Surface of 90% probability sphere

  14. p Orbitals When n = 2, then l = 0 and 1 Therefore, in n = 2 shell there are 2 types of orbitals — 2 subshells For l = 0 ml = 0 this is a s subshell For l = 1 ml = -1, 0, +1 this is a p subshell with 3 orbitals When s = 1, there is a PLANARNODE thru the nucleus.

  15. p Orbitals The three p orbitals lie 90o apart in space

  16. d Orbitals When n = 3, what are the values of l? l = 0, 1, 2 and so there are 3 subshells in the shell. For l = 0, ml = 0 s subshell with single orbital For l = 1, ml = -1, 0, +1 p subshell with 3 orbitals For l = 2, ml = -2, -1, 0, +1, +2 d subshell with 5 orbitals

  17. d Orbitals s orbitals have no planar node (l = 0) and so are spherical. p orbitals have l = 1, and have 1 planar node, and so are “dumbbell” shaped. This means d orbitals (with l = 2) have 2 planar nodes

  18. f Orbitals When n = 4, s = 0, 1, 2, 3 so there are 4 subshells in the shell. For l = 0, ml = 0 s subshell with single orbital For l = 1, ml = -1, 0, +1 p subshell with 3 orbitals For l = 2, ml = -2, -1, 0, +1, +2 d subshell with 5 orbitals For l = 3, ml = -3, -2, -1, 0, +1, +2, +3 f subshell with 7 orbitals

  19. Arrangement of Electrons in Atoms Each orbital can be assigned no more than 2 electrons! This is tied to the existence of a 4th quantum number, the electron spin quantum number, ms.

  20. Electron Spin Quantum Number, ms Can be proved experimentally that electron has an intrinsic property referred to as “spin.” Two spin directions are given by ms where ms = +1/2 and -1/2.

  21. Electron Spin and Magnetism • Diamagnetic: NOT attracted to a magnetic field • Paramagnetic: substance is attracted to a magnetic field. • Substances with unpaired electrons are paramagnetic.

  22. QUANTUM NUMBERS Now there are four! n shell 1, 2, 3, 4, ... l subshell 0, 1, 2, ... n - 1 ml orbital in subshell - l ... 0 ... + l ms electron spin +1/2 and -1/2

  23. Pauli Exclusion Principle No two electrons in the same atom can have the same set of 4 quantum numbers. That is, each electron has a unique address.

  24. Hund’s Rule • You must add electrons to unoccupied orbitals of a subshell before doubly occupying any of them. • Very critical in determining the filling order for electron shells.

  25. Arrangement of Electrons in Atoms Electrons in atoms are arranged as SHELLS (n) SUBSHELLS (l) ORBITALS (ml)

  26. Electrons in Atoms When n = 1, then l = 0 this shell has a single orbital (1s) to which 2e- can be assigned. When n = 2, then l = 0, 1 and ml = -1, 0, +1 2s orbital 2e- three 2p orbitals 6e- TOTAL = 8e-

  27. Electrons in Atoms When n = 3, then l = 0, 1, 2 and ml = -2, -1, 0, +1, +2 3s orbital 2e- three 3p orbitals 6e- five 3d orbitals 10e- TOTAL = 18e-

  28. Electrons in Atoms When n = 4, then l = 0, 1, 2, 3 and ml = -3,-2, -1, 0, +1, +2, +3 4s orbital 2e- three 4p orbitals 6e- five 4d orbitals 10e- seven 4f orbitals 14e- TOTAL = 32e-

  29. Assigning Electrons to Atoms • Electrons generally assigned to orbitals of successively higher energy. • For H atoms, E = - C(1/n2). E depends only on n. • For many-electron atoms, energy depends on both n and l.

  30. Electron Filling OrderPrint this chart out and use it at home!

  31. Electron Filling Order

  32. Effective Nuclear Charge, Zeff • Zeff is the nuclear charge experienced by the outermost electrons. • Explains why E(2s) < E(2p) • Zeff increases across a period owing to incomplete shielding by inner electrons. • Estimate Zeff = [ Z - (no. inner electrons) ] • Charge felt by 2s e- in Li Zeff = 3 - 2 = 1 • Be Zeff = 4 - 2 = 2 • B Zeff = 5 - 2 = 3 and so on!

  33. Electron cloud for 1s electrons Effective Nuclear Charge Zeff is the nuclear charge experienced by the outermost electrons. This will help explain some of the periodic properties we’ll examine later this chapter.

  34. spdf notation for H, atomic number = 1 1 no. of s 1 electrons value of l value of n Writing Atomic Electron Configurations Two ways of writing configs. One is called the spdf notation.

  35. Writing Atomic Electron Configurations Two ways of writing configs. Other is called the orbital box notation. One electron has n = 1, s = 0, ms = 0, ms = + 1/2 Other electron has n = 1, s = 0, ms = 0, ms = - 1/2

  36. Electron Configurations and the Periodic Table See Active Figure 7.4

  37. Lithium Group 1A Atomic number = 3 1s22s1f 3 total electrons

  38. Beryllium Group 2A Atomic number = 4 1s22s2f 4 total electrons

  39. Boron Group 3A Atomic number = 5 1s2 2s2 2p1f 5 total electrons

  40. Carbon Group 4A Atomic number = 6 1s2 2s2 2p2f 6 total electrons Here we see for the first time HUND’S RULE. When placing electrons in a set of orbitals having the same energy, we place them singly as long as possible.

  41. Nitrogen Group 5A Atomic number = 7 1s2 2s2 2p3f 7 total electrons

  42. Oxygen Group 6A Atomic number = 8 1s2 2s2 2p4f 8 total electrons

  43. Fluorine Group 7A Atomic number = 9 1s2 2s2 2p5f 9 total electrons

  44. Neon Group 8A Atomic number = 10 1s2 2s2 2p6f 10 total electrons Note that we have reached the end of the 2nd period, and the 2nd shell is full!

  45. Electron Configurations of p-Block Elements PLAY MOVIE

  46. Sodium Group 1A Atomic number = 11 1s2 2s2 2p6 3s1 or “neon core” + 3s1 [Ne] 3s1 (uses rare gas notation) Note that we have begun a new period. All Group 1A elements have [core]ns1 configurations.

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