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Quantum numbers B Visegrády The Photoelectric Effect • it was observed that many metals emit electrons when a light shines on their surface this is called the Photoelectric Effect • classic wave theory attributed this effect to the light energy being transferred to the electron • according to this theory, if the wavelength of light is made shorter, or the light waves intensity made brighter, more electrons should be ejected remember: the energy of a wave is directly proportional to its amplitude and its frequency if a dim light was used there would be a lag time before electrons were emitted to give the electrons time to absorb enough energy Tro, Chemistry: A Molecular Approach 2 The Photoelectric Effect Tro, Chemistry: A Molecular Approach 3 The Photoelectric Effect The Problem • in experiments with the photoelectric effect, it was observed that there was a maximum wavelength for electrons to be emitted called the threshold frequency regardless of the intensity • it was also observed that high frequency light with a dim source caused electron emission without any lag time Tro, Chemistry: A Molecular Approach 4 Einstein’s Explanation • Einstein proposed that the light energy was delivered to the atoms in packets, called quanta or photons • the energy of a photon of light was directly proportional to its frequency inversely proportional to its wavelength the proportionality constant is called Planck’s Constant, (h) and has the value 6.626 x 10-34 J·s Tro, Chemistry: A Molecular Approach 5 Ejected Electrons • 1 photon at the threshold frequency has just enough energy for an electron to escape the atom binding energy, φ • for higher frequencies, the electron absorbs more energy than is necessary to escape • this excess energy becomes kinetic energy of the ejected electron Kinetic Energy = Ephoton – Ebinding KE = hν - φ Tro, Chemistry: A Molecular Approach 6 Spectra • when atoms or molecules absorb energy, that energy is often released as light energy fireworks, neon lights, etc. • when that light is passed through a prism, a pattern is seen that is unique to that type of atom or molecule – the pattern is called an emission spectrum non-continuous can be used to identify the material flame tests • Rydberg analyzed the spectrum of hydrogen and found that it could be described with an equation that involved an inverse square of integers Tro, Chemistry: A Molecular Approach 7 Emission Spectra Tro, Chemistry: A Molecular Approach 8 Exciting Gas Atoms to Emit Light with Electrical Energy Hg Tro, Chemistry: A Molecular Approach He H 9 Examples of Spectra Oxygen spectrum Neon spectrum Tro, Chemistry: A Molecular Approach 10 Identifying Elements with Flame Tests Na Tro, Chemistry: A Molecular Approach K Li Ba 11 Emission vs. Absorption Spectra Spectra of Mercury Tro, Chemistry: A Molecular Approach 12 Bohr’s Model • Neils Bohr proposed that the electrons could only have very specific amounts of energy fixed amounts = quantized • the electrons traveled in orbits that were a fixed distance from the nucleus stationary states therefore the energy of the electron was proportional the distance the orbital was from the nucleus • electrons emitted radiation when they “jumped” from an orbit with higher energy down to an orbit with lower energy the distance between the orbits determined the energy of the photon of light produced Tro, Chemistry: A Molecular Approach 13 Bohr Model of H Atoms Tro, Chemistry: A Molecular Approach 14 Wave Behavior of Electrons • de Broglie proposed that particles could have wave-like character • because it is so small, the wave character of electrons is significant • electron beams shot at slits show an interference pattern the electron interferes with its own wave • de Broglie predicted that the wavelength of a particle was inversely proportional to its momentum Tro, Chemistry: A Molecular Approach 15 Electron Diffraction Tro, Chemistry: A Molecular Approach if electrons behave like particles, should however,there electrons actually only be two bright spots present an interference onpattern, the target demonstrating the behave like waves 16 Complimentary Properties • when you try to observe the wave nature of the electron, you cannot observe its particle nature – and visa versa wave nature = interference pattern particle nature = position, which slit it is passing through • the wave and particle nature of nature of the electron are complimentary properties as you know more about one you know less about the other Tro, Chemistry: A Molecular Approach 17 Uncertainty Principle • Heisenberg stated that the product of the uncertainties in both the position and speed of a particle was inversely proportional to its mass x = position, Δx = uncertainty in position v = velocity, Δv = uncertainty in velocity m = mass • the means that the more accurately you know the position of a small particle, like an electron, the less you know about its speed and vice versa Tro, Chemistry: A Molecular Approach 18 The Franck-Hertz experiment • The electrons were accelerated by an electric field (Figure 4.) between the cathode (C) and the grid (G) in a tube filled with Hg vapour. After passing through the grid, an opposite field slowed down the electrons and prevented them from reaching the anode (A), unless they gained enough kinetic energy in the previous acceleration. The electrons reaching the anode formed a measurable current, I. Tro, Chemistry: A Molecular Approach 19 The Franck-Hertz experiment • The Hg atoms can not absorb any energy, just well defined values, namely 4.9 eV. This energy excites the ground state electron to the first excited state, e.g. it equals exactly to the energy difference between the E1 and E2 states of the Hg atom. Thus, this experiment gives an excellent proof of the validity of Bohr’s theory. Tro, Chemistry: A Molecular Approach 20 Wave Function, ψ • calculations show that the size, shape and orientation in space of an orbital are determined by three integer terms in the wave function added to quantize the energy of the electron • these integers are called quantum numbers principal quantum number, n angular momentum quantum number, l magnetic quantum number, ml Tro, Chemistry: A Molecular Approach 21 orbital requires n l 3 quantum numbers ml magnetic -l, …, l orientation angular momentum 0, 1, 2, …, (n - 1) shape Principal 1, 2, 3, … size and energy “address” Principal Quantum Number, n • characterizes the energy of the electron in a particular orbital corresponds to Bohr’s energy level • n can be any integer ≥ 1 • the larger the value of n, the more energy the orbital has • energies are defined as being negative an electron would have E = 0 when it just escapes the atom • the larger the value of n, the larger the orbital • as n gets larger, the amount of energy between orbitals gets smaller for an electron in H Tro, Chemistry: A Molecular Approach 23 Principal Energy Levels in Hydrogen Tro, Chemistry: A Molecular Approach 24 Probability & Radial Distribution Functions • ψ2 is the probability density the probability of finding an electron at a particular point in space for s orbital maximum at the nucleus? decreases as you move away from the nucleus • the Radial Distribution function represents the total probability at a certain distance from the nucleus maximum at most probable radius • nodes in the functions are where the probability drops to 0 25 Probability Density Function Tro, Chemistry: A Molecular Approach 26 Radial Distribution Function Tro, Chemistry: A Molecular Approach 27 The Shapes of Atomic Orbitals • the l quantum number primarily determines the shape of the orbital • l can have integer values from 0 to (n – 1) • each value of l is called by a particular letter that designates the shape of the orbital s orbitals are spherical p orbitals are like two balloons tied at the knots d orbitals are mainly like 4 balloons tied at the knot f orbitals are mainly like 8 balloons tied at the knot Tro, Chemistry: A Molecular Approach 28 l = 0, the s orbital • each principal energy state has 1 s orbital • lowest energy orbital in a principal energy state • spherical • number of nodes = (n – 1) Tro, Chemistry: A Molecular Approach 29 2s and 3s 2s n = 2, l=0 3s n = 3, l=0 30 l = 1, p orbitals • each principal energy state above n = 1 has 3 p orbitals ml = -1, 0, +1 • each of the 3 orbitals point along a different axis px, py, pz • 2nd lowest energy orbitals in a principal energy state • two-lobed • node at the nucleus, total of n nodes Tro, Chemistry: A Molecular Approach 31 p orbitals Tro, Chemistry: A Molecular Approach 32 l = 2, d orbitals • each principal energy state above n = 2 has 5 d orbitals ml = -2, -1, 0, +1, +2 • 4 of the 5 orbitals are aligned in a different plane the fifth is aligned with the z axis, dz squared dxy, dyz, dxz, dx squared – y squared • 3rd lowest energy orbitals in a principal energy state • mainly 4-lobed one is two-lobed with a toroid • planar nodes higher principal levels also have spherical nodes Tro, Chemistry: A Molecular Approach 33 d orbitals Tro, Chemistry: A Molecular Approach 34 l = 3, f orbitals • each principal energy state above n = 3 has 7 f orbitals ml = -3, -2, -1, 0, +1, +2, +3 • 4th lowest energy orbitals in a principal energy state • mainly 8-lobed some 2-lobed with a toroid • planar nodes higher principal levels also have spherical nodes Tro, Chemistry: A Molecular Approach 35 f orbitals Tro, Chemistry: A Molecular Approach 36 Stern-Gerlach experiment Particles possess intrinsic angular momentum. Spin angular momentum is quantized (it can only take on discrete values) For the completely filled shells, subshell (4d10) the orbital magnetic momentum is zero; for the 5s orbital ML is also zero. Hypothesis: the argent atom possesses no magnetic momentum >> they move in an inhomogeneous magnetic field along the same path. The electron possesses an intrinsic angular momentum and an intrinsic magnetic momentum: spin S — intrinsic angular momentum Ms — intrinsic magnetic momentum