n m ℓ 1Z a = e 0 - Mercyhurst...

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Normalized Wave Functions for Hydrogen Atom s orbitals Quantum numbers n m 1 0 0 2 0 0 3 0 0 Radial Wave Functions R(r) for Hydrogen Atom Quantum numbers n R(r) 1 0 2 0 3 0 Angular Wave Functions ΘΦ(Өφ) for Hydrogen Atom Quantum numbers m 0 0 a 0 = (4 π ε 0 ħ n 2 ) / (m e e 2 Z) 0 32 Ζ r a 1s 0 1 Z ψ = e a π 0 32 Ζ r 2a 2s 0 0 1 Z Ζ ψ = 2 re a a 4 2π 0 32 Ζ 2 r 3a 2 3s 2 0 0 0 1 Z 18Ζ 2Ζ ψ = 27 r+ r e a a a 81 3π 0 0 32 Ζ r a Z 2 e a 0 32 Ζ r 2a o 0 1 Z Ζ 2 r e a a 2 2 0 32 Ζ 2 r 3a 2 2 o 0 0 2 Z 18Ζ 2Ζ 27 r + r e a a a 81 3 1 2 π

Transcript of n m ℓ 1Z a = e 0 - Mercyhurst...

Page 1: n m ℓ 1Z a = e 0 - Mercyhurst Universityplato.mercyhurst.edu/chemistry/kjircitano/InorgStudysheets/Inorg... · Normalized Wave Functions for Hydrogen Atom s orbitals Quantum numbers

Normalized Wave Functions for Hydrogen Atom s orbitals

Quantum numbers n ℓ mℓ 1 0 0 2 0 0 3 0 0 Radial Wave Functions R(r) for Hydrogen Atom Quantum numbers n ℓ R(r) 1 0 2 0 3 0 Angular Wave Functions ΘΦ(Өφ) for Hydrogen Atom Quantum numbers ℓ mℓ 0 0 a0 = (4 π ε0 ħ n2) / (me e2 Z)

0

3 2 Ζ ra

1s0

1 Zψ = eaπ

0

3 2 Ζ r2a

2s0 0

1 Z Ζψ = 2 r ea a4 2π

0

3 2 Ζ2 r3a2

3s 20 0 0

1 Z 18Ζ 2Ζψ = 27 r + r ea a a81 3π

0

0

3 2 Ζ raZ2 e

a

0

3 2 Ζ r2a

o 0

1 Z Ζ 2 r ea a2 2

0

3 2 Ζ2 r3a2

2o 0 0

2 Z 18Ζ 2Ζ 27 r + r ea a a81 3

12 π

Page 2: n m ℓ 1Z a = e 0 - Mercyhurst Universityplato.mercyhurst.edu/chemistry/kjircitano/InorgStudysheets/Inorg... · Normalized Wave Functions for Hydrogen Atom s orbitals Quantum numbers

Radial Plots of the 1s Orbital

Angular Plot Angular Probability Plot

Electron Density (Contour) Plot 1s 2s 3s 4s 5s 6s

Page 3: n m ℓ 1Z a = e 0 - Mercyhurst Universityplato.mercyhurst.edu/chemistry/kjircitano/InorgStudysheets/Inorg... · Normalized Wave Functions for Hydrogen Atom s orbitals Quantum numbers

Normalized Wave Functions for Hydrogen Atom p orbitals

Quantum numbers n ℓ mℓ 2 1 0 2 1 ±1 3 1 0 3 1 ±1 Radial Wave Functions R(r) for Hydrogen Atom Quantum numbers n ℓ R(r) 2 1 3 1 Angular Wave Functions ΘΦ(Өφ) for Hydrogen Atom Quantum numbers ℓ mℓ 1 0 ±1 a0 = (4 π ε0 ħ n2) / (me e2 Z)

0

z

3 2 Z r2a

2p0 0

1 Z Zψ = re cosθa a4 2π

φ

0

x y

3 2 Z r2a

2p ,2p0 0

1 Z Zψ = re sinθ cosa a4 2π

0

z

3 2 Z2 r3a2

3p 20 0 0

2 Z Z Zψ = r r e cosθa a a81 π

− 6−

φ

0

x y

3 2 Z2 r3a2

3p ,p 20 0 0

2 Z Z Zψ = r r e sinθ cosa a a81 π

− 6−

o

3 2 Z r2a

o o

1 Z Z r ea a2 6

0

3 2 Z2 r3a2

2o 0 0

4 Z Z Z r r ea a a81 6

− 6−

1 3 cos θ2 π

( )φ1 3 sin θ cos2 π

Page 4: n m ℓ 1Z a = e 0 - Mercyhurst Universityplato.mercyhurst.edu/chemistry/kjircitano/InorgStudysheets/Inorg... · Normalized Wave Functions for Hydrogen Atom s orbitals Quantum numbers

Radial Plots of the 2p Orbital

Radial Plots of the 3p Orbital

Angular Plot

Page 5: n m ℓ 1Z a = e 0 - Mercyhurst Universityplato.mercyhurst.edu/chemistry/kjircitano/InorgStudysheets/Inorg... · Normalized Wave Functions for Hydrogen Atom s orbitals Quantum numbers

Angular Probability Plot

Electron Density (Contour) Plot 2p 3p 4p 5p 6p

Page 6: n m ℓ 1Z a = e 0 - Mercyhurst Universityplato.mercyhurst.edu/chemistry/kjircitano/InorgStudysheets/Inorg... · Normalized Wave Functions for Hydrogen Atom s orbitals Quantum numbers

Normalized Wave Functions for Hydrogen Atom d orbitals

Quantum numbers n ℓ mℓ 3 2 0 3 2 ±1 3 2 ±2 Radial Wave Functions R(r) for Hydrogen Atom Quantum numbers n ℓ R(r) 3 2 Angular Wave Functions ΘΦ(Өφ) for Hydrogen Atom Quantum numbers ℓ mℓ 2 0 ±1 ±2 a0 = (4 π ε0 ħ n2) / (me e2 Z)

( )

0

2z

3 2 Z2 r3a2 2

3d 20 0

1 Z Zψ = r e 3cos θ 1a a81 6π

0

xz yz

3 2 Ze2 r3a2

3d ,3d 20 0

2 Z Zψ = r e sinθ cosθ cosφa a81 π

0

xy 2 2x y

3 2 Z2 r3a2 2

3d ,3d 20 0

1 Z Zψ = r e sin θ cos2φa a81 2π

o

3 2 Z23a2

2o 0

4 Z Z r ea a81 30

( )21 5 3cos θ 14 π

1 15 cos θ sin θ cos 2 π

φ

21 15 sin θ cos 24 π

φ

Page 7: n m ℓ 1Z a = e 0 - Mercyhurst Universityplato.mercyhurst.edu/chemistry/kjircitano/InorgStudysheets/Inorg... · Normalized Wave Functions for Hydrogen Atom s orbitals Quantum numbers

Radial Plots of the 3d Orbital

Angular Plot

Angular Probability Plot

Page 8: n m ℓ 1Z a = e 0 - Mercyhurst Universityplato.mercyhurst.edu/chemistry/kjircitano/InorgStudysheets/Inorg... · Normalized Wave Functions for Hydrogen Atom s orbitals Quantum numbers

Electron Density (Contour) Plot

3 dx2 – y2 3 dx2 – y2 3 dz2