1
play

1 3 Nodes Nodes Places where the Probability of finding the - PDF document

Oct 10, 2022 •383 likes •589 views

1 A Closer Look at A Closer Look at Contains Information about the Probability of finding the Quantum Mechanical Entity in a Certain State For atom, know energy so is related to probability of finding electron at a certain

  1. 1 ψ A Closer Look at ψ A Closer Look at • Contains Information about the Probability of finding the Quantum Mechanical Entity in a Certain State – For atom, know energy so ψ is related to probability of finding electron at a certain point in space • The Probability is not ψ , rather ψ 2 – Actually this is ψ * ψ 2 More on Orbitals Orbitals More on • Wavefunctions for Atomic Orbitals can be divided into Two Parts – Radial (depends on distance from nucleus) – Angular (depends on angles φ and θ ) • For Chemistry Angular Part is (most) Important – Molecular shape – Bonding 1

  2. 3 Nodes Nodes • Places where the Probability of finding the Electron is Zero ( ψ = 0 so ψ 2 = 0 ) • When ψ radial is zero, called a radial (or spherical) node – There are n - l - 1 radial nodes • When ψ angular is zero, called an angular node (or a nodal plane ) – There are l angular nodes 4 1s Radial Wavefunction Wavefunction 1s Radial 1.2 3 / 2 ⎛ ⎞ Z ⎜ ⎟ ψ = − Zr / a 1.0 2 e ⎜ ⎟ 0 radial ⎝ ⎠ a 0 0.8 ψ (arbitrary units) l There are n - - 1 = 1 - 0 - 1 = 0 radial nodes. 0.6 Note that ψ ≠ 0 at x = 0. 0.4 0.2 0.0 2 4 6 8 10 12 14 16 -0.2 Distance from Nucleus (arbitrary units) 2

  3. 5 1s Orbital 1s Orbital 6 2s Radial Wavefunction 2s Radial Wavefunction 1.2 3 / 2 ⎛ ⎞ ⎛ − ⎞ 1 Z Z ⎜ ⎟ ⎜ ⎟ − ψ = Zr / 2 a 1.0 2 r e ⎜ ⎟ ⎜ ⎟ 0 radial ⎝ ⎠ ⎝ ⎠ a a 2 2 0 0 0.8 ψ (arbitrary units) l There are n - - 1 = 2 - 0 - 1 = 1 radial node. 0.6 Note that ψ ≠ 0 at x = 0. 0.4 0.2 0.0 2 4 6 8 10 12 14 16 -0.2 Distance from Nucleus (arbitrary units) 3

  4. 7 2s Orbital 2s Orbital 8 3s Radial Wavefunction 3s Radial Wavefunction 1.2 ⎛ ⎞ 3 / 2 2 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 Z ⎜ 4 Z 2 Z ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ − ψ = − + Zr a 2 / 3 6 r r e ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0 1.0 ⎜ ⎟ radial ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ a a 3 a 9 3 ⎝ ⎠ 0 0 0 0.8 ψ (arbitrary units) l There are n - - 1 = 3 - 0 - 1 = 2 radial nodes. 0.6 Note that ψ ≠ 0 at x = 0. 0.4 0.2 0.0 2 4 6 8 10 12 14 16 -0.2 Distance from Nucleus (arbitrary units) 4

  5. 9 3s Orbital 3s Orbital 10 Probability of Finding an Electron Probability of Finding an Electron • Remember ψ 2 , not ψ, is Probability 1.2 ψ 2 for 1s, 2s and 3s orbitals ψ 2 (arbitrary units) 1.0 Note s orbitals have a non-zero 0.8 probability of being found at the nucleus. 0.6 0.4 0.2 0.0 0 2 4 6 8 10 12 14 16 Distance from the Nucleus (arbitrary units) 5

  6. 11 Probability of Finding an Electron Probability of Finding an Electron 0.10 1s Note wherever there was a node ψ 2 = 0, there is no probability that 0.08 2 (arbitrary units) 2s the electron can be found there. 0.06 3s If the electron can’t be in a certain place, how does it get across? 0.04 Ψ 0.02 0.00 0 2 4 6 8 10 12 14 16 Distance from the Nucleus (arbitrary units) 12 Radial Distribution Function Radial Distribution Function • Problem with ψ 2 , it over estimates Probability Close to Nucleus and under estimates it Further Out • Correct by multiplying ψ 2 by 4 π r 2 – Takes into account that a wedge is smaller toward the center than ends – This correction only works for s orbitals 6

  7. 13 Radial Distribution Function Radial Distribution Function Electrons in orbitals with higher n are 6 usually found further from nucleus. 5 4 π r 2 ψ 2 (arbitrary units) 2s Note that 2s and 3s 4 3s electrons have probability of being 3 closer to nucleus than 1s! 1s 2 1 0 0 2 4 6 8 10 12 14 16 Distance from Nucleus (arbitrary units) 14 1s, 2s, and 3s orbitals orbitals 1s, 2s, and 3s 7

  8. 15 Angular Part of Wavefunction Wavefunction Angular Part of • Every Time ψ goes through a node Sign of Wavefunction changes – s orbital has same angular sign throughout – p orbital lobes have different signs – Lobes alternate signs in a d orbital • Difference in Phase p orbital d orbital d z 2 orbital 16 p Orbitals p Orbitals Typical p orbital Typical p orbital When n = 2, then l When n = 2, then l = 0 and 1 = 0 and 1 When n = 2, then l = 0 and 1 Therefore, in n = 2 shell there Therefore, in n = 2 shell there Therefore, in n = 2 shell there are 2 types of orbitals (2 are 2 types of orbitals (2 are 2 types of orbitals (2 subshells) ) subshells subshells) planar node planar node For l l = 0 = 0 m l = 0 For m l = 0 For l = 0 m l = 0 this is a s subshell this is a s subshell this is a s subshell When l When l = 1, there is a = 1, there is a For l For l = 1 m = 1 m l l = = - -1, 0, +1 1, 0, +1 For l = 1 m l = -1, 0, +1 PLANAR NODE thru PLANAR NODE thru this is a p subshell this is a p subshell this is a p subshell the nucleus. the nucleus. with 3 orbitals with 3 orbitals with 3 orbitals 8

  9. 17 p Orbitals Orbitals p The three p The three p orbitals lie 90 o o orbitals lie 90 A p orbital A p orbital apart in space apart in space 18 2p Radial Wavefunction 2p Radial Wavefunction 3 / 2 ⎛ ⎞ ⎛ ⎞ 1 Z 2 Z ⎜ ⎟ ⎜ ⎟ − ψ = Zr / 2 a r e ⎜ ⎟ ⎜ ⎟ 0 1.2 radial ⎝ ⎠ ⎝ ⎠ a a 4 6 0 0 1.0 l There are n - - 1 = 2 - 1 - 1 = 0 radial nodes. 0.8 ψ (arbitrary units) Note that ψ = 0 at x = 0. Only s orbitals have any probability 0.6 density at the nucleus. 0.4 0.2 0.0 2 4 6 8 10 12 14 16 -0.2 Distance from Nucleus (arbitrary units) 9

  10. 19 2p x Orbital 2p x Orbital 20 2p y Orbital 2p y Orbital 10

  11. 21 2p z Orbital 2p z Orbital 22 Degenerate 2p Orbitals Orbitals Degenerate 2p l ), • All 3 orbitals have the same energy (n and l but differ in orientation (m l l ) 11

  12. 23 3p Radial Wavefunction Wavefunction 3p Radial 3 / 2 ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 0.10 1 Z 2 Z 2 Z ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ − ψ = − Zr / a 4 r r e ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 0 ⎜ ⎟ radial ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ a a a ⎝ 3 ⎠ 27 6 0.08 0 0 0 l Number of radial nodes = n - - 1 0.06 Ψ radial ((Z/a 0 ) 3/2 ) Number of radial nodes = 3 - 1 - 1 = 1 0.04 0.02 Distance from Nucleus (Zr/a 0 ) 0.00 5 10 15 20 25 30 -0.02 -0.04 24 3p x Orbital 3p x Orbital 12

  13. 25 3p y Orbital 3p y Orbital 26 3p z Orbital 3p z Orbital 13

  14. 27 d Orbitals Orbitals d When n = 3, what are the values of l ? l = 0, 1, 2 so there are 3 subshells in the shell. For l = 0, m l = 0 ---> s subshell with single orbital For l = 1, m l = -1, 0, +1 ---> p subshell with 3 orbitals For l = 2, m l = -2, -1, 0, +1, +2 ---> d subshell with 5 orbitals 28 d Orbitals Orbitals d typical d orbital planar node planar node s orbitals have no planar nodes ( l = 0) and are spherical. p orbitals have l = 1, have 1 planar node, and are “dumbbell” shaped. This means d orbitals ( l = 2) have 2 planar nodes 14

  15. 29 3d Radial Wavefunction Wavefunction 3d Radial 0.05 3 / 2 2 ⎛ ⎞ ⎛ ⎞ 1 Z 2 Zr ⎜ ⎟ ⎜ ⎟ − ψ = Zr / 3 a e 0.04 ⎜ ⎟ ⎜ ⎟ 0 radial ⎝ ⎠ ⎝ ⎠ a a 81 30 0 0 Ψ radial ((Z/a 0 ) 3/2 ) 0.03 Number of radial nodes = 0 0.02 0.01 0.00 0 5 10 15 20 25 30 35 Distance from Nucleus (Zr/a 0 ) 30 3d xy Orbital 3d xy Orbital 15

  16. 31 3d xz Orbital 3d xz Orbital 32 3d yz Orbital 3d yz Orbital 16

  17. 33 3d z 2 Orbital Orbital 3d z 2 34 3d x 2 Orbital Orbital 3d x 2 2 - y 2 -y 17

  18. 35 d Orbitals Orbitals d 36 f orbitals orbitals f When n = 4, l = 0, 1, 2, 3 so there are 4 subshells in the shell. For l = 0, m l = 0 → s subshell with single orbital For l = 1, m l = -1, 0, +1 → p subshell with 3 orbitals For l = 2, m l = -2, -1, 0, +1, +2 → d subshell with 5 orbitals For l = 3, m l = -3, -2, -1, 0, +1, +2, +3 → f subshell with 7 orbitals 18

  19. 37 f orbitals orbitals f 19

Recommend

More recommend