Quantum Numbers  n,l,ml :

1. Quantum Numbers n,l,ml: n = 1  l = 0  l = 0 therefore ml = 0 Giving 1s or 1,0,0 or 1s orbital n = 1 has 1 orbital n = 2  l = 0, 1 l = 0 therefore ml = 0 Giving 2s or 2,0,0 or 2s orbital l = 1 therefore ml = -1,0,1 Giving 2,1,0 or 2pz orbital 2,1,1 AND2,1,-1or 2pX and 2pY n = 2 has 4 orbitals n = 3  l = 0, 1, 2 l = 0 therefore ml = 0 Giving 3s or 3,0,0 or 3s l = 1 therefore ml = -1,0,1 Giving 3pz, 3pX and 3pY l = 2 therefore ml = -2,-1,0,1,2 Giving 5 orbitals termed 3d n = 3 has 9 orbitals

2. Quantum Numbers n,l,ml: n = 4  l = 0, 1, 2, 3  l = 0 therefore ml = 0 Giving 4s or 4,0,0 or 4s l = 1 therefore ml = -1,0,1 Giving 4pz, 4pX and 4pY l = 2 therefore ml = -2,-1,0,1,2 Giving 5 orbitals termed 4d l = 3 therefore ml = -3,-2,-1,0,1,2,3 Giving 7 orbitals termed 4f n = 4 has 16 orbitals (N) l =2  4d (5 orbitals) • l =0  4s (1orbital) • l =3  4f(7 orbitals) • l =1  4p(3 orbitals)

3. 1 4 9 16

4. The energy levels for H-atom Energy is quantized: En = f (n) Where n is principal quantum number

5. 2 orbitals are degenerate = 2 orbitals have same energy Degeneracy = number of orbitals with same energy = n2

6. Orbital SHAPES S-ORBITAL

7. Orbital Representation • An orbital is represented most clearly by a probability distribution • An orbital is also represented by a boundary surface that surrounds 90% of the total electron density z Spherical coordinates y x

8. Nodal Surfaces A nodal surface (or a node) is a surface where the probability of finding the electron is zero for a particular orbital (or wavefunction)-- e-probability distribution Solve for  = 0 you obtain the positions of the nodes boundary surface

9. TAKE-HOME EXCERCISE r=? r=? 3s orbital: 2 radial nodes

10. Solution to Node Problem 3s = 0 for which r Solve the quadratic equation:

11. The p-orbitals l=1 2Px 2Py 2Pz + - - + + - Each has one nodal surface (nodal plane) The plane between the lobes containing the nucleus

12. Representation of the 2p orbitals. (a) The electron probability distribution for a 2p orbital. (b) The boundary surface representations of all three 2p orbitals. Boundary surface Electron probability distribution

13. Nodal Surface of the 3p-orbital X- axis

14. Nodal Surface of the 3p-orbital X- axis Yz plane Angular Node Radial Node: at a particular r

15. Total number of nodes = n-1 • Number of angular nodes = l • Number of radial nodes = n-1-l • 3p orbitals: • Total number of nodes = n-1=2 • Number of angular nodes = l=1 • Number of radial nodes = n-1-l=1 • 3d orbitals: • Total number of nodes = n-1=2 • Number of angular nodes = l=2 • Number of radial nodes = n-1-l=0

16. The d-orbitals: l= 2First one at n=3 (impossible state at n=1 and n=2) + - - - + + + - - - + + + + - - - + +

17. The f-orbitals: l=3 First one at n=4 (impossible state at n=1, 2, 3)

18. Summary of the H-atom (1) 1. In the quantum mechanical model, the electron is viewed as a standing wave. This representation leads to a series of wave functions (orbitals) that describe the possible energies and spatial distribution available for the electron

19. Summary of the H-atom (2) 2. In agreement with the Heisenberg Uncertainty Principle the model cannot specify the detailed electron motions. Instead, the square of the wavefunction represents the probability distribution of the electron in that orbital. Orbitals are pictured in terms of probability distributions or electron density maps

20. Summary of the H-atom (3) 3. The size of the orbital is arbitrarily defined as the surface that encloses 90% of the total electron probability

21. Summary of the H-atom (4) 4. The Hydrogen atom has many types of orbitals. In the ground state, the single electron resides in the 1s orbital. The electron can be excited to higher- energy orbitals if energy is put into the atom