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Chemistry: A Molecular Approach, 1st Ed. Nivaldo Tro Chapter 7 The QuantumMechanical Model of the Atom Roy Kennedy Massachusetts Bay Community College Wellesley Hills, MA 2007, Prentice Hall Electron Energy Ηψ E ψ electron energy and position are complimentary because KE = ½mv2 for an electron with a given energy, the best we can do is describe a region in the atom of high probability of finding it – called an orbital a probability distribution map of a region where the electron is likely to be found distance vs. y2 many of the properties of atoms are related to the energies of the electrons Tro, Chemistry: A Molecular Approach 2 Wave Function, y calculations show that the size, shape and orientation in space of an orbital are determined be 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 3 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 E n -2.1810 Tro, Chemistry: A Molecular Approach -18 1 J 2 n for an electron in H 4 Principal Energy Levels in Hydrogen Tro, Chemistry: A Molecular Approach 5 Electron Transitions in order to transition to a higher energy state, the electron must gain the correct amount of energy corresponding to the difference in energy between the final and initial states electrons in high energy states are unstable and tend to lose energy and transition to lower energy states energy released as a photon of light each line in the emission spectrum corresponds to the difference in energy between two energy states 6 Predicting the Spectrum of Hydrogen the wavelengths of lines in the emission spectrum of hydrogen can be predicted by calculating the difference in energy between any two states for an electron in energy state n, there are (n – 1) energy states it can transition to, therefore (n – 1) lines it can generate both the Bohr and Quantum Mechanical Models can predict these lines very accurately E phot on released E hydrogen 18 h 2 . 18 10 hc Tro, Chemistry: A Molecular Approach elect ron E 1 J 2 n final final E initial 2 . 18 10 18 1 J n initial 2 7 Hydrogen Energy Transitions 8 Calculate the wavelength of light emitted when the hydrogen electron transitions from n = 2 to n = 1 Given: ni = 2, nf = 1 Find: ,m Concept Plan: Relationships: ni, nf 1 E R H 2 n Ephoton Eatom hc Eatom = -Ephoton E E=hc/,En = -2.18 x 10-18 J (1/n2) Solve: E atom 2 . 18 10 Ephoton = -(-1.64 x 10-18 J) = 1.64 x 10-18 J hc E 18 1 1 18 J 2 2 1 . 64 10 J 2 1 6 .626 10 3 .00 10 1 . 21 10 1 .64 10 34 J 8 m s s -18 7 J Check: the unit is correct, the wavelength is in the UV, which is appropriate because more energy than 3→2 (in the visible) m Probability & Radial Distribution Functions 2 y 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 10 Probability Density Function Tro, Chemistry: A Molecular Approach 11 Radial Distribution Function Tro, Chemistry: A Molecular Approach 12 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 13 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 14 2s and 3s 2s n = 2, l=0 3s n = 3, l=0 15 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 16 p orbitals Tro, Chemistry: A Molecular Approach 17 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 18 d orbitals Tro, Chemistry: A Molecular Approach 19 l = 3, f orbitals each principal energy state above n = 3 has 7 d 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 20 f orbitals Tro, Chemistry: A Molecular Approach 21 Why are Atoms Spherical? Tro, Chemistry: A Molecular Approach 22 Energy Shells and Subshells Tro, Chemistry: A Molecular Approach 23