Final Exam Tuesday, May 8, 2012 Starting at 8:30 a.m., Hoyt Hall Duration: 2h 30m

Transcription

1 Final Exam Tuesday, May 8, 2012 Starting at 8:30 a.m., Hoyt Hall Duration: 2h 30m

2 Chapter 39 Quantum Mechanics of Atoms

3 Units of Chapter Quantum-Mechanical View of Atoms 39-2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers 39-3 Hydrogen Atom Wave Functions 39-4 Complex Atoms; the Exclusion Principle 39-5 The Periodic Table of Elements 39-7 Magnetic Dipole Moments; Total Angular Momentum

4 39.2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers Potential energy for the hydrogen atom:

5 39.2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers The time-independent Schrödinger equation in three dimensions is then: where

6 37-11 The Bohr Model In each of its orbits, the electron would have a definite energy: E n = (13.6 ev) Z 2 n 2, n = 1,2,3,...

7 37-10 Atomic Spectra: Key to the Structure of the Atom A portion of the complete spectrum of hydrogen is shown here. The lines cannot be explained by the Rutherford theory.

8 39.2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers There are four different quantum numbers needed to specify the state of an electron in an atom. 1. The principal quantum number n gives the total energy. 2. The orbital quantum number gives the angular momentum; can take on integer values from 0 to n - 1.

9 39.2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers 3. The magnetic quantum number, m gives the direction of the electron s angular momentum, and can take on integer values from to +.

10 39.2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers This plot indicates the quantization of angular momentum direction for = 2. The other two components of the angular momentum are not defined.

11 39.2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers 4. The spin quantum number, m s, for an electron can take on the values +½ and -½. The need for this quantum number was found by experiment; spin is an intrinsically quantum mechanical quantity, although it mathematically behaves as a form of angular momentum.

12 39.7 Magnetic Dipole Moments; Total Angular Momentum A magnetic dipole moment is associated to the electron orbital motion: µ = e 2m where L is the quantized angular momentum. The magnitude of the dipole moment is then: L L = l(l +1) µ = e 2m l(l +1)

13 39.7 Magnetic Dipole Moments; Total Angular Momentum The z component of the dipole moment, where z is defined to be the direction of an external magnetic field, is given by: µ = e z 2m L z Because L z is also quantized L z = m l µ z = e with µ B the Bohr magneton 2m m l = µ B m l

14 39.7 Magnetic Dipole Moments; Total Angular Momentum An atom placed in a magnetic field will have its energy levels shifted depending on the value of m l. This is called the Zeeman effect: B z x y U = µ B = µ B B m l

15 39.2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers In a magnetic field, the energy levels of an atom split depending on m l (Zeeman effect).

16 The Zeeman effect appears as the splitting of spectral lines in a strong magnetic field. Zeeman Effect In this case the upper level, with l = 1, splits into three different levels

17 39.7 Magnetic Dipole Moments; Total Angular Momentum In the Stern-Gerlach experiment, atoms are sent through a nonuniform magnetic field. This field deflects atoms differently depending on their magnetic dipole moments. Classically, one would expect a continuum of deflection angles.

18 39.7 Magnetic Dipole Moments; Total Angular Momentum Instead, the angles are quantized, corresponding to the quantized values of the magnetic moment.

19 39.7 Magnetic Dipole Moments For H atoms in the ground state we see only two lines: The orbital momentum is zero (l = 0), therefore the effect must be due to an intrinsic momentum of the electron. µ z = µ B m l µ z = g µ B m s Stern-Gerlach experiment

20 39.2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers This table summarizes the four quantum numbers.

21 39.2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers Conceptual Example 39-1: Possible states for n = 3 in hydrogen. How many different states are possible in hydrogen for an electron whose principal quantum number is n = 3?

22 39.2 Hydrogen Atom: Quantum Numbers Problem 39-9: Show that for the hydrogen atom the number of different electron states of a given value of n is 2n 2. Solution: For a given n, 0 l (n-1). Since for each l the number of possible states is 2(2l + 1), the number of possible states for a given n is as follows: ( ) n 1 n 1 n 1 2( 2l + 1) = 4 l + 2 = 4 n n 1 l=0 l=0 l= n = 2n 2

23 39.2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers Example 39-2: E and L for n = 3. Determine (a) the energy and (b) the orbital angular momentum for a hydrogen electron in each of the hydrogen atom states of Example 39 1.

24 39.2 Hydrogen Atom: Schrödinger Equation and Quantum Numbers Selection Rules: Allowed transitions between energy levels occur between states whose value of differ by one: Other, forbidden, transitions also occur but with much lower probability.

25 39.2 Hydrogen Atom: Quantum Numbers Problem 39-10: A hydrogen atom is excited in a 5d state. (a) Name all the states to which the atom is dipole-allowed to decay trough emission of a photon. (b) How many different wavelengths are there (ignoring fine structure)?

26 39.3 Hydrogen Atom Wave Functions The ground state wave function of the hydrogen atom is: n = 1 l = 0 m l = 0 ψ 100 = 2 r 0 3 e r r 0 1 4π r 0 = nm

27 39.3 Hydrogen Atom Wave Functions The ground state is spherically symmetric; the probability of finding the electron at a distance between r and r + dr from the nucleus is: P(r)dr = ψ 2 dv = ψ 2 4πr 2 dr P(r) = 4πr 2 ψ = 4 r 0 3 r 2 e 2r r 0

28 39.3 Hydrogen Atom Wave Functions Example 39-3: Most probable electron radius in hydrogen. Determine the most probable distance r from the nucleus at which to find the electron in the ground state of hydrogen.

29 39.3 Hydrogen Atom Wave Functions Example 39-4: Calculating probability. Determine the probability of finding the electron in the ground state of hydrogen within two Bohr radii (r 0 = nm) of the nucleus.

30 Spherical Coordinates x = r sinθ cosϕ y = r sinθ sinϕ z = r cosθ 2 f x + 2 f 2 y f z 2 =

31 39.3 Hydrogen Atom Wave Functions n = 1 l = 0 m l = 0 ψ 100 = 2 r 0 3 e r r 0 1 4π n = 2 l = 0 m l = 0 ψ 200 = n = 2 l = 1 m l = +1 ψ 211 = n = 2 l = 1 m l = 0 ψ 210 = n = 2 l = 1 m l = 1 ψ 21 1 = 1 2r r r r r 2r 0 r r 0 r r 0 r r 0 1 e r 2r 0 4π 3 e r 2r 0 8π sinθ eiϕ 3 e r 2r 0 4π cosθ 3 e r 2r 0 8π sinθ e iϕ

32 Comparison of 1s and 2s States The plot of the radial probability density for the 2s state has two peaks The highest value of P corresponds to the most probable value

33 39.3 Hydrogen Atom Wave Functions Probability distributions for n = 2 and = 1 (the distributions for m = +1 and m = -1 are the same), and the radial distribution for all n = 2 states. P(r) dr = ( ψ ψ ψ ) πr 2 dr

34 39.4 Complex Atoms; the Exclusion Complex atoms contain more than one electron, so the interaction between electrons must be accounted for in the energy levels. This means that the energy depends on both n and. Principle A neutral atom has Z electrons, as well as Z protons in its nucleus. Z is called the atomic number.

35 39.4 Complex Atoms; the Exclusion Principle In order to understand the electron distributions in atoms, another principle is needed. This is the Pauli exclusion principle: No two electrons in an atom can occupy the same quantum state. The quantum state is specified by the four quantum numbers; no two electrons can have the same set.

36 39.4 Complex Atoms; the Exclusion Principle This chart shows the occupied and some unoccupied states in He, Li, and Na.

37 39.5 The Periodic Table of the Elements This table shows the configuration of the outer electrons only.

38 39.5 The Periodic Table of the Elements We can now understand the organization of the periodic table. Electrons with the same n are in the same shell. Electrons with the same n and are in the same subshell. The exclusion principle limits the maximum number of electrons in each subshell to 2(2 + 1).

39 39.5 The Periodic Table of the Elements Electron configurations are written by giving the value for n, the letter code for, and the number of electrons in the subshell as a superscript. For example, here is the ground state configuration of sodium (Na): 1s 2 2s 2 2p 6 3s 1

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41 39.5 The Periodic Table of the Elements Conceptual Example 39-5: Electron configurations. Which of the following electron configurations are possible, and which are not: (a) 1s 2 2s 2 2p 6 3s 3 ; (b) 1s 2 2s 2 2p 6 3s 2 3p 5 4s 2 ; (c) 1s 2 2s 2 2p 6 2d 1?

42 39.5 The Periodic Table of the Elements Atoms with the same number of electrons in their outer shells have similar chemical behaviors. They appear in the same column of the periodic table. The outer columns those with full, almost full, or almost empty outer shells are the most distinctive. The inner columns, with partly filled shells, have more similar chemical properties.

43 39.5 The Periodic Table of the Elements Example: Let us apply the exclusion principle to the infinitely high square well. Let there be five electrons confined to this rigid box whose width is l. Each energy level can have a maximum of two electrons, since the only quantum numbers are n and m s. Thus the lowest energy level will have two electrons in the n = 1 state, two electrons in the n = 2 state, and 1 electron in the n = 3 state. E = 2E 1 + 2E 2 + E 3 = 2 1 ( ) ( ) +1 3 ( ) 2 h 2 8ml 2 h2 = ml