Chemistry: Quantum Mechanics & Spectroscopy PDF

J.E. Parker Chemistry: Quantum Mechanics and Spectroscopy 2 Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy 1st edition © 2015 J.E. Parker & bookboon.com ISBN 978-87-403-1182-2 3 Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Contents Contents Acknowledgements 7 1 Quantum Mechanics 8 1.1 The Failures of Classical Mechanics 9 1.2 Wave-Particle Duality of Light 16 1.3 The Bohr Model for the Hydrogen Atom 18 1.4 The Wave-Particle Duality of Matter, the de Broglie Equation 20 1.5 Heisenberg’s Uncertainty Principles 22 1.6 Physical Meaning of the Wavefunction of a Particle 24 1.7 Schrödinger’s Wave Equation 26 1.8 Comparison of Matter and Light 43 1.9 Spectroscopy and Specific Selection Rules 44 2 Pure Rotational Spectroscopy 47 2.1 Rigid Rotor Model for a Diatomic Molecule 48 2.2 Specific Selection Rule for Pure Rotational Spectroscopy 54 2.3 Gross Section Rule for Pure Rotational Spectroscopy 57 GET THERE FASTER Some people know precisely where they want to go. Others seek the adventure of discovering uncharted territory. 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OUR WORLD 4 Click on the ad to read more Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Contents 2.4 Rotational Motion of Polyatomic Molecules 58 2.5 Intensities of Rotational Lines 59 3 Pure Vibrational Spectroscopy 62 3.1 Simple Harmonic Oscillator (SHO) Model for a Vibrating Bond 63 3.2 Anharmonic Model for a Vibrating Molecule 68 3.3 Hot Band Transitions 71 3.4 Vibrational Spectra of Polyatomic Molecules 73 4 Vibration-Rotation Spectroscopy 78 4.1 Selection Rules for Vibration-Rotation Transitions 79 4.2 Rotations and Nuclear Statistics 83 5 Raman Spectroscopy 89 5.1 Rotational Raman Scattering 93 5.2 Vibrational Raman Scattering 95 5.3 Advantages and Applications of Raman Scattering 96 5 Click on the ad to read more Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Contents 6 Atomic Spectroscopy 98 6.1 Analytical Applications of Atomic Spectroscopy 99 6.2 Atomic Quantum Numbers 100 6.3 Term Symbols, Selection Rules and Spectra of Atoms 105 6.4 Hund’s Rules for Finding the Lower Energy Terms 111 7 Electronic Spectra 113 7.1 Term Symbols and Selection Rules for Diatomic Molecules 115 7.2 Vibrational Progressions 124 7.3 Electronic Spectra of Polyatomic Molecules 139 7.4 Decay of Electronically Excited Molecules 141 7.5 Ultraviolet Photoelectron Spectroscopy of Molecules 149 8 References 153 9 List of Formulae 154 In the past four years we have drilled 81,000 km That’s more than twice around the world. Who are we? We are the world’s leading oilfield services company. Working globally—often in remote and challenging locations—we invent, design, engineer, manufacture, apply, and maintain technology to help customers find and produce oil and gas safely. Who are we looking for? We offer countless opportunities in the following domains: n Engineering, Research, and Operations n Geoscience and Petrotechnical n Commercial and Business If you are a self-motivated graduate looking for a dynamic career, apply to join our team. What will you be? careers.slb.com 6 Click on the ad to read more Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Acknowledgements Acknowledgements I was pleased to respond to requests to write a textbook that introduces quantum mechanics and the physical-chemistry aspects of spectroscopy. I developed and presented lectures and tutorials on quantum mechanics and spectroscopy to our first and second year Chemistry students over many years at the Chemistry Department, Heriot-Watt University, Edinburgh, Scotland. These lectures and tutorials have formed the basis for this textbook. I would like to thank the staff of Heriot-Watt University Chemistry Department for their help and thank the students who gave me valuable feedback on these lectures and tutorials. I hope this textbook will help future students with their Chemistry, Physics, Chemical Engineering, Biology or Biochemistry degrees and then in their later careers. Most of all I would like to thank my wife Jennifer for her encouragement and help over many years. I shall be delighted to hear from readers who have comments and suggestions to make, please email me. So that I can respond in the most helpful manner I will need your full name, your University, the name of your degree and which level (year) of the degree you are studying. I hope you find this book helpful and I wish you good luck with your studies. Dr John Parker, BSc, PhD, CChem, FRSC Honorary Senior Lecturer Chemistry Department Heriot-Watt University Edinburgh June 2015 [email protected] http://johnericparker.wordpress.com/ 7 Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Quantum Mechanics 1 Quantum Mechanics Chemistry and the related subjects of physics, chemical engineering, biology and biochemistry deal with molecules, their properties, their reactions and their uses. The “molecular” world of electrons, atoms, molecules, ions, chemical bonding and chemical reaction is dominated by quantum mechanics and its consequences so that’s where we start our journey. One of our most powerful ways of exploring molecules is by using spectroscopy which is the interaction of light with molecules, so we will also need to understand some things about light. In this book I will be using the term “light” as a short-hand term to mean not just visible light but the whole spectral range of electromagnetic radiation at our disposal. We will be using maths and drawing graphs as we explore quantum mechanics and spectroscopy and I recommend the free PDF books which cover the maths required in a first year chemistry, a related science or engineering degree. The three books are Introductory, Intermediate and Advanced Maths for Chemists (Parker 2013b), (Parker 2012) and (Parker 2013a), respectively. There are links on my website to download these books from the publisher http://johnericparker.wordpress.com/ also you will want to look at other chemistry books for more details and I have listed some recommendations in the References. This textbook is accompanied by a companion book (Parker 2015) Chemistry: Quantum Mechanics and Spectroscopy, Tutorial Questions and Solutions, which should be used together with the current book, chapter by chapter. When anyone first meets quantum mechanics they find it strange and against their everyday experience. This is because we live in a macroscopic, classical mechanics world because humans have a mass of around 60–90 kg and heights of about 1.3–2.0 m and other objects such as your car is perhaps 2–4 m long and has a mass of between 800 to 1800 kg. As we will see, quantum effects are so small as to be unmeasurable, even in principle, for macroscopic objects such as you, me and other everyday objects. Newton Laws of motion (classical mechanics) is an extremely accurate approximation to quantum mechanics for macroscopic objects. Figure 1.1: haem B, O red, N blue, Fe brown, C grey, H light grey. 8 Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Quantum Mechanics Molecules, however, have small masses and dimensions ranging from the hydrogen atom at 1.674×10−27 kg and a diameter of about 1.6×10−10 m; to large molecules, e.g. haem B (or heme B) at 1.0237×10−24 kg and a diameter of about 16×10−10 m. The non-SI unit angstrom (symbol Å where 1 Å = 1×10−10 m) is commonly used as it is of the order of the lengths of chemical bonds and atomic radii. Quantum mechanics dominates the properties of the microscopic matter such as electrons, atoms and molecules and also the interaction of matter with light and so it is of major importance for chemistry, molecular biology, and much of chemical engineering and physics. 1.1 The Failures of Classical Mechanics Around 1880 it was believed: (1) that classical mechanics explained how the universe behaved; (2) that energy behaved rather like a fluid; (3) that atoms and molecules were hard-sphere objects and (4) that light was a continuous wave. Together these ideas of classical mechanics could explain all of physics, chemistry and biology. Between about 1880 and about 1900 several experimental results showed that this complacent view was untrue and the quantum mechanics revolution was founded from 1900–1927 and is still developing. Atom, molecules and quantum mechanics then moved into centre stage with major changes in chemistry, physics and then new subjects of molecular biology and genetics. These early failures of classical mechanics and their quantum mechanics solutions are briefly covered now. Although much of what follows may appears to be “physics” to you, it is not, it is at the foundations of chemistry and biology! It is central to chemistry and our understanding of electrons, atoms, molecules and bonding; so “hang-on in there” and things will become clear. 1.1.1 Blackbody Radiation and the Quantization of Energy When a solid material is heated it emits light, e.g. red-hot iron in a blacksmiths, the Sun and the stars. A perfect emitter and a perfect absorber of radiation would not reflect any light at all and is called a blackbody because at room temperature it looks black because most of the radiation emitted from it is in the infrared (IR). Such a perfect body does not exist but a very good approximation is a thermostatically heated hollow insulating vessel inside which the radiation is continually emitted and absorbed but with a pin-hole in it to allow some light to escape so that we can measure its spectrum. Figure 1.2: cross-section through a blackbody apparatus. 9 Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Quantum Mechanics The blackbody spectrum depends only on the temperature and not on the material of the solid. The spectrum could not be explained at all by the classical view that the light could emit and absorb continuously at all frequencies ν (Greek italic “nu” don’t confuse it with the Roman italic “vee”). The frequency ν and the wavelength λ (Greek italic “lambda”) are related to the speed of light c = 2.9979×108 m s−1 through the wave nature of light (see section 1.2). 3, Classically it was thought that the atoms making up the solid of the blackbody have a continuous distribution of frequencies of vibration about their mean positions in the solid, which would give a radiant energy density ρ (Greek italic “rho”) of " + where ρ is the energy at a frequency ν per unit volume of the blackbody per unit frequency range and kB is Boltzmann’s constant kB = 1.3806×10−23 J K−1. Unfortunately this equation completely disagrees with the experimental result which showed a maximum in the radiant energy density at a certain wavelength and this maximum wavelength varied with the temperature. Max Planck in 1900 realized that the solution to this failure was that the atoms making up the solid could only vibrate around their equilibrium positions with certain frequencies and not a continuum of possible frequencies. That is the atoms have quantized vibration energies which can only acquire an integer number (n) of the discrete unit of energy (hν). % => >!> $> " ? C )C Where n is a quantum number and can only have the integer values shown, Planck’s constant h = 6.6261×10−34 J s and the ellipsis … means “and so on”. The atoms can only vibrate with an energy of 0hν, 1hν, 2hν, 3hν and so on. The quantum mechanics equation for the radiant energy density ρ is shown below. / 8& " 8 10 Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Quantum Mechanics Planck fitted his blackbody equation to the experimental spectra to find an experimental value for h Planck’s constant. The quantum mechanical blackbody equation was a perfect fit at all temperatures for all wavelengths for all solids, this was a major triumph for quantum mechanics. Fig. 1.3 shows the blackbody spectrum for a temperature of 5000 K. , ! %& :' ($# ) ( + * #$$$ $ $ #$$ ($$$ (#$$ *$$$ *#$$ Figure 1.3: plot of radiant energy density against wavelength for a blackbody at T = 5000 K. 1.1.2 The Photoelectric Effect and the Quantization of Light Planck (1900) had introduced the idea of quantization of energy, the next step was the realization by Einstein in 1905 that light could be quantized as well to explain a problem that had existed for several years. %&! ! ! ! Figure 1.4: schematic of photoelectric apparatus. The photoelectric effect uses monochromatic light of a chosen frequency ν to hit a metal target inside a vacuum chamber and the ejected electrons have their kinetic energies measured by a grid electrode to which a negative voltage was applied to stop the electrons and so measure their kinetic energy EKE. Fig. 1.4 shows the schematic apparatus but for clarity without showing the stopping electrode. Fig. 1.5 is a schematic of experimental results for sodium which shows that a threshold frequency Φ (Greek italic capital “phi”) of light is required to emit any electrons at all and that the photoelectrons’ kinetic energy increases linearly with light frequency above this threshold frequency. Different metals give graphs with lines parallel to one another with identical gradients but different threshold frequencies. Increasing the light intensity did not alter the results in Fig. 1.5 at all. Sodium has a threshold frequency of Φ = 5.56×1014 s−1 at a wavelength of 539 nm in the green visible region. 11 Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Quantum Mechanics - ) ( Figure 1.5: schematic plot of the photoelectron KE versus light frequency. However, varying the intensity of the light does alter the number of electrons (the electric current I) ejected from the metal target, Fig. 1.6. Again for a given metal there is a threshold frequency below which no current is obtained. Above the threshold the number of electrons (current) plateaus off and is constant. 1 ( Figure 1.6: schematic of the photoelectron intensity (current) versus light frequency. Could you think of 101 new things to do with eggs and oil? Hellmann’s is one of Unilever’s oldest brands having been popular for over 100 years. If you too share a passion for discovery and innovation we will give you the tools and opportunities to provide you with a challenging career. Are you a great scientist who would like to be at the forefront of scientiﬁc innovations and developments? Then you will enjoy a career within Unilever Research & Development. For challenging job opportunities, please visit www.unilever.com/rdjobs. 12 Click on the ad to read more Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Quantum Mechanics The photoelectric experimental results summarized in Figs. 1.5 and 1.6 could not be explained at all by the wave theory of light. Einstein could explain these experimental results if the light was quantized as a particle called a photon with an energy E. % & This equation shows both particle and wave properties, E is the energy of the particle, the photon, but ν is the frequency of the wave. Metals are conductors because the valence electrons of the metal atoms are shared between all the atoms, i.e. they behave like a “mobile electron gas” which are attracted to the lattice of metal cations. There is a minimum energy of the photon for a given metal for it to eject an electron from the solid, this minimum energy is the work function of the metal Φ which is the solid state equivalent of the ionization energy of a gas phase atom. The photon is annihilated when it is absorbed and its energy hν is used to overcome the work function of the metal and eject the electron from the solid with the photon’s excess energy appearing as the kinetic energy EKE of the ejected electron. If we increase the number of photons of the same frequency, the light intensity, then we increase the number of electrons ejected from the metal target. % V X ! - Figure 1.7: energy balance of the photoelectric effect. So we have a duality of light properties, with the continuous wave theory relevant for interference, refraction and diffraction but the particle photon theory is needed for the photoelectric effect and as we shall see shortly for the heat capacity of solids and for the absorption and emission of light by atoms and molecules. Self Test Question and Solution on the photoelectric effect (Parker 2013b, p. 74). 1.1.3 Heat Capacities of Solids and the Quantization of Atom Vibrations in a Solid The heat capacity of a solid at constant volume CV was believed to be given by the equipartition principal and to be constant and independent of temperature (Dulong and Petit’s law 1819) which for a metallic crystalline solid is CV = 3R ≈ 25 J K−1 mol−1. 13 Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Quantum Mechanics Although approximately true for many solids at or above room temperature, the heat capacity of all solids decreases with decreasing temperature and approaches zero at zero kelvin. This decrease with temperature could not be explained classically. Einstein (1906) with improvements by Peter Debye (1912) realized that the solid is made of molecules which are vibrating in three dimensions around their average positions. The molecules do not have a continuous range of vibration frequencies but have quantized vibration energies (nhν) of n quanta of energy hν with quantized frequencies ν ranging from zero up to a maximum frequency νM. The molecular rotations and translations are quenched by collisions with the neighbouring molecules in the crystal lattice for nearly all solids. This quantized vibration mechanism, using Planck’s vibration energy quantization of blackbody radiation, explained quantitatively the T variation in CV for all solids. 2$ ) ( ( 2 *# *$ (# ($ # $ $ ($$ *$$ 2$$ +$$ #$$ ,$$ 3$$ Figure 1.8: CV versus T for copper. The gas constant R = 8.3145 J K−1 mol−1 appears in many situations in science that don’t involve gases! Its name is purely historical and the reason for its appearance is that the gas constant is a disguise for the much more fundamental Boltzmann’s constant R = NAkB (NA is Avogadro’s constant 6.0221×1023 mol−1). Boltzmann’s constant kB applies to individual atoms or molecules whether they be gas, liquid or solid and the gas constant is “Boltzmann’s constant for a mole” of atoms or molecules whether they be gas, liquid or solid. Self Test Question and Solution on heat capacity (Parker 2012, p. 76). 1.1.4 Atomic and Molecular Spectra and their Quantization of Electronic Levels Classical mechanics says that when we excite an object it should emit or absorb light in a continuous series of wavelengths. However, we know that fireworks emit light from isolated atoms or ions and are or various different colours depending upon the metal salt used to make them. The Sun’s spectrum has a blackbody distribution of light of an object at 5778 K, but superimposed upon this the Sun has about 1000 absorption lines from the isolated atoms in the Sun’s atmosphere. These lines’ wavelengths were accurately measured by Fraunhofer and Ångstrom. Spectra in the lab from isolated atoms and molecules can be measured using electric discharges or flames and they showed the presence of lines, e.g. a sodium salt held in a Bunsen burner flame is yellow and consists of two closely spaced lines at 588.9950 nm and 589.5924 nm, these yellow lines are also the light from a Na street lamp. Note the typical precision of spectroscopic measurements typically to seven significant figures. 14 Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Quantum Mechanics +$$ #6$ 3$$ Figure 1.9: Na atom emission spectrum, gap between the Na lines is exaggerated. The unit for the wavelength of visible and UV light is the nanometre (symbol nm) with 1 nm = 10−9 m. These line emissions and absorptions mean that isolated atoms and molecules have quantized absorption and emission, but why? The simplest explanation is that atoms and molecules can only absorb energy into quantized energy levels rather than continuously and so can only emit the energy as discrete quanta of light. 360° thinking. Figure 1.10: continuous (left) and quantized (right) energy levels. 360° thinking. 360° thinking. Discover the truth at www.deloitte.ca/careers Dis © Deloitte & Touche LLP and affiliated entities. Discover the truth at www.deloitte.ca/careers © Deloitte & Touche LLP and affiliated entities. Deloitte & Touche LLP and affiliated entities. Discover the truth 15 at www.deloitte.ca/careers Click on the ad to read more Download free eBooks at bookboon.com © Deloitte & Touche LLP and affiliated entities. Chemistry: Quantum Mechanics and Spectroscopy Quantum Mechanics 1.2 Wave-Particle Duality of Light 1.2.1 The Wave Nature of Light Light acts as a wave in: refraction (e.g. the apparent bending of a stick at an air-water interface); interference (e.g. the colours of a thin film of oil on water); and diffraction (e.g. X-ray diffraction). Self Test Question and Solution on X-ray diffraction (Parker 2013a, p. 70). The light wave has a wavelength λ (the distance between two equivalent points on the wave) and a frequency ν (the number of waves per second). A light wave consists of an electric field (E) and a magnetic field (B) whose intensities vary like sine waves but at right angles to one another. These electric and magnetic fields are moving in-phase at the speed of light c = 2.9979×108 m s−1 in a vacuum. In other media such as air or water the electromagnetic wave moves at a lower velocity, the wavelength λ decreases but the frequency ν remains constant. When we say the speed of light we mean in a vacuum unless specified otherwise. 3, Figure 1.11: electromagnetic radiation (light). To characterize the light we can use either the frequency or wavelength, it is also useful to quote the number of complete waves (peak plus trough) per centimetre which is called the wavenumber ῡ (“nu bar”). The wavenumber unit is cm–1 pronounced as “centimetres to the minus one” (or “reciprocal centimetres”) and wavenumber is the commonly used practical unit, even though it is a non-SI unit. , ) 9 3, ) ) The regions of the electromagnetic spectrum vary only in their wavelengths, frequencies or wavenumbers. In Fig. 1.12 frequency increases to the right and wavelength increases to the left. 16 Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Quantum Mechanics ( ($ 2$ ( 3$$ +$$ ($ $$# ' 78 :' %& 9 ( # , 3 : 6 ($ (( (* (2 (+ (# (, (3 (: (6 *$ *( Figure 1.12: the electromagnetic spectrum. The various regions are “separated” only because they use different instrumental techniques and also they affect atoms and molecules in different ways. These effects are: radiofrequency (RF) the magnetism of atomic nuclei; microwave (MW) the rotations of molecules around their centre of mass; infrared (IR) the vibrations of bond lengths and angles; visible and ultraviolet (UV) the excitation and ionization of outer shell electrons; X-rays the excitation and ionization of inner shell electrons; and gamma rays (γ-rays) the excitation of atomic nuclei energy levels. Self Test Question and Solution on light waves (Parker 2013b, p. 14). 1.2.2 The Particle Nature of Light Blackbody radiation, the heat capacity of solids, the photoelectric effect and the line spectra in the absorption and emission of light by atoms and molecules can only be explained by light being made of irreducible packets of electromagnetic energy (a particle of light), the photon. The photon travels at the speed of light c, it has a rest mass of zero but has measurable linear momentum p = hυ/c, it exhibits deflection by a gravitational field, and it can exert a force. It has no electric charge, has an indefinitely long lifetime, and has a spin (s = 1) of one unit of h/2π. The component of the spin vector ms = ±1. These correspond to right-handed or left-handed circularly polarized light depending upon whether the spin direction is the same as the direction of motion c (Fig. 1.13, left) or against the speed of light vector c (see the right-hand rule, section 1.7.6). The photon spin angular momentum quantum number of unity and its components ms = ±1 are important when we consider spectroscopy. For help with vectors see Parker (Parker 2013a, p. 127). ( ( Figure 1.13: right-handed and left-handed circularly polarized photons. Self Test Question and Solution on the particle and wave properties of light see Parker (Parker 2013b, p. 29). 17 Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Quantum Mechanics 1.3 The Bohr Model for the Hydrogen Atom The explanation of the line spectra of atoms and molecules has its origins from Balmer, Lyman, Paschen and Brackett measuring the wavelengths of the visible, UV and IR emission lines of excited hydrogen atoms formed in electric discharges through H2. +($* +2+( +:,( ,#,2 Figure 1.14: Balmer visible emission spectrum of the hydrogen atom. Wavelength is not proportional to the energy of the light, we could use frequency which is proportional to energy, but 400 nm = 7.495×1014 s−1 to 700 nm = 4.283×1014 s−1 which is not user friendly. Wavenumber ῡ is proportional to the energy of light and 400 nm = 25,000 cm−1 to 700 nm = 14,285 cm−1 is more convenient, hence the use of this practical but non-SI unit. Rydberg brought the experimental work together by finding that the wavenumber ῡ of the spectral lines of the H-atom in the UV, visible and IR could all be fitted perfectly to an equation involving only two integer terms, n1 and n2. , *% ! 8 ! ! 3 > !>$> C 18 Click on the ad to read more Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Quantum Mechanics where Rydberg’s constant is RH = 109,737 cm−1. Niels Bohr in 1913 noted that the Rydberg equation could be modelled if the H-atom had a quantized electron energy and the electron could only occupy one of several stable circular orbits which are characterized by the principal quantum number n = 1, 2, 3, … (but not n = 0). +:,( ( * 2 + Figure 1.15: the Bohr model of the H-atom. Fig. 1.15 shows the second Balmer line n = 4 to n = 2 (cyan) at 486.1 nm. The UV Lyman lines are transitions to n = 1; the visible Balmer to n = 2; the IR Paschen and Brackett are to n = 3 and 4, respectively. Absorption arises when the H-atom converts the energy of the absorbed photon into electronic excitation energy and the electron makes a quantum jump up to a higher energy level with a higher quantum number n. Emission arises when an excited H-atom, e.g. from an electric discharge, converts its electronic excitation energy into light by the electron making a quantum jump down to a lower energy level with a lower quantum number n. The difference in electronic energy is converted to a photon of light given by Planck’s equation. Figure 1.16: quantized light absorption (left) and emission (right). The Bohr model explained the line spectra of the H-atom and introduced the idea of atoms and molecules having quantized electronic energy levels and that emission and absorption involved jumps between these quantum states. But it is only a first approximation as the Rydberg equation only worked for the H-atom and other one-electron atoms such as He+ or Li2+ and the Bohr theory was superseded in 1924 as quantum mechanics developed. Self Test Question and Solution on the Rydberg equation (Parker 2013b, p. 32). 19 Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Quantum Mechanics 1.4 The Wave-Particle Duality of Matter, the de Broglie Equation Louis de Broglie (1924 in his PhD thesis, awarded the Nobel prize in 1929!) had an inspirational insight when he realized that if light has both wave and particle properties then perhaps particles of matter such as electrons, atoms and molecules should also have wave properties. Up to then matter had always been though of as particles which behaved like hard-spheres. A particle of mass m moving with a velocity v in a straight line acts like a “matter wave” with a wavelength λ given by the de Broglie equation. C where h is Planck’s constant, p is the momentum p = mv with m the mass and v the velocity. Let’s calculate some de Broglie wavelengths, h is in J s, m in kg and v in m s−1 giving λ in metres. I will also quote the wavelengths in ångstroms where 1 Å = 1×10−10 m to give us a feel for the size of the de Broglie wavelength in terms of the size of atoms and molecules. Worked Example: what is the de Broglie wavelength of a 70 eV electron which is a typical electron energy used in a mass spectrometer? We must calculate its kinetic energy, then its velocity, then its de Broglie wavelength. % V 4=!=8 0= * !=8 0 G 8 0 ! % V !!= G % V X ! 8 $ + 4=4 ) 8 = = 44!4=8 $+ G 8 = 8 $ 4 8 +44= ) = = + 4= ) The de Broglie wavelength of the 70 eV electron is 1.5 Å and is of atomic dimensions. See for yourself if you can calculate the de Broglie wavelengths for the objects shown below? & ) ) R $ / * = / = != ! $ ==* = / = 4 != = ! =* = $ / = 0 != =! ==*& 40= !0 $=/ ! = ! ==! !0 $ % ! !== $$= !+= += =+ !! != $ $!= == $$ == !$ +/= $ $= + = $+ + = !+ $4 !4 ) 0= $+ 0= 0= 20 Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Quantum Mechanics As one increases the velocity but keeps the mass constant, e.g. the electrons, the wavelength decreases but is still of atomic dimensions. Increasing the mass, e.g. from the electron to the proton and the hydrogen molecule, the wavelengths decrease but they are still of an atomic and molecular size. However, for macroscopic objects, e.g. the bullet, golf ball and humans, the wavelength is 10−23 to 10−26 m. These wavelengths are so small as to be meaningless in the sense that they cannot be measured, compare them to the radius of an electron at about 10−15 m and an atomic nucleus about 10−14 to 10−15 m. So macroscopic matter may be safely treated with the classical mechanics of Newton but we must use quantum mechanics for microscopic matter. What experimental evidence is there that matter does have wave properties? In 1927 Davisson and Germer, Fig.1.17 left, observed in the US a diffraction pattern when an electron beam was “reflected” from the front surface of a metal consisting of large nickel crystals. Independently, whilst in Aberdeen GP Thomson, Fig. 1.17 right, found a diffraction pattern when an electron beam was passed through a thin polycrystalline film of gold or aluminium, the end-on view of the pattern is shown on the extreme right. The Wake the only emission we want to leave behind.QYURGGF'PIKPGU/GFKWOURGGF'PIKPGU6WTDQEJCTIGTU2TQRGNNGTU2TQRWNUKQP2CEMCIGU2TKOG5GTX 6JGFGUKIPQHGEQHTKGPFN[OCTKPGRQYGTCPFRTQRWNUKQPUQNWVKQPUKUETWEKCNHQT/#0&KGUGN6WTDQ 2QYGTEQORGVGPEKGUCTGQHHGTGFYKVJVJGYQTNFoUNCTIGUVGPIKPGRTQITCOOGsJCXKPIQWVRWVUURCPPKPI HTQOVQM9RGTGPIKPG)GVWRHTQPV (KPFQWVOQTGCVYYYOCPFKGUGNVWTDQEQO 21 Click on the ad to read more Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Quantum Mechanics ! ! 1 1 Figure 1.17: electron diffraction patterns: Davisson and Germer (left) and GP Thomson (right). These experiments agreed with previous X-ray diffraction measurements of the samples (X-rays are light acting as waves in diffraction experiments). Interestingly, JJ Thomson received one of the first Noble Prizes in 1906 for proving the existence of the electron and showing it was a particle and his son GP Thomson received a Nobel Prize in 1937 (sharing it with Davisson) for showing that the electron could also have wave properties! 1.5 Heisenberg’s Uncertainty Principles The physical meaning of these particle-waves was supplied by Max Born whose explanation is based on the work of Louis de Broglie, Arthur Compton and Werner Heisenberg. Arthur Compton (1923) had showed that photons, which have zero mass, nevertheless have momentum p from scattering experiments of a beam of X-ray photons by a beam of electrons where the X-rays lost energy and had a longer wavelength than the initial X-rays. &)) ) Werner Heisenberg (1927) realized that there was a fundamental limit to the information you could obtain about particle-waves. Imagine a particle, e.g. an electron, moving freely through space with a constant wavelength λ and its wavefunction (section 1.6) is a sine wave sin(2π/λ) and the wavefunction of the electron is spread through space and it has no definite position. From de Broglie’s equation (λ = h/mv) the particle has a definite momentum p = h/λ. So if we know the momentum exactly we have no idea of its position. Conversely, we may know the position of a particle exactly as in Fig. 1.18. ! * Figure 1.18: Heisenberg’s uncertainty principle. 22 Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Quantum Mechanics The wavefunction for Fig. 1.18 is zero everywhere except at the particle’s position. In order to get this result we need to add an infinite number of wavefunctions each of sin(2π/λ) with wavelengths ranging from λ = 0 to λ = ∞. So we know nothing about the momentum of the particle p = h/λ as λ is unknown but we know the particle position exactly. Heisenberg summarized the situation. : )& & 3 ' & )) )& & ) ; ) )=== ==>=== , = 8 $ $== ++== ) = 8 =! 4= ) > )!/p = 8 = $=$ $! * = 8 +>=== +=>=== $ & & = +>=== !/>=== The period of rotation for a typical molecule is about 10−11 s and the time between collisions at one atmosphere pressure is about 10−10 s so a molecule will on average undergo several full rotations before it is disturbed by a collision. The rotational state of the molecule is thus “well defined”, particularly at pressures of less than one atmosphere in the gas phase when the time between collisions will increase. In the liquid or solid phases the time between collisions is much smaller than 10−10 s and the rotational states of molecules are not well defined so MW spectra must be taken in the gas phase. The rotations are “pure” in the sense that MW photons of an energy of ~0.2 to ~60 cm−1 are not energetic enough to excite either the vibrations of chemical bonds (IR) or the electrons in their molecular orbitals (UV-visible). Fig. 2.1 shows the potential energy plot of the ground and excited electronic states of a diatomic molecule as a function of the internuclear distance, this is called a potential energy diagram, with superimposed energy levels of the vibrational and rotational energies. Note the convention that the upper quantum states are indicated with a single prime (ε′, v′ and J′) and the lower quantum states with a double prime (ε″, v″ and J″). 2 * ( $ 2 * ( $ Figure 2.1: PE curve for a diatomic molecule, not to scale. 47 Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Pure Rotational Spectroscopy Considering the masses and velocities of the various species involved in the different motions of a molecule we find that the coupling or interaction of the motions is small and to a first approximation the different types of motion may be treated separately (the Born-Oppenheimer approximation). ! % " % % %, % 65&& ) (&& 8 ) I will mention later where the Born-Oppenheimer approximation breaks down. Translational motion is the motion of the centre of mass (CM) of the whole molecule and has already been discussed in the allowed levels for a particle in a three-dimensional box in which the gas is contained (section 1.7.4). Let’s first look at the rotations of molecules with MW spectroscopy which allows us to measure very precise bond lengths and bond angles, i.e. to obtain accurate chemical structures for molecules. Fig. 2.2 shows a simple absorption spectrometer, which can be used for stable molecules. A klystron (similar to the one in your MW oven) emits a narrow range of wavelengths and it can be tuned to give monochromatic MW of variable wavelength, so the klystron acts as its own monochromator. The MW radiation is focussed by an evacuated copper, or silver, rectangular tube waveguide which is under vacuum. A low pressure of gas or the vapour above a solid or liquid is held inside the waveguide between two thin mica windows as mica is MW transparent. A solid state detector has its signal amplified, processed and stored. Increasingly, however, Fourier transform MW (FTMW) spectrometers are being used to study at low temperatures the weakly bound complexes of molecules. FTMW gives measurements at high resolution and at high sensitivity. An introduction to the Maths behind the FT process is available (Parker 2012, pp. 60 and 69). ! ! 78 Figure 2.2: a MW absorption spectrometer. 2.1 Rigid Rotor Model for a Diatomic Molecule Fig. 2.3 shows a simulation of the pure MW spectrum of carbon monoxide. The spectrum has approximately equally spaced absorption (or emission) lines when plotted on an energy scale of wavenumbers or frequency. The intensities of the lines initially increase, pass through a maximum and then decrease slowly. Let’s firstly look at the positions of the spectral lines. (*(( ($ *#*+ $ *$ +$ ,$ :$ ($$ , ( Figure 2.3: simulation of the MW absorption spectrum of 12C16O. 48 Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Pure Rotational Spectroscopy The term “rigid-rotor” means that the bond length does not vary as the molecule rotates faster, i.e. there is no centrifugal distortion (stretching) of the bond length. Of course the bond is still vibrating, so the measured bond distance r is the bond length averaged over ~100 vibrational periods. The rigid-rotor model is a good approximation to the true situation. If the diatomic molecule was only rotating in two-dimensional space (like the particle on a ring) it would have only one degree of freedom and one quantum number J. However, the molecule is rotating in three-dimensional space, x y z, and it is equivalent to the reduced mass (defined below) confined on the surface of a sphere. So the diatomic molecule has two degrees of freedom, equivalent to our position east-west and north-south on the surface of the earth, with two quantum numbers J and MJ. Solving the Schrödinger equation for the allowed wavefunctions of these spherical harmonics (Atkins & de Paula 2006, p. 301) gives functions for the angular momentum Iω in terms of the angular momentum quantum number J and the components of the angular momentum quantum number MJ along the rotation axis. @ @ @ => > !> $> # ! @ => $> $!> $$> $ @ Technical training on WHAT you need, WHEN you need it At IDC Technologies we can tailor our technical and engineering training workshops to suit your needs. We have extensive OIL & GAS experience in training technical and engineering staff and ENGINEERING have trained people in organisations such as General ELECTRONICS Motors, Shell, Siemens, BHP and Honeywell to name a few. 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ELECTRICAL For a no obligation proposal, contact us today POWER at [email protected] or visit our website for more information: www.idc-online.com/onsite/ Phone: +61 8 9321 1702 Email: [email protected] Website: www.idc-online.com 49 Click on the ad to read more Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Pure Rotational Spectroscopy Let’s look at J = 1 and its the three components. ( A A ( ( ( * * ( $ $ A A (+(+ * ( * A A Figure 2.4: J = 1 and the MJ = 0, ±1 components. On the left of Fig. 2.4 are the components of the angular momenta and on the right are the angular momenta. Each of the angular momenta (the sides of the cones) are of length |Iω| = 1.414(h/2π) and the heights of the cones are the components MJ = 0 or ±1(h/2π). What are these cones? We know both the angular momentum Iω and its z-component exactly so from Heisenberg’s uncertainty principle the position of the angular momentum vector has infinite x and y uncertainty, i.e. the angular momentum vector is pointing somewhere to the tip of the cone but we don’t know where, so these are “cones of uncertainty”. ( * Figure 2.5: a diatomic molecule rotating around its centre of mass. We can now calculate the quantized energy levels of our rotating diatomic, Fig. 2.5 shows the rotation of a diatomic molecule of bond length r around its centre of mass (or centre of gravity). From the analogy of a single particle’s moment of inertia, I = mr2 (section 1.7.6) the diatomic molecule’s moment of inertia will be. % ! & & )) 8 The moment of inertia I is the sum of the separate atoms’ moment of inertia. 1! ! 8 1 ! 3, , 1 ! 8 1 The moment of inertia I with the level rule (as in a child’s see-saw) allows us to eliminate the unknown x. ! ! ! ! 8 ! ! ) ) ) ) ! ! 50 Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Pure Rotational Spectroscopy From the moment of inertia I and the quantized angular momentum we can find the allowed rotational energy levels EJ which are equal to the rotational kinetic energies as the potential energy Vrot = 0. The analogy with translational energy Etrans = ½mv2 gives us the rotational equivalent. ! %@ ! %@ @ @ ! ! ! ! % @ @ @ G ) ) 6 ! EJ is in joules, a unit with a symbol J (symbols are in roman type), don’t confuse this unit with the quantum number J (a variable in italic type). Help with physical quantities, variables, units, symbols, labels and equations see (Parker 2013b, p. 9). In the absence of any external electrical or magnetic fields the rotational energy depends only on J and the molecular property I but the rotational energy does not depend upon the component of the angular momentum along the rotation axis MJ. So each rotational energy level J has a degeneracy of (2J+1). The rotational energy are conventionally measured in wavenumbers (symbol FJ) which comes by dividing EJ in joules by hc with c in cm s−1 rather than m s−1. 8 B @ @ @ ! ) ) ) 6 Note the difference between the equations for EJ and FJ. In FJ the h is not squared and it is divided by the speed of light in cm s−1. We define the rotational constant B cm−1 below which is characteristic of the molecule as it depends upon the masses of the atoms and the geometry of the molecule through the molecular variable, the moment of inertia I. 8 ! ) 8 ) ) The FJ is usually written as shown below. B @ @ @ 8 )8 Worked example: calculate the reduced mass and moment of inertia of D35Cl given m(D) = 2.01410 g mol−1, m(35Cl) = 34.96885 g mol−1 and r0 = 1.275 Å. 51 Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Pure Rotational Spectroscopy “Molecule” is not an SI unit but nevertheless it is still a very useful reminder of exactly what we are doing. Also the SI base unit of mass is the kg not the g (Parker 2013b p. 9). The orders of magnitude for reduced masses of many simple molecules is 10−27 kg with the order of magnitude for moments of inertia of 10−47 kg m2. The quantized rotational energy levels for a rigid-rotor diatomic molecule are shown in Fig. 2.6. + + *$ +2 : ( 2 2 (* 2* , * * , *( + ( ( * ($ * $ $ $ Figure 2.6: rigid-rotor energy levels and transitions between neighbouring levels. We note the energies of the spectral lines are at 2B, 4B, 6B, 8B ∙∙∙ and so the rigid-rotor model gives the spacings between spectrum lines as a constant 2B agreeing with experiment. ZZZVWXG\DWWXGHOIWQO 5DQNHGWKLQWKHZRUOG 7+(67HFKQRORJ\UDQNLQJ $OPRVW\HDUVRISUREOHPVROYLQJ H[SHULHQFH ([FHOOHQW6SRUWV &XOWXUHIDFLOLWLHV &KHFNRXWZKDWDQGKRZZHWHDFKDW ZZZRFZWXGHOIWQO 52 Click on the ad to read more Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Pure Rotational Spectroscopy Worked example: if a rotational absorption occurs at 220 cm−1 for a molecule with B = 10 cm−1 what are the quantum states involved in the transition? 2.1.1 Centrifugal Distortion and the Non-Rigid Rotor The rigid rotor model is a good approximation and centrifugal distortion is only normally significant for light molecules or when spectra are taken at high resolution then the spectrum lines close up slightly as J increases, the gap becomes less than the constant 2B. This is due to centrifugal distortion of both the bond lengths and angles at high rotational velocities. We need to modify our quantum energies to allow for the effect at high J values. B @ @ @ 8 8 @ ! @ ! = ) 8 6 , Note that the negative centrifugal distortion term increases faster than J 4 compared with the rigid rotor term increasing faster than J 2 and hence centrifugal distortion is important at high values of J. The centrifugal constant D is characteristic for a given molecule and depends upon the flexibility of the chemical bond. + 8$ 8 = ! ) , Where ῡe (nu “e” bar) is the equilibrium vibration wavenumber which measures the flexibility of the bond and D is typically about 1/1000 of the magnitude of B, i.e. centrifugal distortion is generally only a small second order effect. For example, B and D for 1H2 are B = 60.85 cm−1 and D = 0.046 cm−1 but for C16O we have B = 1.931 cm−1 and D = 6.1×10−6 cm−1. If anharmonicity (see section 3.2) is to be taken 12 into account for the flexibility of the bonds then terms in higher powers of J(J+1) should be added to the expressions for the energy levels and line positions. B @ 8 @ @ 9 = @ ! @ ! C @ $ @ $ 9 9 @ + @ + @ / @ / Most of the terms are negligible compared with D except are high rotational levels and for measurements at very high resolution. A striking example is the rotational energy of hydrogen fluoride which has been fitted to the fifth-order polynomial in J(J+1) for transitions up to ΔJ(33,32) (Jennings et al. 1987). 53 Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Pure Rotational Spectroscopy 75 ( ,(,2,#* *$##632 ,2## *(*($ 2 2 ! +6($ $(,2($ , 3 ++($ $(#($ ($ (( *:($ $6($ (# Table 2.1: HF rotational constants for the ground vibrational level. For most molecules at normal resolution the quadratic polynomial is an excellent fit to the experimental microwave data. The positions of the (J+1)←J transitions between neighbouring levels are then B @ @ @ 8 8 @ ! @ ! = )8 B @ @ @ ! 8 8 @ ! @ !! = ) 8 ! ! ! 8 B @ > @ @ @ ! 8 @ 8 8 @ @ + @ + 8 @ = ) $ 8 B @ > @ ! @ 8 8 + @ = ) As there are three unknowns in this equation, J, B and D we need to use three consecutive lines in the spectrum with this equation to find B (gives r0), D (gives ῡe) and J which assigns the J values to the three consecutive lines in the spectrum. 2.2 Specific Selection Rule for Pure Rotational Spectroscopy A specific selection rule deals with the changes in quantum numbers that are experimentally observed. Observed transitions are called “allowed” and those not observed or only weakly observed are called “forbidden”. In the MW spectrum we only see absorption or emission between neighbouring energy levels, why? We have to consider angular momentum both of the molecule and of the photon. Angular momentum is a conserved quantity in any process (in the same way as energy is conserved in a process). Fig. 2.7 shows absorption (top) and emission (bottom) of a photon by a rotating molecule. ( ( Figure 2.7: rotational spectroscopy specific selection rule and angular momentum. For a linear molecule the specific selection rule for the component of J along an external axis MJ can be visualized by considering the standing waves of a sphere, the spherical harmonics. For a linear molecule, @ $ @ => $ ) & 54 Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Pure Rotational Spectroscopy ΔMJ = 0, ±1 selection rule is important if the molecule is in an external electric field where the (2J+1) fold-degeneracy is then lifted (Atkins & de Paula 2006, p. 445). The (2J+1) degeneracy is important for the intensity of the microwave lines. The second way of looking at ΔJ = ±1 is to consider the symmetry of the rotational wavefunctions? From our particle on a ring discussion we can extend this to our diatomic rotor if we consider the particle to be the reduced mass μ of our diatomic molecule rotating around the centre of mass, Fig. 2.8. The symmetry of the wavefunctions J = 0, 2, 4, … is even (symmetric) and for J = 1, 3, 5, … is odd (antisymmetric) with respect to 180° rotation around the axis. * * ! + * : * * ( ( ( $ $ ! $ Figure 2.8: symmetry of diatomic molecule’s rotational wavefunctions. 55 Click on the ad to read more Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Pure Rotational Spectroscopy We have previously seen that the transition moment μmn is non-zero, even or symmetric, and hence allowed for jumps between even ↔ odd wavefunctions as the component of the molecule’s dipole moment operator (“hat” mu) is odd in the direction of the light’s electric vector (section 1.9). ) f) 8 ) , , ) , , So symmetry would potentially allow transitions of ΔJ = ±1, ±3, ±5 … but the angular momentum of the photon restricts the allowed transitions to only ΔJ = ±1 for pure rotational spectroscopy. In microwave absorption spectra the molecule is excited and they jump up a level ΔJ = +1. Rotationally excited molecules emit microwave dropping down a level with ΔJ = −1. Worked Example: what is the bond length of 1H79Br if its rotational spectrum has peaks separated by 16.72 cm−1 given m(1H) = 1.0008 g mol−1 and m(79Br) = 78.92 g mol−1? Spectrum line spacing is 2B so B = 8.36 cm−1 and the rotational constant B = h/8π2Ic. 44!4=8 $+ G 8 +0 ! ! 8 = 8 $$+= ) $4 ) ! = ) ==0 ! 8 =8 $ 8 ) 4/!0=8 !0 == 0 ! 4=!!=!$ ) 8 ! $$+=8 +0 )! 8 !0 4/!0= & +!$=8 = ) +!$ ( Notice the use of physical quantities and units (Parker 2013b, p. 9) and we also use Newton’s laws of motion for the units conversion of joule = kg m2 s−2. There are a couple of points to note. Firstly, that chemical bond lengths don’t vary very much for small molecules they are generally 1-2.5 Å, e.g. H2 is 0.74 Å and Cl2 is 1.99 Å. Secondly, a simple spectroscopic measurement has led to an accurate value for the average internuclear distance of the molecule in the v = 0 vibrational level. So the bond length is normally written as r0 = 1.423 Å. 56 Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Pure Rotational Spectroscopy 2.3 Gross Section Rule for Pure Rotational Spectroscopy A gross selection rule deals with the physical property that a molecule must possess in order to undergo a given type of spectroscopic transition. ) ) , & )) - So homonuclear diatomic molecules e.g. N2 or O2 and molecules with a centre of symmetry e.g. CH4, CO2, C2H2 are transparent to MW radiation. Molecules without a centre of symmetry, e.g. heteronuclear diatomics such as HCl or CO and molecules such as CHCl3 or HCCBr, absorb in the microwave region. ! Figure 2.9: vertical component of the dipole moment of a rotating polar molecule. A rotating molecule with a permanent dipole moment produces an oscillating electric field along a chosen direction, Fig. 2.9. Classically this oscillating electric field can exchange energy with the oscillating electric field of the MW radiation. A rotating molecule without a permanent dipole does does not produce an oscillating electric field although it is still rotating with a quantum number J. Such non-polar molecules may still change their value of J by collisional energy transfer with other molecules but not by photon absorption or emission. So in the Earth’s atmosphere the major constituent gases N2, O2, Ar, CH4, CO2 are transparent to MW but H2O and N2O are MW absorbers. This allows us to observe from the Earth’s surface using MW telescopes, e.g. the cosmic microwave background radiation which corresponds to a blackbody temperature of about 2.7 K. Also, from the Earth’s surface, around 210 different molecules in interstellar gas clouds have been detected using microwave spectroscopy, the largest being C70 a 70-fullerene, and the molecules’ temperatures have also been measured from their line intensities giving the various gas clouds’ temperatures. 57 Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Pure Rotational Spectroscopy 2.4 Rotational Motion of Polyatomic Molecules A molecule’s random rotation may be resolved into three components of independent rotation around the x, y, and z axes. " " " " " " Figure 2.10: independent rotations for some polyatomic molecules. Any molecule has three moments of inertia which are conventionally defined in terms of their magnitudes as Ia ≤ Ib ≤ Ic. Remember, the difference between rotation and spin (section 1.7.6) that a rotating molecule must change at least two of its three coordinates and a spinning molecule does not change any of its coordinates. We can then classify molecules into five different shapes or symmetries with the following moments of inertia. Study at one of Europe’s leading universities DTU, Technical University of Denmark, is ranked as one closely under the expert supervision of top international of the best technical universities in Europe, and offers researchers. internationally recognised Master of Science degrees in 39 English-taught programmes. DTU’s central campus is located just north of Copenhagen and life at the University is engaging and vibrant. At DTU, DTU offers a unique environment where students have we ensure that your goals and ambitions are met. Tuition hands-on access to cutting edge facilities and work is free for EU/EEA citizens. Visit us at www.dtu.dk 58 Click on the ad to read more Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Pure Rotational Spectroscopy = 5>5'>.! >% >5 ! )) &;&< 4 %4 >.% $ & )) &; < %$ > & & %+ > ' + > '4 )) & l l %! 5>.5! Non-linear molecule are said to have 3 degrees of rotational freedom but linear molecules only have 2 degrees of rotational freedom as Ia is zero and it is a spin not a rotation as none of the x, y, z coordinates change. The moments of inertia of non-linear polyatomic molecules are summarized in Atkins and de Paula (Atkins & de Paula 2013, p 480). Although spherical top molecules such CH4 or SiH4 have no permanent dipole moment, rotations around any of the C-H bonds will centrifugally distort the remaining three C-H bonds and give a small dipole moment. An extremely weak rotational spectrum which may be observed with Fourier transform instruments using long path length cells of 10s metres, but for ordinary purposes such spherical top molecules may be considered transparent to MW. 2.5 Intensities of Rotational Lines *,*# +3+, ($ , ( $ # ($ (# *$ *# 2$ 2# +$ Figure 2.11: simulation of the microwave absorption spectrum of N2O. The intensities of the lines I(J+1,J) are normally all measured relative to the intensity of the lowest transition I(1,0). In Fig. 2.11 the relative intensities of the lines increase with J then pass through a maximum before a slow decrease. @ > @ 5 @ > @ @ @ >= 5>= = = The quantum mechanical probabilities of the transitions P(J+1,J) all have ΔJ = +1 are equal to one another, so the ratio of P(J+1,J)/P(1,0) is unity or one. The degeneracy gJ of level J (the number of states MJ with the same energy) is 2J+1 and g0 is unity so the ratio of degeneracies is 2J+1, see Fig. 2.4. The ratio of the populations of molecules in two quantum energy levels, Ei and Ef, is given by Boltzmann’s distribution law. Notice the difference (due to degeneracies) between counting states and counting energy levels. 59 Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Pure Rotational Spectroscopy 8& 8 " % 8 % ) 3 8& 8 % 8% " ) 3 Don’t confuse the distribution law with the non-quantized (classical) Maxwell-Boltzmann equation for translational motion. The Boltzmann distribution law is true for any quantized motion not just rotations. In our case the rotational energy terms in the exponential are F(J) = BJ(J+1) cm−1 and F(0) = 0 cm−1. Converting kBT from joules to wavenumbers it becomes kBT /hc with the speed of light in cm s−1. The ratio of rotational spectrum lines is then @ > @ >= ! @ 8& 8 " 8 @ @ & ) The pre-exponential term increases linearly with J and the exponential term decreases slowly with J(J+1) and the product of the two terms models the intensities of the rotational spectrum. Worked example: what is the ratio of the populations of rotational energy levels J = 3 to J = 0 for 1H35Cl at 25°C if B = 10.59 cm−1 for 1H35Cl? Note that the useful conversion factor kBT/hc = 207.23 cm−1 at 298.15 K with c in cm s−1 this factor also allows us to easily calculate the kBT/hc at any other temperature. Increase your impact with MSM Executive Education For almost 60 years Maastricht School of Management has been enhancing the management capacity of professionals and organizations around the world through state-of-the-art management education. Our broad range of Open Enrollment Executive Programs offers you a unique interactive, stimulating and multicultural learning experience. Be prepared for tomorrow’s management challenges and apply today. For more information, visit www.msm.nl or contact us at +31 43 38 70 808 or via [email protected] the globally networked management school For more information, visit www.msm.nl or contact us at +31 43 38 70 808 or via [email protected] Executive Education-170x115-B2.indd 1 18-08-11 15:13 60 Click on the ad to read more Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Pure Rotational Spectroscopy Fig. 2.12 shows the populations of rotational quantum level populations for 12C16O with a rotational constant of B of 1.9313 cm−1 at three temperatures. The highest populated J level at 100 K is J = 4, at 300 K J = 7 and at 500 K J = 9. (* #$$ ($ $ : 2$$ , + * ($$ $ $ ($ *$ 2$ +$ Figure 2.12: 12C16O relative rotational populations. Worked example: if the maximum intensity of the CO spectrum is ΔJ = (12,11) what is the temperature, assuming it is a rigid-rotor? B = 1.931 cm−1 for CO, use the differential of a product (Parker 2013b, p. 89). @ > @ >= ! @ 8& 8 8 @ ! @ " @ ! 8& 8 8 @ ! @ " 8 ! @ 8 " ! @ 8& 8 8 @ ! @ " = " 8 ! $ ) 8 ! 8 ! @ )8 ! /=0 ) ! ! 8 " !=0! ) ! /V 0$/ V The Boltzmann distribution law applies what is called Boltzmann statistics, i.e. the effect of any nuclear spin is ignored. This is valid for most molecules but we will return to the effect of nuclear spin on rotational populations and spectra later on (section 4.2). 61 Download free eBooks at bookboon.com Chemistry: Quantum Mechanics and Spectroscopy Pure Vibrational Spectroscopy 3 Pure Vibrational Spectroscopy G' 4 ( + * 2 , 3 # ! Figure 3.1: schematic FTIR Michelson interferometer. The Fourier transform FTIR based on the Michelson interferometer is the normal instrument for measuring IR spectra. The non-dispersed I

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