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Lecture 6 (Ch. 5): Quantum Numbers and Electron Configurations

Lecture 6 (Ch. 5): Quantum Numbers and Electron Configurations. Dr Harris Suggested HW : ( Ch 5) 5, 7, 8, 9, 10, 17, 20, 43, 44. Bohr’s Theory Thrown Out. In chapter 4, we used Bohr’s model of the atom to describe atomic behavior

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Lecture 6 (Ch. 5): Quantum Numbers and Electron Configurations

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  1. Lecture 6 (Ch. 5): Quantum Numbers and Electron Configurations Dr Harris Suggested HW: (Ch 5) 5, 7, 8, 9, 10, 17, 20, 43, 44

  2. Bohr’s Theory Thrown Out • In chapter 4, we used Bohr’s model of the atom to describe atomic behavior • Unfortunately, Bohr’s mathematical interpretation fails when an atom has more than 1 electron(nonetheless, it still serves as a useful visual representation of an atom) • Also, Bohr has no explanation of why electrons simply don’t fall into the positively charged nucleus. • This failure is due to violation of the Uncertainty principle. 2πr = nλ n= 1, 2, 3… Bohr’s model

  3. Heisenburg’s Uncertainty Principle • The uncertainty principle is a cornerstone of quantum theory • It asserts that “You can NOT measure accuratelyboth the position and momentum of an electron simultaneously, and this uncertainty is a fundamental property of the act of measurement itself” • This limitation is a direct consequence of the wave-nature of electrons

  4. Heisenburg’s Uncertainty Principle • Consider an electron • If you wish to locate the electron within a distance Δx, we must use a photon with a wavelength which is equal to or less than Δx • When the electron and photon interact, there is a change in momentum of the electron due to collision with the photon • Thus, the act of measuring the position results in a change in its momentum λ’ λ

  5. Why does Bohr’s Model Fail? • Bohr’s model conflicts with the uncertainty principle because if the electron is set within a confined orbit, you know both its momentum and position at a given moment. Therefore, it can not hold true.

  6. Contrast Between Bohr’s Theory and Quantum Mechanics • The primary differences between Bohr’s theory and quantum mechanics are: • Bohr restricts the motion of electrons to exact, well-defined orbits • In quantum mechanics, the location of the electron is not known. Instead, “wavefunctions” (Ψ) are used to describe an electron’s wave characteristics • The square of the wavefunction, Ψ2, gives the PROBABILTY DENSITY, or the probability that an electron will be found in some region of volume around the nucleus.

  7. Probability Density??? • This is directly in line with the uncertainty principle. • We CAN NOT locate an electron accurately • We CAN calculate a probability of an electron being in a certain region of space in the atom • From Ψ2, we get ORBITALS • An orbital is a theoretical, 3-D “map” of the places where an electron could be.

  8. Orbitals and the Quantum Numbers • An orbital is defined by 4 quantum numbers • n (principle quantum number) • L (azimuthal quantum number) • mL (magnetic quantum number) • ms (magnetic spin quantum number)

  9. 1st quantum number ‘n’: Principal Quantum Number • n = 1, 2, 3…..etc These numbers correlate to the distance of an electron from the nucleus. In Bohr’s model, these corresponded to the “shell” orbiting the nucleus. • n determines the energies of the electrons • n also determines the orbital size. As n increases, the orbital becomes larger and the electron is more likelyto be found farther from the nucleus

  10. 2nd Quantum Number ‘L’: Aziumthal Quantum Number • L dictates the orbital shape • L is restricted to values of 0, 1….(n-1) • Each value of L has a letter designation. This is how we label orbitals.

  11. Orbital Labeling • Orbitals are labeled by first writing the principal quantum number, n, followed by the letter representation of L

  12. 3rd quantum number ‘mL’: Magnetic quantum number • The 3rd quantum number, mL, relates to the spatial orientation of an orbital • mL can assume all integer values between –L and +L • Number of possible values of mL gives the number of each orbital type in a given shell

  13. Magnetic Quantum Number

  14. S-orbitals • When n=1, the wavefunction that describes this state (Ψ1) only depends on r, the distance from the nucleus. • Because the probability of finding an electron only depends on r and not the direction, the probability density is spherically symmetric • n=1 can only be an s -orbital Since L= 0 for s orbital, mL can only be 0. Hence, each “shell” only has one s-orbital

  15. z z z y y y Spheres shown above represent the volume of space around the nucleus in which an electron may be found. Regions where an electron has 0% probability of existing is called a NODE. x x x node 2 nodes 3s 1s 2s

  16. P-orbitals • P orbitals exist in all shells where n> 2. • For a p orbital, L =1. • Therefore, mL = -1, 0, 1. Hence, there are 3 arrangements of p-orbitals in each shell n> 2 pz • Unlike s-orbitals, p-orbitals are not spherically symmetric. Instead, electron density is concentrated in lobes around the nucleus along either the x, y, or z axis. py px

  17. D-orbitals • D-orbitals exist in all shells where n>3 • L= 2, so mL can be any of the following: -2,-1,0,1,2 • Thus, there are five d-orbitals in every shell where n>3 • 4 of them have a ‘four-leaf clover’ shape. The 5th is shown as two lobes along the z axis with ring of electron density around the nucleus

  18. Questions: • Can a 2d orbital exist? • Can a 1p orbital exist? • Can a 4s orbital exist?

  19. The 4th and final quantum number ‘ms’: Magnetic Spin Number Stern- Gerlach experiment • A beam of Ag atoms was passed through an uneven magnetic field. Some of the atoms were pulled toward the curved pole, others were repelled. • All of the atoms are the same, and have the same charge. Why does this happen?

  20. The 4th and final quantum number ‘ms’: Magnetic Spin Number • Spinning electrons have magnetic fields. The direction of spin changes the direction of the field. If the field of the electron does not align with the magnetic field, it is repelled. • Thus, because the beam splits two ways, electrons must spin in TWO opposite directions with equal probability. We label these “spin-up” and “spin-down” Two possible orientations: ms =

  21. Example: What are the allowed quantum numbers of a 2p electron? 1 1 0 2 n l -1 ml ms Spin up: + ½ Spin down: - ½ -1 0 1 Representation of the 3 p-orbitals

  22. Pauli Exclusion Principle NO TWO ELECTRONS IN THE SAME ATOM CAN HAVE THE SAME 4 QUANTUM NUMBERS!!! 1s 1s Quantum numbers: 1, 0 , 0, + ½ 1, 0, 0, – ½ * Allowed Quantum numbers: 1, 0, 0, + ½ 1, 0, 0 , + ½ * Forbidden !!

  23. Example • List all possible sets of quantum numbers in the n=2 shell? • n = 2 • L = 0, 1 • mL = 0 (L=0) • = -1, 0, 1 (L=1) • ms = +/- ½ S P

  24. Electron Configurations • As previously stated, the energy of an electron depends on n. • Orbitals having the same n, but different L (like 3s, 3p, 3d) have different energies. • When we write the electron configuration of an atom, we list the orbitals in order of energy according to the diagram shown on the left. REMEMBER: S-orbitals can hold no more than TWO electrons. P- orbitals can hold no more than SIX, and D-orbitals can hold no more than TEN electrons.

  25. Example • Write the electron configurations of N, Cl, and Ca Energy

  26. Example, contd. • If we drew the orbital representations of N based on the configuration in the previous slide, we would obtain: Energy For any set of orbitals of the same energy, fill the orbitals one electron at a time with parallel spins. (Hund’s Rule) 2p 2s 1s

  27. Noble Gas Configurations • In chapter 4, we learned how to write Lewis dot configurations. Now that we can assign orbitals to electrons, we can write proper valence electron configurations.

  28. Examples • Give the noble gas configurations of: • K • K+ • Cl- • Zn • Sr

  29. Excited States • When we fill orbitals in order, we obtain the ground state (lowest energy) configuration of an atom. • What happens to the electron configuration when we excite an atom? • Absorbing light with enough energy will bump a valence electron into an excited state. • The electron will move up to the next available orbital. This is the 1st excited state.

  30. Example: Multiple Excited states of Li • Ground state Li: 1s2 2s1 • 1st excited state Li: 1s2 2s0 2p1 • 2nd excited state Li: 1s2 2s0 2p0 3s1 Energy 2nd excited state 3s 2p 1st excited state Ground state 2s 1s

  31. Valence Electron Configurations ns2 np6 ns1 ns2 np2 ns2 np3 ns2 np4 ns2 np1 ns2 np5 1 2 3 4 5 6 7 ns2 ns2 (n-1)dx

  32. Transition Metals • As you know, the d-orbitals hold a max of 10 electrons • These d-orbitals, when possible, will assume a half-filled, or fully-filled configuration by taking an electron from the ns orbital • This occurs when a transition metal has 4 or 9 valence d electrons Example: Cr  [Ar] 4s2 3d4 [Ar] 4s1 3d5 3d 3d Unfavorable Favorable 4s 4s

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