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**Lecture 18: The Hydrogen Atom**

Reading: Zumdahl Outline The wavefunction for the H atom Quantum numbers and nomenclature Orbital shapes and energies

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H-atom wavefunctions Recall from the previous lecture that the Hamiltonian is composite of kinetic (KE) and potential (PE) energy. • The hydrogen atom potential energy is given by:

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**The Coulombic potential can be generalized:**

• Z = atomic number (= 1 for hydrogen)

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H-atom wavefunctions The radial dependence of the potential invites us to switch from Cartesian to spherical polar coordinates to facilitate the q. m. solution (“Separation of variables”) r = interparticle distance (0 ≤ r ≤ ) e- = angle from “xy plane” (/2 ≤ ≤ - /2) p+ = rotation in “xy plane” (0 ≤ ≤ 2)

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H-atom wavefunctions If we solve the Schrodinger equation using this potential, we find that the energy levels are quantized: ‘n’ is the principle quantum number, and ranges in value from 1 to infinity.

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**H-atom wavefunctions • In solving the Schrodinger Equation, two other**

quantum numbers become evident: l, the orbital angular momentum quantum number. Ranges in value 0, 1, 2, … (n-1). ml, the “z component” of orbital angular momentum. Ranges in value from - l to 0 to l. We can then characterize the wavefunctions based on the quantum numbers (n, l., m).

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Orbital Shapes Let’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 = Å 1

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**Note that the “1s” wavefunction has no angular dependence (i. e**

Note that the “1s” wavefunction has no angular dependence (i.e., Q and F do not appear). Probability = • Probability is spherical

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**Naming orbitals is done as follows:**

Principle q.n. ‘n’ is simply referred to as 1,2,3,… etc The quantum number l (0 to (n-1)) is given a letter value as follows: l 0 = s 1 = p 2 = d 3 = f ml (- l …,0,… l) subscript is usually dropped

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**Orbital Shapes (cont.) • Table 12.3: Quantum Numbers and Orbitals**

n l Orbital ml # of Orb. s s p , 0, s p , 0, d -2, -1, 0, 1,

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**Which of the following sets of quantum numbers (n, l, m)**

is not allowed? (3, 2, 2). B. (0, 0, 0). C. (1, 0, 0). D. (2, 1, 0).

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Orbital Shapes (cont.) • Example: Write down the orbitals associated with n = 4. Ans: n = 4 l = 0 to (n-1) = 0, 1, 2, and 3 = 4s, 4p, 4d, and 4f 4s (1 ml sublevel) 4p (3 ml sublevels) 4d (5 ml sublevels 4f (7 ml sublevels)

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**Orbital Shapes (cont.) s (l = 0) orbitals • r dependence only**

• as n increases, orbitals demonstrate n-1 nodes.

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**Orbital Shapes (cont.) 2p (l = 1) orbitals • not spherical, but lobed.**

• labeled with respect to orientation along x, y, and z.

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**Orbital Shapes (cont.) 3p orbitals**

• more nodes as compared to 2p (expected.). • still can be represented by a “dumbbell” contour.

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**Orbital Shapes (cont.) 3d (l = 2) orbitals**

• labeled as dxz, dyz, dxy, dx2-y2 and dz2.

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Orbital Shapes (cont.) 3d (l = 2) orbitals • dxy • dx2-y2

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Orbital Shapes (cont.) 3d (l = 2) orbitals • dz2

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**What orbital is depicted in the following animation?**

A. 3dxy B. 3dz2 C. 3dxz D. 2s

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**Orbital Shapes (cont.) 4f (l = 3) orbitals**

• exceedingly complex probability distributions.

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**Orbital Energies l are considered to be of equal**

energy increases as 1/n2 orbitals of same n, but different l are considered to be of equal energy (“degenerate”). the “ground state” or lowest energy orbital is the 1s.

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**Which orbital is expected to have the greatest energy?**

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