Presentation on theme: "Lecture 17: The Hydrogen Atom"— Presentation transcript:
1 Lecture 17: The Hydrogen Atom Reading: ZuhdahlOutlineThe wavefunction for the H atomQuantum numbers and nomenclatureOrbital shapes and energies
2 Schrodinger EquationErwin Schrodinger develops a mathematical formalism that incorporates the wave nature of matter:Kinetic EnergyThe Hamiltonian:d2/dx2xThe Wavefunction:E = energy
3 Potentials and Quantization (cont.) • What if the position of the particle is constrained by a potential:“Particle in a Box”Potential E= 0 for 0 ≤ x ≤ L= all other x
4 Potentials and Quantization (cont.) • What does the energy look like?n = 1, 2, …Energy is quantizedEyy*y
5 Potentials and Quantization (cont.) • Consider the following dye molecule, the length of which can be considered the length of the “box” an electron is limited to:L = 8 ÅWhat wavelength of light corresponds to DE from n=1 to n=2?(should be 680 nm)
6 H-atom wavefunctionsRecall that the Hamiltonian is composite of kinetic (KE) and potential (PE) energy.• The hydrogen atom potential energy is given by:
7 H-atom wavefunctions (cont.) • The Coulombic potential can be generalized:Z• Z = atomic number (= 1 for hydrogen)
8 H-atom wavefunctions (cont.) • The radial dependence of the potential suggeststhat we should from Cartesian coordinates to sphericalpolar coordinates.r = interparticle distance(0 ≤ r ≤ )e-= angle from “xy plane”(/2 ≤ ≤ - /2)p+ = rotation in “xy plane”(0 ≤ ≤ 2)
9 H-atom wavefunctions (cont.) • If we solve the Schrodinger equation using thispotential, we find that the energy levels arequantized:• n is the principle quantum number, and rangesfrom 1 to infinity.
10 H-atom wavefunctions (cont.) • In solving the Schrodinger Equation, two otherquantum numbers become evident:l, the orbital angular momentum quantum number.Ranges in value from 0 to (n-1).m, the “z component” of orbital angular momentum.Ranges in value from -l to 0 to l.• We can then characterize the wavefunctions based onthe quantum numbers (n, l, m).
11 Orbital ShapesLet’s take a look at the lowest energy orbital, the “1s” orbital (n = 1, l = 0, m = 0)a0 is referred to as the Bohr radius, and = Å11
12 Orbital Shapes (cont.)Note that the “1s” wavefunction has no angular dependence (i.e., Q and F do not appear).Probability =• Probability is spherical
13 Orbital Shapes (cont.)Radial probability (likelihood of finding the electron in each spherical shell
14 Orbital Shapes (cont.) Naming orbitals is done as follows n is simply referred to by the quantum numberl (0 to (n-1)) is given a letter value as follows:0 = s1 = p2 = d3 = f- ml (-l…0…l) is usually “dropped”
15 Orbital Shapes (cont.) • Table 12.3: Quantum Numbers and Orbitals n l Orbital ml # of Orb.ssp , 0,sp , 0,d -2, -1, 0, 1,
16 Orbital Shapes (cont.)• Example: Write down the orbitals associated with n = 4.Ans: n = 4l = 0 to (n-1)= 0, 1, 2, and 3= 4s, 4p, 4d, and 4f4s (1 ml sublevel)4p (3 ml sublevels)4d (5 ml sublevels4f (7 ml sublevels)
17 Orbital Shapes (cont.) s (l = 0) orbitals • r dependence only • as n increases, orbitalsdemonstrate n-1 nodes.
18 Orbital Shapes (cont.) 2p (l = 1) orbitals • not spherical, but lobed. • labeled with respect to orientation along x, y, and z.
19 Orbital Shapes (cont.) 3p orbitals • more nodes as compared to 2p (expected.).• still can be represented by a “dumbbell” contour.
20 Orbital Shapes (cont.) 3d (l = 2) orbitals • labeled as dxz, dyz, dxy, dx2-y2 and dz2.
21 Orbital Shapes (cont.) 4f (l = 3) orbitals • exceedingly complex probability distributions.
22 Orbital Energies • energy increases as 1/n2 • orbitals of same n, but differentl are considered to be of equalenergy (“degenerate”).• the “ground” or lowest energyorbital is the 1s.