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Electronic Structure of Atoms, Cont’d The Quantum Mechanical Hydrogen Atom De Broglie, 1923: If radiant energy, which is thought to be a wave, could behave as a particle (Einstein), could matter, which is thought to be a particle, behave as a wave? wave-particle duality: could the electron, in its orbit about the hydrogen nucleus, be thought of as a wave with an associated wavelength? De Broglie proposes the existence of a matter wavelength for a particle of mass m and velocity v given by h mv 48 given the magnitude of h, when will the matter wavelength become important? E.g., calculate the matter wavelength of: a 50-g golf ball moving at 400 m/s a Li atom moving at 6.5 x 105 m/s 49 The Uncertainty Principle (Heisenberg) Ascribing wave-like properties to the electron gives rise to uncertainties in our ability to make measurements on the electron If we give electron wave-like properties, then it is no longer appropriate to imagine the electron as moving in a welldefined circular orbit Implications? 50 Quantum Mechanics (Schrödinger, 1926) Mechanics: branch of physics which deals with motion Classical mechanics (Newton): laws of motion of macroscopic bodies (planets, tennis balls etc) Quantum mechanics: laws of motion of microscopic bodies Solution of Schrödinger's equation for H atom leads to quantum mechanical wave functions for H atom: orbitals A quantum mechanical orbital is an allowed energy state of an electron in a H atom The square of the orbital represents a probability of locating the electron in a given energy state 51 Bohr model: one quantum number (n) Quantum mechanical H atom: three quantum numbers to describe an orbital principal quantum number, n restricted to integer values: n = 1, 2, 3... larger n: larger orbital, electron spends more time farther from nucleus note that for the quantum mechanical H atom, 1 E 2.18x10 J n 18 n 2 which is the same as the Bohr result 52 azimuthal quantum number, l restricted to integer values from 0 ... (n-1) describes shape of orbital magnetic quantum number, ml restricted to integer values from - l...0...l for each l value, there are (2l+1) values of ml describes orientation of orbital (i.e., along x, y, z, etc.) e.g., what are the allowed values of l for n = 2? what are the allowed values of ml for this case? 53 Electron shells and subshells shell: orbitals with same value of n subshell: orbitals with same values of n and l subshell designations: l = 0: s l = 1: p l = 2: d l = 3: f A subshell is designated by a number (the value of n) and a letter (corresponding to the value of l) 54 E.g, what is the subshell name for n=3, l=2? n=2, l=0? Note that the restrictions on the allowed values of the quantum numbers give rise to a very important pattern: The first shell (n=1) consists of only the 1s subshell The second shell (n=2) consists of two subshells (2s and 2p) The third shell (n=3) consists of three subshells (3s, 3p, 3d) What about the n=4 shell? Each subshell is further divided into orbitals each s subshell consists of 1 orbital each p subshell consists of 3 orbitals each d subshell consists of 5 orbitals 55 Why is this so??? How many orbitals are in an f subshell? finally, notice in the H atom that orbitals in the same shell have the same energy.... Why is this??? We construct an orbital diagram for H atom: 56 Problems du Jour For n=5, what are the possible values of l ? Give the values of n, l, and ml for each orbital in the: n = 3 shell 4f subshell 57