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Principal Quantum Number, n

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Principal Quantum Number, n

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  1. Colloquium Speaker Physics and AstronomyFriday, November 7, 2008, 4 pm, CF 316Refreshments 3:30 Physics/Astronomy Conference Room CF 386Charles ClarkWWU Physics BS AlumniChief, Electron and Optical Physics DivisionNational Institute of Standards and TechnologyMatter Wave MagicLouis de Broglie suggested in 1924 that particles, when observed close up, behave like waves. I present some elementary visual demonstrations of matter waves, and explore the connections between matter and light

  2. In 1923, Louis de Broglie, a French physicist, reasoned that particles (matter) might also have wave properties. • The wavelength of a particle of mass, m (kg), and velocity, v (m/s), is given by the de Broglie relation:

  3. A wave function for an electron in an atom is called an atomic orbital (described by three quantum numbers—n, l, ml). It describes a region of space with a definite shape where there is a high probability of finding the electron. • We will study the quantum numbers first, and then look at atomic orbitals.

  4. Principal Quantum Number, n • This quantum number is the one on which the energy of an electron in an atom primarily depends. The smaller the value of n, the lower the energy and the smaller the orbital. • The principal quantum number can have any positive value: 1, 2, 3, . . . • Orbitals with the same value for n are said to be in the same shell.

  5. Shells are sometimes designated by uppercase letters: Letter n K 1 L 2 M 3 N 4 . . .

  6. Angular Momentum Quantum Number, l • This quantum number distinguishes orbitals of a given n (shell) having different shapes. • It can have values from 0, 1, 2, 3, . . . to a maximum of (n– 1). • For a given n, there will be n different values of l, or n types of subshells. • Orbitals with the same values for n and l are said to be in the same shell and subshell.

  7. Subshells are sometimes designated by lowercase letters: l Letter 0 s 1 p 2 d 3 f . . . n≥ 1 2 3 4 Not every subshell type exists in every shell. The minimum value of n for each type of subshell is shown above.

  8. Magnetic Quantum Number, ml • This quantum number distinguishes orbitals of a given n and l—that is, of a given energy and shape but having different orientations. • The magnetic quantum number depends on the value of l and can have any integer value from –l to 0 to +l. Each different value represents a different orbital. For a given subshell, there will be (2l + 1) values and therefore (2l + 1) orbitals.

  9. Let’s summarize: • When n = 1, l has only one value, 0. • When l = 0, mlhas only one value, 0. • So the first shell (n = 1) has one subshell, an s-subshell, 1s. That subshell, in turn, has one orbital.

  10. When n = 2, lhas two values, 0 and 1. • When l = 0, mlhas only one value, 0. So there is a 2s subshell with one orbital. • When l = 1, mlhas only three values, -1, 0, 1. So there is a 2p subshell with three orbitals.

  11. When n = 3, lhas three values, 0, 1, and 2. • When l = 0, ml has only one value, 0. So there is a 3s subshell with one orbital. • When l = 1, mlhas only three values, -1, 0, 1. So there is a 3p subshell with three orbitals. • When l = 2, mlhas only five values, -2, -1, 0, 1, 2. So there is a 3d subshell with five orbitals.

  12. We could continue with n =4 and 5. Each would gain an additional subshell (f and g, respectively). • In an f subshell, there are seven orbitals; in a g subshell, there are nine orbitals. • Table 7.1 gives the complete list of permitted values for n, l, and ml up to the fourth shell. It is on the next slide.

  13. The figure shows relative energies for the hydrogen atom shells and subshells; each orbital is indicated by a dashed-line.

  14. Spin Quantum Number, ms • This quantum number refers to the two possible orientations of the spin axis of an electron. • It may have a value of either +1/2 or -1/2.

  15. Which of the following are permissible sets of quantum numbers? • n = 4, l = 4, ml = 0, ms = ½ • n = 3, l = 2, ml = 1, ms = -½ • n = 2, l = 0, ml = 0, ms = ³/² • n = 5, l = 3, ml = -3, ms = ½ (a) Not permitted. When n = 4, the maximum value of l is 3. (b) Permitted. (c) Not permitted; ms can only be +½ or –½. (b) Permitted.

  16. Atomic Orbital Shapes • An s orbital is spherical. • A p orbital has two lobes along a straight line through the nucleus, with one lobe on either side. • A d orbital has a more complicated shape.

  17. The cross-sectional view of a 1s orbital and a 2s orbital highlights the difference in the two orbitals’ sizes.

  18. The cutaway diagrams of the 1s and 2s orbitals give a better sense of them in three dimensions.

  19. The next slide illustrates p orbitals. • Figure A shows the general shape of a p orbital. • Figure B shows the orientations of each of the three p orbitals.

  20. The complexity of the d orbitals can be seen below

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