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The Electronic Structure of Atoms

The Electronic Structure of Atoms. Electromagnetic Radiation. A wave is a vibration by which energy is transmitted The wavelength, , is the distance between two identical points on the wave The frequency, , is the number of identical points on a wave that pass a given point per second

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The Electronic Structure of Atoms

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  1. The Electronic Structure of Atoms

  2. Electromagnetic Radiation • A wave is a vibration by which energy is transmitted • The wavelength, , is the distance between two identical points on the wave • The frequency, , is the number of identical points on a wave that pass a given point per second • The amplitude is the height measured from the median line to a crest or a trough

  3. Electromagnetic Radiation • The velocity of the propagation of a wave, v, is given by • Electromagnetic radiation travels at 3.00 x 108 m s-1, the speed of light, c, • Example: What is the length (in meters) of an electromagnetic wave that has a frequency of 3.64 x 107 Hz (1 Hz = 1 s-1) • Solution:

  4. Electromagnetic Radiation

  5. Planck’s Quantum Theory • When a solid is heated, it emits radiation • With classical physics, we cannot adequately describe this phenomenon • Planck’s hypothesis was that matter could emit (or absorb) energy in discrete amounts , i.e., quanta • A quantum is the smallest amount of energy that can be emitted (or absorbed) in the form of electromagnetic radiation

  6. Planck’s Quantum Theory • The energy of a quantum, E, is given by • h is the Planck constant and its value is 6.63 x 10-34 J s • According to Planck’s theory, emitted energy (or absorbed) is an integer multiple of h (h, 2h, 3h, …) and never a fraction of h • Planck could not explain why energy is quantized, but his theory was able to accurately describe the radiation emitted by a hot object • Nobel Prize in 1918

  7. The Photoelectric Effect • Classical physics cannot describe the photoelectric effect, i.e., a certain minimum frequency (frequency threshold) is required to eject an electron from a metal and the energy of the ejected electrons does not depend on the intensity of light • Not possible if the light is simply a wave • Einstein proposed that light is a stream of particles, called photons, and the energy of a photon, E, is given by • N.B. Same formula as the Planck equation

  8. The Photoelectric Effect • To eject an electron from a metal, a photon with enough energy (with a high enough frequency) must hit the metal • If the metal is irradiated with light of too low of a frequency (even if the light is intense), the photons do not have enough energy to eject an electron • The energy of the ejected electron is equal to the "surplus” energy that the photon had • If the light is intense, more photons strike the metal, and more electrons are ejected (if the frequency is high enough), but their energy remain the same • Nobel Prize 1921

  9. The Photoelectric Effect • Einstein's work forced scientists of his time to accept the fact that light: • Acts sometimes like a wave • Acts sometimes like a beam of particles • The wave/particle duality is not unique to light as matter also has it (we will see this shortly). • Example: The energy of a photon is 5.87 x 10-20 J. What is its wavelength (in nanometers)? • Solution:

  10. Bohr Model of the Atom • The emission spectrum of a substance is a spectrum, continuous or discontinuous, of the radiation emitted by the substance • We can observe the emission spectrum by heating the substance at very high temperature or by hitting it with a beam of energetic electrons • A condensed phase will typically have a continuous emission spectrum • However, atoms in a gaseous state emit light at specific wavelengths (a line spectrum)

  11. Bohr Model of the Atom • In the Bohr model for the hydrogen atom, Bohr assumed that the electron revolves around the proton in a circular orbit like a planet around the Sun. • However, in the Bohr model, the electron's circular orbit can just do well defined radii • The radiation emitted by a hydrogen atom is attributed to the release of a quantum of energy when the electron jumps from a higher orbit to a lower orbit (and vice versa for the absorption of light)

  12. Bohr Model of the Atom • Each orbit is associated with a principal quantum number, n, which must be a positive integer (n = 1, 2, 3, …) • The energy of an electron in the orbit with the principle quantum number n is given by where RH is the Rydberg constant (2.18 x 10-18 J) • The lowest energy level (n = 1) is the ground level (or state) • All of the other levels are excited levels (or states) • The energy of the electron increases as it is farther from the nucleus and therefore more weakly retained by the nucleus

  13. The Dual Nature of the Electron • In 1924, de Broglie proposed that if light had a wave/particle duality, why not matter as well • According to de Broglie, the electron of the hydrogen atom behaves as a stationary wave, i.e., the positions of the nodes are fixed • Therefore: where λ is the wavelength, r is the radius of orbit n, and n is a positive integer (n = 1, 2, 3, ….)

  14. The Dual Nature of the Electron • The work of Bohr found that where me and v are the mass and velocity of the electron and n is the principle quantum number (n = 1, 2, 3, ….) • Thus, de Broglie proposed that for the electron, and, in general, for any particle

  15. The Dual Nature of the Electron • de Broglie demonstrated that any moving particle has wave properties • Nobel Prize 1929 • Example: Calculate the wavelength (a) of a tennis ball (60 g) that travels at 62 m/s and (b) of an electron that travels at 62 m/s. • Solution:

  16. Heisenberg’s Uncertainty Principle • If a particle such as an electron, has an important wave behavior, how can we describe its movement? • Heisenberg proposed the Uncertainty Principle: • It is impossible to know simultaneously, with certainty, the momentum and the position of a particle • Heisenberg’s Uncertainty Principle says that the more we know about the position, the less we know about the momentum, and vice versa • N.B. Heisenberg’s Uncertainty Principle is not due to experimental limitations but rather a fundamental law of nature • Nobel Prize 1932

  17. The Schrödinger Equation • In 1926, Schrödinger proposed a general method for describing the behavior of microscopic particles • The Schrödinger equation is H = E where  is the wave function, E is the energy of the system, and H is the Hamiltonian of the system (this concept is beyond this course) • 2gives us the probability of finding an electron at a point in space • We can only speak of probability because of Heisenberg’s uncertainty principle • N.B. Classical Newtonian mechanics can describe the motion of a particle with perfect precision, but Schrödinger’s quantum mechanics can only talk about probabilities • Nobel Prize 1933

  18. Quantum Mechanics Applied to the Hydrogen Atom • The Schrödinger equation for a hydrogen atom gives all possible wavefunctions and energies for the hydrogen atom • The electronic density, 2, is the probability density of the presence of an electron per unit volume for a given state described by the wavefunction • Rather than talking about the orbit of an electron, we’re talking about an atomic orbital • An atomic orbital is like the wavefunction of an electron in an atom

  19. Polyelectronic Atoms • No exact solution to the Schrödinger equation is known for systems with two or more electrons • We make the approximation that the electrons in a polyelectronic atom are in atomic orbitals that resemble those found within the hydrogen atom (for which the exact solutions are known) • N.B. The 1s, 2s, 2p, 3s, 3p, 3d, 4s, …., orbitals exist only in the hydrogen atom (or an atom with only one electron, like He+) • It is only an approximation when discussing such orbitals in a polyelectronic atom (even for something as simple as He or H-)

  20. Quantum Numbers • In the solutions to the Schrödinger equation for the hydrogen atom , there are integers which define the solution • These are the quantum numbers • If we know the value of the three quantum numbers of an orbital, we can describe the structure/shape/orientation of the orbital • The quantum numbers obey very specific relationships (we will see them soon) • If we try to use a fractional number in the place of a whole number or we do not obey a given specific relationship, we obtain an "orbital" which is not a solution to the Schrödinger equation and is therefore not truly an orbital

  21. The Principal Quantum Number • The principle quantum number (n) must be an integer value 1, 2, 3, …. • For the hydrogen atom, the value of n determines the energy of the orbital • This is not strictly the case in a polyelectronic atom • The principal quantum number determines the average distance between an electron in a given orbital and the nucleus • The greater the value of n, the greater the average distance of an electron from the nucleus

  22. The Secondary or Azimuthal Quantum Number • The secondary or azimuthal quantum number (l) defines the shape of the orbital • The possible values of l depend on the value of the principal quantum number (n) • l is any integer between 0 and n-1 • If n = 1, l = 0 • If n = 2, l = 0, 1 • If n = 3, l = 0, 1, 2 • etc.

  23. The Secondary or Azimuthal Quantum Number • The value of l is often designated by a letter, • l = 0 is an s orbital • l = 1 is a p orbital • l = 2 is a d orbital • l = 3 is an f orbital • l = 4 is a g orbital • etc. • A set of orbitals with the same value of n is a shell • A set of orbitals having the same values ​​of n and l is a subshell • The fact that l < n explains why there aren’t 1p, 1d, 2d, 3f, etc. orbitals

  24. The Magnetic Quantum Number • The magnetic quantum number (m) describes the orientation of an orbital in space (eg.; m distinguishes between the px, py, and pz orbitals) • The possible values of m depend on the value of the azimuthal quantum number (l) • m is any integer value between -l and +l • If l = 0, m = 0 • If l = 1, m = -1, 0, +1 • If l = 2, m = -2, -1, 0, +1, +2 • If l = 3, m = -3, -2, -1, 0, +1, +2, +3 • etc. • This explains why subshells have only 1 s orbital, or 3 p orbitals, or 5 d orbitals, or 7 f orbitals, etc.

  25. The Spin Quantum Number • The first three quantum numbers define the orbital occupied by the electron • Experiments indicate that a fourth quantum number exists; the spin quantum number (s) • Electrons act like microscopic magnets • The spin quantum number describes the direction that the electron spins • The possible values are +1/2 and -1/2

  26. The s Orbitals • An s orbital is a spherical structure • Since we are dealing with probabilities, it is difficult to describe the size of an orbital and give it a specific form • In principle each orbital extends from the nucleus to infinity • The probability of finding an electron increases when approaching the nucleus

  27. The s Orbitals • Often, an orbital is shown with a contour surface (surface of constant probability of finding the electron) which defines a border encompassing 90% (or 95%, or any percentage) of the electron density for the orbital in question • All of the s orbitals are spherical • Their sizes increase as the principal quantum number increases

  28. The p Orbitals • The p orbitals exist only if the principal quantum number is greater than or equal to 2 • Each p subshell has three orbitals: px, py, and pz • The subscript indicates the axis along which each orbital is oriented • Apart from their orientation, the three orbitals are identical • Each p orbital consists of two lobes, with the nucleus being positioned where the two lobes join

  29. The d and f Orbitals • The d orbitals only exist if the principal quantum number is equal to or greater than 3 • Each d subshell has five orbitals: • Each f subshell has seven orbitals • Their structure is difficult to visually represent • The electrons in f orbitals only play an important role in the elements with atomic numbers greater than 57

  30. Atomic Orbitals • Example: Give the values of the quantum numbers associated with the orbitals in the 3p subshell. • Solution: • Example: What is the total number of orbitals associated with the principal quantum number n = 4? • Solution: There is one 4s orbital, three 4p orbitals, five 4d orbitals, and seven 4f orbitals. It is impossible to have 4g, 4h, …. orbitals since the value of l would be equal or greater than n, and this is not allowed. The total number of orbitals associated with the principle quantum number n = 4 is therefore 1 + 3 + 5 + 7 = 16. For one orbital: For one orbital: For one orbital:

  31. Orbital Energies • In a hydrogen atom, the energy of an orbital is entirely determined by the principal quantum number, n, even if the forms of the orbitals are different 1s < 2s = 2p < 3s = 3p = 3d < 4s = ...

  32. Orbital Energies • In a polyelectronic atom, the energies of orbitals with the same value of n but different values of l are not identical s < p < d < f < …. • This order is observed because the other electrons hide the nucleus and this “screening effect” becomes more important when going from s to p to d to f…

  33. Orbital Energies • The screening effect becomes so important that the energetic order of the orbitals depends primarily on the value of (n+1) rather than just n • Klechkowski’s Rule: • The filling of subshells in a polyelectronic atom is always in the order of increasing sum of n and l • If two subshells have the same (n+1) sum, the subshell with the smallest value of n is filled first

  34. Electronic Configurations • Each atomic orbital has three quantum numbers: n, l, m • Each electron has four quantum numbers: n, l, m, s • A compact notation for giving the values of the quantum numbers of an electron is the following: (n, l, m, s) • Example: Give the possible quantum numbers for an electron in a 5p orbital. • Solution:

  35. Electronic Configurations • The electronic configuration of an atom indicates how the electrons are distributed in different atomic orbitals • The ground state is the electronic configuration that provides the lowest possible energy for that atom • N.B. For an atom, the number of electrons it contains is equal to its atomic number • N.B. To indicate the spin of an electron, we use  and  rather that +1/2 and -1/2 • N.B. If we have a single unpaired electron,  and  are energetically equivalent • N.B. If we have an electron and three empty p orbitals, or 5 empty d orbitals, or 7 empty f orbitals, …., we can choose to fill any orbital in the subshell first, i.e., each orbital in the subshell has the same energy

  36. The Pauli Exclusion Principle • The Pauli Exclusion principle states that two electrons in the same atom can not be represented by the same set of quantum numbers • A consequence of the Pauli Exclusion principle is that an atomic orbital can contain only two electrons and one electron must has a spin of +1/2 and the other a spin of -1/2 • Because an atomic orbital can only have two electrons and the two electrons have opposite spins, the electronic configurations of the first five elements are • H (1s1) 1s  • He (1s2) 1s  • Li (1s22s1) 1s  2s  • Be (1s22s2) 1s  2s  • B (1s22s22p1) 1s  2s  2px 

  37. Hund’s Rule • When we get to carbon, we have three options • C (1s22s22p2) 1s  2s  2px  • C (1s22s22p2) 1s  2s  2px  2py  • C (1s22s22p2) 1s  2s  2px  2py  • Hund’s Rule states that the most stable electronic arrangement of a subshell is that which contains the greatest number of parallel spins • The third option respects Hund’s rule • The configurations of the other elements in the second row are, therefore: • C (1s22s22p2) 1s  2s  2px  2py  • N (1s22s22p3) 1s  2s  2px  2py  2pz  • O (1s22s22p4) 1s  2s  2px  2py  2pz  • F (1s22s22p5) 1s  2s  2px  2py  2pz  • Ne (1s22s22p6) 1s  2s  2px  2py  2pz 

  38. Diamagnetism and Paramagnetism • An atom or molecule is paramagnetic if it contains one or more unpaired electrons • A paramagnetic species is attracted by the magnetic field of a magnet • An atom or a molecule is diamagnetic if it has no unpaired electrons • A diamagnetic species is repelled by the magnetic field of a magnet • H, Li, B, C, N, O, F are paramagnetic • He, Be, Ne are diamagnetic

  39. Aufbau Principle • The Aufbau principle (German for “building up”) means that the electronic configuration of an element is achieved by filling the subshells one after the other • To represent the electronic configuration in a compact fashion, an inert gas configuration is represented by bracketing the atomic symbol of the noble gas • For example, the configuration of Al is changed from 1s22s22p63s23p1 to [Ne] 3s23p1 as [Ne] replaces 1s22s22p6 • On the periodic table, each element in a group (a column) shares the same configuration for its valence subshells • For example, for the first column (alkali metals) • Li: [He]2s1 • Na: [Ne]3s1 • K: [Ar]4s1 • Rb: [Kr]5s1 • Cs: [Xe]6s1

  40. Transition Metals • Transition metals have incomplete d subshells or they easily form cations with incomplete d subshells • In the first row (Sc to Cu), the Aufbau principle and Hund’s rule are respected except in two cases: • Cr should be [Ar]4s23d4 but instead it is [Ar]4s13d5 • Cu should be [Ar]4s23d9 but instead it is [Ar]4s13d10 • These deviations for Cr and Cu are attributed to the particular stability of a half-filled d subshell (the electrons are all the same spin) or a filled d subshells • The other rows of transition metals also show such particularities

  41. For the following atoms/ions, give the number of electrons that would have the following quantum numbers: • P: m = +1 • Ar: l = 1, m = 0 • Pt: l = 0, m = +1 • As: m = -1 • Br- = m = 0, s = -½

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