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# Physical Chemistry CHEM 3520

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This 13 page Class Notes was uploaded by Rebeka Zemlak MD on Wednesday October 21, 2015. The Class Notes belongs to CHEM 3520 at Tennessee Tech University taught by Staff in Fall. Since its upload, it has received 20 views. For similar materials see /class/225696/chem-3520-tennessee-tech-university in Chemistry at Tennessee Tech University.

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Date Created: 10/21/15

CHEM 3520 Spring 2008 Unit I Introduction A Review of Quantum Mechanics 1 Molecular energy levels N nuclei a Consider a system with n electrons b This system will be completely described by a wavefunction Pr R depending on the coordinates of all the nuclei and all the electrons c According to BornOppenheimer approximation the wavefunction is separated PrR we r39Rlm R III Solve for electronic wavefunction Vel considering a fixed position of the nuclei ie at one nuclear configuration III Electrons create a potential in which nuclei move and this potential is called a potential energy surface for polyatomic molecule or a potential energy curve for a diatomic molecule III The nuclear motion which is also quantized can be separated in vibrational rotational and translation motions or degrees of freedom l nucleus 2 nuclei N gt2 nuclei atom diatomic polyatomic we depends on we depends on we depends on 3n coordinates 3n coordinates 3n coordinates 14 depends on 14 depends on 3N 6 coordinates 3N coordinates Linear Nonlinear 3 translational 3 translational 3 translational 2 rotational 2 rotational 3 rotational l vibrational 3N 5 vibrational 3N 6 vibrational CHEM 35 20 Spring 2003 UI R R 2 Translational energy levels a The energy levels predicted by the onedimensional particleinabox model h n2 Equot 2 Wheren 12 Sma o a is the length ofthe box b The energy levels predicted by the threedimensional particleinabox model Equotxquotyquotz a a2 b2 02 o a b and c are the lengths of the threedimensional box 3 Rotational energy levels for diatomic molecules a The energy levels predicted by the rigidrotator model h E 211mm Where 14123 and 012 0 The degeneracy ofJLh level is g 21 1 b The energies in wavenumbers predicted by the rigidrotator model FJ JJ 1 Where E 2 87 c CHEM 3520 Spring 2008 c The energies predicted by the nonrigidrotator model FJ JJ 1 5J2 J 12 0 E is the rotational constant 0 5 is the centrifugal distortion constant 4 Vibrational energy levels for diatomic molecules a The energy levels predicted by the harmonicoscillator model En 11th Where n 012 0 y w and V i i is the fundamental frequency mA m3 27 y b The energies in wavenumbers predicted by the harmonicoscillator model Gn amp 11 1 Where 17 L i and n 012 hc 2 27m y c The energies predicted by the anharmonicoscillator model Gn 17611 9 617411 Where n 012 0 x6 is the anharmonicity constant 0 Anharmonic corrections are much smaller than the harmonic term xe ltlt 1 d The energy levels are not evenly spaced in anharmonic oscillator model and the energy difference gets smaller as quantum number 11 increases CHEM 3520 Spring 2008 B Basic Concepts of Thermodynamics l Thermodynamics a b Literally thermodynamics means heat movement It deals mainly with the energy and also with other observables too El An important aspect of thermodynamics is understanding of how these measurables relate to each other It is the study of various properties and particularly the relations between the various properties of systems in equilibrium It is primarily an experimental science that was developed in the 19th century CI The development of thermodynamics along these lines is called classical thermodynamics El It is governed by few laws that are followed by all systems El Its development did not involve knowledge about the molecular structure but also could not give insight about the system at the molecular level CI The statistical thermodynamics does that however We will cover the material as a mixture of classical and statistical thermodynamics 2 Terms and definitions System Surrounding A system can be described by indicating El El El El El CHEM 3520 3 Pres a Spring 2008 Equations of state are The quantities variables or properties describing macroscopic systems can be divided in D Extensive properties 0 0 Examples D Intensive properties 0 0 Examples The molar properties are indicated by a bar over the symbol D VLmol versus VL 7 versus U E versusE sure It is defined as force divided by the area to which the force is applied Movable wall Vacuum a Pressure 3 The pressure exerted by the Novel Equal pyessmes atmosphere 1s measured w1th a barometer the pressure of High New Law Pressure a sample of gas inside a container is measured with a manometer P g Mg p1hg phg orPPexphg The units of pressure 1Pa12k g21bar105pa m ms 1atm 101325gtlt105 Pa 101325 kPa 101325 bar 760 mmHg 1 tor 1 mmHg 1760 atm 5 CHEM 3520 Unit I Spring 2007 Introduction A Review of Quantum Mechanics 1 Molecular energy levels a Consider a system with N nuclei 7 electrons b This system will be completely described by a wavefunction Pr R depending on the coordinates of all the nuclei and all the electrons c According to BornOppenheimer approximation the wavefunction is separated PrR we r39Rlm R III Solve for electronic wavefunction Vel considering a fixed position of the nuclei ie at one nuclear configuration III Electrons create a potential in which nuclei move and this potential is called a potential energy surface for polyatomic molecule or a potential energy curve for a diatomic molecule III The nuclear motion which is also quantized can be separated in vibrational rotational and translation motions or degrees of freedom l nucleus 2 nuclei N gt2 nuclei atom diatomic polyatomic we depends on we depends on we depends on 3n coordinates 3n coordinates 3n coordinates 14 depends on 14 depends on 3N 6 coordinates 3N coordinates Linear Nonlinear 3 translational 2 rotational l vibrational 3 translational 3 translational 2 rotational 3 rotational 3N 5 vibrational 3N 6 vibrational CHEM 35 20 Spring 2007 UI R R 2 Translational energy levels a The energy levels predicted by the onedimensional particleinabox model h n2 Equot 2 Wheren 12 Sma o a is the length ofthe box b The energy levels predicted by the threedimensional particleinabox model Equotxquotyquotz a a2 b2 02 o a b and c are the lengths of the threedimensional box 3 Rotational energy levels for diatomic molecules a The energy levels predicted by the rigidrotator model h E 211mm Where 14123 and 012 0 The degeneracy ofJLh level is g 21 1 b The energies in wavenumbers predicted by the rigidrotator model FJ JJ 1 Where E 2 87 c CHEM 3520 Spring 2007 c The energies predicted by the nonrigidrotator model FJ JJ 1 5J2 J 12 0 E is the rotational constant 0 5 is the centrifugal distortion constant 4 Vibrational energy levels for diatomic molecules a The energy levels predicted by the harmonicoscillator model En 11th Where n 012 0 y w and V i i is the fundamental frequency mA m3 27 y b The energies in wavenumbers predicted by the harmonicoscillator model Gn amp 11 1 Where 17 L i and n 012 hc 2 27m y c The energies predicted by the anharmonicoscillator model Gn 17611 9 617411 Where n 012 0 x6 is the anharmonicity constant 0 Anharmonic corrections are much smaller than the harmonic term xe ltlt 1 d The energy levels are not evenly spaced in anharmonic oscillator model and the energy difference gets smaller as quantum number 11 increases CHEM 3520 Spring 2007 B Basic Concepts of Thermodynamics l Thermodynamics a b Literally thermodynamics means heat movement It deals mainly with the energy and also with other observables too El An important aspect of thermodynamics is understanding of how these measurables relate to each other It is the study of various properties and particularly the relations between the various properties of systems in equilibrium It is primarily an experimental science that was developed in the 19th century CI The development of thermodynamics along these lines is called classical thermodynamics El It is governed by few laws that are followed by all systems El Its development did not involve knowledge about the molecular structure but also could not give insight about the system at the molecular level CI The statistical thermodynamics does that however We will cover the material as a mixture of classical and statistical thermodynamics 2 Terms and definitions System Surrounding A system can be described by indicating El El El El El CHEM 3520 3 Pres a Spring 2007 Equations of state are The quantities variables or properties describing macroscopic systems can be divided in D Extensive properties 0 0 Examples D Intensive properties 0 0 Examples The molar properties are indicated by a bar over the symbol D VLmol versus VL 7 versus U E versusE sure It is defined as force divided by the area to which the force is applied Movable wall Vacuum a Pressure 3 The pressure exerted by the Novel Equal pyessmes atmosphere 1s measured w1th a barometer the pressure of High New Law Pressure a sample of gas inside a container is measured with a manometer P g Mg p1hg phg orPPexphg The units of pressure 1Pa12k g21bar105pa m ms 1atm 101325gtlt105 Pa 101325 kPa 101325 bar 760 mmHg 1 tor 1 mmHg 1760 atm 5 CHEM 3520 Spring 2004 Review N nuclei Consider a system with n electrons 7 This will be described by a wave function I I R depending on the coordinates of all the nuclei and all the electrons 7 According to BomOppenheimer approximation the wave function can be separated 1 F R We1rRlnu R 7 Solve for we considering a fixed position of the nuclei at one nuclear con guration 7 Electrons create a potential in which nuclei move This is called a potential energy surface for polyatomic and potential energy curve for a diatomic l nucleus 2 nuclei N gt2 nuclei atom diatomic polyatomic we depends on We depends on we depends on 3n coordinates 3n coordinates 3n coordinates 11quot depends on 11quot depends on 3N 6 coordinates 3N coordinates Linear Nonlinear 3 translational 3 translational 3 translational 2 rotational 2 rotational 3 rotational l Vibrational 3N 5 Vibrational 3N 6 Vibrational swam Emu 39 l 39 3 CHEM 3520 Spring 2004 Translational Energy Levels 7 The energy levels predicted by the onedimensional particleinabox model 2 2 h Equot quot2 wherenl2 8ma where n 12 and a is the length ofthe box 7 The energy levels predicted by the threedimensional particleina box model 2 2 n2 2 En n n 11 nx y quot 2 r y 2 8m a2 b2 62 where a b and c are the lengths of the threedimensional box Rotational Energy Levels for Diatomic Molecules 7 The energy levels predicted by the rigidrotator model E J KJ J 1 2 where I uRSZ andJ0l2 7 The degeneracy ofJm level is gJ 2 1 7 The energies in wavenumbers predicted by the rigidrotator model FJ EJU 1 where E 2 87239 c 7 The energies predicted by the nonrigidrotator model FJ JJ 1 5120 12 where E is the rotational constant 5 is the centrifugal distortion constant Vibrational Energy Levels for Diatomic Molecules 7 The energy levels predicted by the harmonicoscillator model E v v 1 V 2 where V is called fundamental frequency u mAmB 7139 I mAmB andv0l2 CHEM 3520 Spring 2004 7 The energies in wavenumbers predicted by the harmonicoscillator model E l N N l k Gv V v V where V andv0l2 he 2 no u 7 The energies predicted by the anharmonicoscillator model Gv 178 v 9 978178 v 2 where V 012 7 where x9 is the anharmonicity constant 7 anharmonic corrections are much smaller than the harmonic term xe ltlt 1 7 The energy levels are not evenly spaced in anharmonic oscillator model and the energy difference gets smaller as quantum number v increases Consider a macroscopic system 7 For the case of an ideal gas the total energy is a sum of energies of individual molecules EJNV 8182 MEN 7 The energy of individual molecules is a sum of electronic vibrational rotational and translational 7 Probability that a system will be in the state j having energy E J N V eTBE p J W the Boltzmann d1str1bution j 7 The denominator is denoted by Q and is a very important quantity called the partition function QNV Ze 139 75E W 7 Correlations between the partition function and physical observables lt E gt kBT 2 K 35 Ny 6T NV ltPgt kBT kBT 61HQ QNVa 6V N 3V MB

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