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THE STRUCTURE OF THE ATOMS/QUANTUM NUMBERS September 2014 UP, Medical School, Dept. Biophysics ATOMOS = INDIVISIBLE ‘All that exists are atoms and empty space; everything else is merely thought to exist.’ Democritus, 415 B.C. 1 THOMSON MODEL (1902) Joseph John Thomson 1897 - electron „Plum pudding” THE ATOM MODELS Atoms are stable Their chemical properties show periodicity (Mendeleev 1869) After excitation they emit light, and their emission spectra is linear Johann Jacob Balmer’s empirical formula (1885): 1 1 R 2 4 n 1 n: 3,4,5…… R: Rydberg constant (R = 10 973 731.6 m-1) 2 RUTHERFORD MODEL (1911) Ernest Rutherford RUTHERFORD’S CONCLUSIONS 1. The majority of matter is „empty space”! 2. The positive charge is concentrated into a tiny space (nucleus ~10 -15 m). 3. Electrons are revolving around the nucleus, like planets around the Sun. 3 BOHR’S MODEL Niels Bohr Bohr’s postulates: 1. Electrons in an atom can only have defined orbits. The formula defining the radius of the allowed orbits is: L mrv n h 2 Stationary wave! h 2r n n mv 2. When the electron jumps from one allowed orbit to another, the energy difference of the two states is emitted as a photon with the energy of hν: h f E2 E1 h 6.6 10 34 Js Planck constant THE CONCLUSIONS OF THE BOHR’S MODEL 1. Radius of the 1st orbit: r1 = 5.3 ·10-11 m (Bohr-radius) r2 = 4r1, r3 = 9r1….. rn = n 2 r1 2. Energy of the first orbit: E1 = -13.6 eV (because it is bound) E2 E1 4 En E3 E1 n2 E1 9 4 THE FRANK-HERTZ EXPERIMENT The proof of the Bohr’s model Atoms can absorb only precisely given amounts of energy. The Hg atoms e.g. 4,9 eV. The 4,9 eV is equals to the energy difference between the ground state and the first excited state of a Hg atom. QUANTUM MECHANICAL ATOM MODEL Matter wave – wave function () Described by the Schrödinger’s equation Probability of occurrence of an electron: 2 The position of the ground state electron of a hydrogen atom, around the nucleus. The density of the spots is proportional to the finding probability of the electron. The graph shows Ψ2 in the function of the distance measured from the nucleus. 5 Heisenberg’s uncertainty principle (1927) It is impossible to precisely determine the position and the momentum of the particle at the same time. The multiplication of the uncertainty (error) of two measurements at the same time is always higher than h / 4 : x p x h 4 The relation gives a limit of principle: the multiplication of the measured uncertainty of the two quantities can not be smaller than h / 4. QUANTUM MECHANICAL ATOM MODEL QUANTUM NUMBERS Quantum numbers describe values of conserved quantities in the dynamics of the quantum system. They often describe specifically the energies of electrons in atoms, but other possibilities include angular momentum, spin etc. It is already known from the Bohr’s atom model that the energy of the electrons is quantized so they can have only one value. The energy values are determined by the n principal quantum number. The quantum mechanics is proved that there are sublevels of the given energy levels that is why the n principal quantum number is not enough and more other quantum numbers are needed. 6 QUANTUM NUMBERS The principal quantum number (n) It is known that the principal quantum number defines the energy, and an energy value belongs to every n value ( n → En ). The electrons with given n values are forming shells which are named with K, L, M, etc. letters. There can be more other states inside a shell which states are determined by the orbital quantum number. Bohr had predicted the positions of orbits with amazing accuracy but did not take count that this is not the only position of electron, this is the place where the electron can be found with the highest probability. En E1 n2 QUANTUM NUMBERS The orbital quantum number (l) It defines the magnitude of the angular momentum of an electron. Angular momentum: The angular momentum of a body which is revolving around an r radius orbital with v speed is a vectored quantity. Its value is L = mvr. Its direction is perpendicular to the plane of the velocity. The angular momentum resulting from the movements of the electrons on their orbital can only be: L l (l 1) h 2 where h is the Planck constant and l is the orbital quantum number, which can be an integer between 0 and n-1. Example: n = 2; l = 0 (2s state): L = 0 h l = 1 (2p state): L 2 2 7 QUANTUM NUMBERS The magnetic quantum number (m) It defines the direction of the angular momentum of an electron. That is why the angular momentum can be set only in given directions. The projection of the angular momentum on the direction of the outer magnetic field can only be: Lz m h 2 where m is the magnetic quantum number which values are whole numbers between -l and +l. This determines the direction of the angular momentum definitely. How can it define the angular momentum: Example: if n = 2; l = 0, 1; m = -1, 0, +1 ZEEMAN EFFECT I When an atom turns from an initial higher energy level to a stationary level with lower energy then the energy difference can be emitted as a photon. This may give a line in the visible spectrum. In the presence of an external magnetic field, these different states will have different energies due to having different orientations of the magnetic dipoles in the external field, so the atomic energy levels are split into a larger number of levels and the spectral lines are also split. The rate of split is proportional to the applied magnetic field. The new lines appear symmetrically on the right and on the left side of the original line. This is the so-called Zeeman effect (normal Zeeman effect). 8 QUANTUM NUMBERS The spin quantum number (s) It defines the value of the spin angular momentum of the electron. It is imagined as the electron (like the Earth) not just revolving around its orbit but it is spinning around its own axis. The electron’s own angular momentums can only be: h S s ( s 1) 2 where s is the spin quantum number. The spin quantum number can only be ½. It does not defines other sublevels. QUANTUM NUMBERS The magnetic spin quantum number (ms) It defines the direction of spin angular momentum of an electron. The projection of the angular momentum on the direction of the outer magnetic field (z) can only be: h S z ms 2 where ms is the magnetic spin quantum number which is ½ or -½, so the spin (owned angular momentum) can be set only in two directions. 9 THE STERN-GERLACH EXPERIMENT The Stern-Gerlach experiment involves sending a beam of particles through an inhomogeneous magnetic field and observing their deflection. The particles passing through the Stern-Gerlach apparatus are deflected either up or down by a specific amount. This result indicates that spin angular momentum is quantized (it can only take on discrete values), so that there is not a continuous distribution of possible angular momenta. http://www.youtube.com/watch?v=rg4Fnag4V-E THE STERN-GERLACH EXPERIMENT 1922 Conclusions: 1st The experiment proves that the angular momentum is quantized. 2nd Why is the beam deflected into two beams? if l=0 => m=0 => no deflection if l=1 => m=0, 1 => deflects into three beams (that is why a two-beam deflection can not caused by direction of the angular momentum) Phipps and Taylor reproduced the effect using hydrogen atoms in their ground state in 1927. 1925 – Uhlenbeck and Goudsmit formulated their hypothesis of the existence of the electron spin. 10 THE EINSTEIN-DE HAAS EFFECT A freely suspended body consisting of a ferromagnetic material acquires a rotation when its magnetization changes. Because of the change of the external magnetic field mechanical rotation of the ferromagnetic material is happened associated with the mechanical angular momentum, which, by the law of conservation of angular momentum, must be compensated by an equally large and oppositely directed angular momentum inside the ferromagnetic material. spin angular momentum is indeed of the same nature as the angular momentum of rotating bodies QUANTUM NUMBERS Quantum number Symbol Quantized value Values Principle n Energy 1,2,3… Orbital l Value of angular momentum 0,1……n-1 Magnetic m Direction of angular momentum -l, -l+1…0…l-1, l Spin s Value of own angular momentum ½ ms Direction of own angular momentum Magnetic spin –½, +½ 11 QUANTUM NUMBERS http://dilc.upd.edu.ph/images/lo/chem/Quantum/quantum.swf 12