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The world of Atoms

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Quantum Mechanics Theory that describes the physical properties of smallest particles (atoms, protons, electrons, photons) "A scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die and a new generation grows up that is familiar with it." Max Planck Erwin Schrödinger "I don't like it and I'm sorry I ever had anything to do with it." "An expert is someone who knows some of the worst mistakes that can be made in his subject, and how to avoid them" Werner Heisenberg "It is true that many scientists are not philosophically minded and have hitherto shown much skill and ingenuity but little wisdom." Max Born

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The hydrogen atom Niels Bohr (1885-1962) - electron orbits around the nucleus like a wave - orbit is described by wavefunction - wavefunction is discrete solution of wave equation - only certain orbits are allowed - orbits correspond to energy levels of atom

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The hydrogen atom In the Bohr model of the atom, the hydrogen atom is like a planetary system with the electron in certain allowed circular orbits. The Bohr model does not work for more complicated systems!

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Quantum numbers Each orbital is characterized by a set of quantum numbers. Principal quantum number (n): integral values (1,2,3). Related to the size and energy of the orbital. Angular momentum quantum number ( l ): integral values from 0 to (n-1) for each value of n. Magnetic quantum number (m l ): integral values from - l to l for each value of n.

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Quantum numbers How many orbitals are there for each principle quantum number n = 2 and n = 3? For each n, there are n different l-levels and (2l+1) different m l levels for each l. n=2:n = 2different l-levels (2l+1) = 2 x 0 + 1= 1 m l -levels for l = 0 l = 0, 1 (2l+1) = 2 x 1 + 1= 3 m l -levels for l = 1 Total: 1 + 3 = 4 levels for n = 2

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Quantum numbers How many orbitals are there for each principle quantum number n = 2 and n = 3? For each n, there are n different l-levels and (2l+1) different m l levels for each l. n=3:n = 3different l-levels (2l+1) = 2 x 0 + 1= 1 m l -levels for l = 0 l = 0, 1,2 (2l+1) = 2 x 1 + 1= 3 m l -levels for l = 1 Total: 1 + 3 + 5 = 9 levels for n = 3 (2l+1) = 2 x 2 + 1= 5 m l -levels for l = 2 The total number of levels for each n is n 2

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Quantum numbers Names of atomic orbitals are derived from value of l :

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Quantum numbers Quantum numbers for the first four levels in the hydrogen atom.

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What is the meaning of ? Wavefunction itself is not an observable! Square of wavefunction is proportional to probability density “I cannot but confess that I attach only a transitory importance to this interpretation. I still believe in the possibility of a model of reality - that is to say, of a theory which represents things themselves and not merely the probability of their occurrence. On the other hand, it seems to me certain that we must give up the idea of complete localization of the particle in a theoretical model. This seems to me the permanent upshot of Heisenberg's principle of uncertainty. (Albert Einstein, on Quantum Theory, 1934”

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Wavefunction and probability ‘function’ r ‘probability’

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Quantum numbers A subshell is a set of orbitals with the same value of l. They have a number for n and a letter indicating the value of l. l = 0 (s) l = 1 (p) l = 2 (d) l = 3 (f) l = 4 (g)

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Orbital Shapes

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Heisenberg uncertainty principle Life is uncertain! Where’s the electron? Werner Heisenberg That’s quite uncertain!

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Heisenberg uncertainty principle It is not possible to know both the position and momentum of an electron at the same time with infinite precision. x is the uncertainty in position. (mv) is the uncertainty in momentum. h is Planck’s constant.

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Heisenberg

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The s orbitals in hydrogen The higher energy orbitals have nodes, or regions of zero electron density. orbital surfaces probability distributions s-orbitals have n-1 nodes. The 1s orbital is the ground state for hydrogen. The orbital is defined as the surface that contains 90% or the total electron probability ( ).

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Pauli exclusion principle How many electrons fit into 1 orbital? m s = +1/2m s = -1/2 Only 2 electrons fit into 1 orbital:1 spin up 1 spin down

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Pauli exclusion principle As the temperature is lowered, bosons pack much closer together, while fermions remain spread out. Electrons are fermions. There are also bosons

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Energy Levels n =1 n =2 n =3 n =4 n =5 n =∞ E R H = 2.178 x 10 -18 J Z = atomic number n = energy level

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Energy Transitions For the energy change when moving from one level to another: n =1 n =2 n =3 n =4 n =5 n =∞ E transition

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Lines and Colors Change in energy corresponds to a photon of a certain wavelength: Change in energy Frequency of emitted light Wavelength of light emitted

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Lines and Colors What is the wavelength of the photon that is emitted when the hydrogen atom falls from n=3 into n=2? nm

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Light out of Molecules n =1 n =2 n =3 n =4 n =5 n =∞ E transition hydrogen Rhodamine 532 nm 570 nm ‘Fluorescence’

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Degeneracy Orbital energy levels for the hydrogen atom.

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Beyond hydrogen Hydrogen is the simplest element of the periodic table. Exact solutions to the wave equations for other elements do not exist!

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Polyelectric Atoms What do the orbitals of non-hydrogen atoms look like? Multiple electrons: electron correlation Due to electron correlation, the orbitals in non-hydrogen atoms have slightly different energies

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Polyelectric Atoms Screening: due to electron repulsion, electrons in different orbits ‘feel’ a different attractive force from the nucleus 11 + e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- Sees a different effective charge! Screening changes the energy of the electron orbital; the electron is less tightly bound.

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Polyelectric Atoms Penetration: within a subshell (n), the orbital with the lower quantum number l will have higher probability closer to the nucleus n =2 orbital n=3 orbital

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Polyelectric Atoms Hydrogen Polyelectric atom Orbitals with the same quantum number n are degenerate Degeneracy is gone: E ns < E np < E nd < E nf

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Spectra of Polyelectric Atoms Due to lifting of degeneracy, many more lines are possible in the spectra of polyelectric atoms

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