An overview of the use of the Marcus Theory to calculate the energies of transition states.
Contributed by: Elizabeth Greenhalgh, Amanda Bischoff, and Matthew Sigman, University of Utah, 2015
Bonding and Antibonding interactions; Idea about σ, σ*, π, π *, n – MOs; HOMO, LUMO and SOMO; Energy levels of π MOs of different conjugated acyclic and cyclic systems; Hückel’s rules for aromaticity; Frost diagram
A ppt compiled by Yaseen Aziz Wani pursuing M.Sc Chemistry at University of Kashmir, J&K, India and Naveed Bashir Dar, a student of electrical engg. at NIT Srinagar.
Warm regards to Munnazir Bashir also for providing us with refreshing tea while we were compiling ppt.
The document summarizes the Franck-Condon principle, which states that during an electronic transition between two states of a molecule, the transition occurs so rapidly that the positions of the nuclei remain almost unchanged. It describes the different types of molecular energy levels and vibrational transitions. It also provides three cases that illustrate how the Franck-Condon principle determines the relative intensities of vibrational transitions between electronic states based on differences in the equilibrium internuclear distances of the states.
1. Organometallic compounds are defined as compounds in which carbon atoms are directly bonded to metal atoms. Organometallic compounds can be classified as ionic, covalent, or electron deficient based on the nature of the metal-carbon bond.
2. Metal carbonyls are compounds where carbon monoxide ligands are bonded directly to metal centers. They can be mononuclear, containing one metal center, or polynuclear, containing multiple metal centers. Common mononuclear metal carbonyls prepared include nickel tetracarbonyl, iron pentacarbonyl, and chromium hexacarbonyl.
3. The document provides details on the nomenclature, classification, preparation, and structures
APPLICATIONS OF ESR SPECTROSCOPY TO METAL COMPLEXESSANTHANAM V
This document discusses the applications of electron spin resonance (ESR) spectroscopy to study metal complexes. It outlines several key factors that influence the ESR spectra of metal complexes, including the nature of the metal ion, ligands, geometry, number of d electrons, and crystal field effects. It also describes how zero-field splitting and Jahn-Teller distortions can lead to splitting of electronic levels and influence the number and pattern of transitions observed in ESR spectra. Understanding these various effects is important for extracting information about electronic structure and bonding from ESR data of metal complexes.
Metal carbonyls are coordination complexes of transition metals with carbon monoxide ligands. They were first synthesized in 1868 by passing carbon monoxide over platinum. Metal carbonyls typically obey the 18 electron rule and are often diamagnetic. They have applications as catalysts in organic synthesis and in producing pure metals like nickel. Precautions must be taken when using metal carbonyls due to their toxicity.
An overview of the use of the Marcus Theory to calculate the energies of transition states.
Contributed by: Elizabeth Greenhalgh, Amanda Bischoff, and Matthew Sigman, University of Utah, 2015
Bonding and Antibonding interactions; Idea about σ, σ*, π, π *, n – MOs; HOMO, LUMO and SOMO; Energy levels of π MOs of different conjugated acyclic and cyclic systems; Hückel’s rules for aromaticity; Frost diagram
A ppt compiled by Yaseen Aziz Wani pursuing M.Sc Chemistry at University of Kashmir, J&K, India and Naveed Bashir Dar, a student of electrical engg. at NIT Srinagar.
Warm regards to Munnazir Bashir also for providing us with refreshing tea while we were compiling ppt.
The document summarizes the Franck-Condon principle, which states that during an electronic transition between two states of a molecule, the transition occurs so rapidly that the positions of the nuclei remain almost unchanged. It describes the different types of molecular energy levels and vibrational transitions. It also provides three cases that illustrate how the Franck-Condon principle determines the relative intensities of vibrational transitions between electronic states based on differences in the equilibrium internuclear distances of the states.
1. Organometallic compounds are defined as compounds in which carbon atoms are directly bonded to metal atoms. Organometallic compounds can be classified as ionic, covalent, or electron deficient based on the nature of the metal-carbon bond.
2. Metal carbonyls are compounds where carbon monoxide ligands are bonded directly to metal centers. They can be mononuclear, containing one metal center, or polynuclear, containing multiple metal centers. Common mononuclear metal carbonyls prepared include nickel tetracarbonyl, iron pentacarbonyl, and chromium hexacarbonyl.
3. The document provides details on the nomenclature, classification, preparation, and structures
APPLICATIONS OF ESR SPECTROSCOPY TO METAL COMPLEXESSANTHANAM V
This document discusses the applications of electron spin resonance (ESR) spectroscopy to study metal complexes. It outlines several key factors that influence the ESR spectra of metal complexes, including the nature of the metal ion, ligands, geometry, number of d electrons, and crystal field effects. It also describes how zero-field splitting and Jahn-Teller distortions can lead to splitting of electronic levels and influence the number and pattern of transitions observed in ESR spectra. Understanding these various effects is important for extracting information about electronic structure and bonding from ESR data of metal complexes.
Metal carbonyls are coordination complexes of transition metals with carbon monoxide ligands. They were first synthesized in 1868 by passing carbon monoxide over platinum. Metal carbonyls typically obey the 18 electron rule and are often diamagnetic. They have applications as catalysts in organic synthesis and in producing pure metals like nickel. Precautions must be taken when using metal carbonyls due to their toxicity.
For UG students of All Engineering Branches (Mechanical Engg., Chemical Engg., Instrumentation Engg., Food Technology) and PG students of Chemistry, Physics, Biochemistry, Pharmacy
The link of the video lecture at YouTube is
https://www.youtube.com/watch?v=t3QDG8ZIX-8
Nuclear Quadrupole Resonance Spectroscopy (NQR) is a chemical analysis technique that detects nuclear energy level transitions in the absence of a magnetic field through the absorption of radio frequency radiation. NQR is applicable to solids due to the quadrupole moment averaging to zero in liquids and gases. The interaction between a nucleus's quadrupole moment and the electric field gradient of its surroundings results in quantized energy levels. Transitions between these levels are detected as NQR spectra and provide information about electronic structure, hybridization, and charge distribution. NQR finds applications in studying charge transfer complexes, detecting crystal imperfections, and locating land mines.
Electron spin resonance (ESR) spectroscopy is a technique used to study compounds with unpaired electrons. In ESR, a sample is placed in a static magnetic field and irradiated with microwaves. This causes transitions between the electron spin energy levels. The absorption of microwave energy is detected to obtain an ESR spectrum. ESR spectra provide information about electron environments through parameters like g-values and hyperfine splitting patterns. ESR finds applications in studying transition metal complexes and unstable free radicals.
CONTENTS
INTRODUCTION
CONCEPTS OF WALSH DIAGRAM
APPLICATION IN TRIATOMIC MOLECULES
[IN AH₂ TYPE OF MOLECULES(BeH₂,BH₂,H₂O)]
INTRODUCTION
Arthur Donald Walsh FRS The introducer of walsh diagram (8 August 1916-23 April 1977) was a British chemist, professor of chemistry at the University of Dundee . He was elected FRS in 1964. He was educated at Loughborough Grammar School.
Walsh diagrams were first introduced in a series of ten papers in one issue of the Journal of the Chemical Society . Here, he aimed to rationalize the shapes adopted by polyatomic molecules in the ground state as well as in excited states, by applying theoretical contributions made by Mulliken .
It contains what are the shift reagents, and how they will use in NMR spectroscopy. It includes lanthanide shift reagents and their effect using NMR spectroscopy. It has mostly used shift reagents like Europium and their importance. paramagnetic species that affect the NMR spectra are also explained in detail. What are contact shift and pseudo-contact shift also explained. It contains what are the chiral shift reagent, and the advantages, and disadvantages of lanthanide shift reagents. Reference books are also included.
Hyperfine splitting occurs due to the interaction between an electron's spin and the nucleus' spin. This interaction causes each electron spin state to split into 2I+1 levels, where I is the nuclear spin quantum number. As examples, the document discusses the hyperfine splitting in hydrogen, where the nuclear spin is 1/2, and deuterium, where the nuclear spin is 1. Hyperfine splitting has applications in radio astronomy, nuclear technology such as laser isotope separation, and atomic clocks.
This document provides an overview of the application of phase rule to a three component system of acetic acid, chloroform, and water. It defines key terms like phases, components, and degrees of freedom. It explains Gibbs phase rule and how it applies to a three component system. Specifically, it discusses how the water-acetic acid-chloroform system can be represented on a triangular phase diagram, with acetic acid enhancing the miscibility of water and chloroform. The document outlines how the system transitions from two heterogeneous phases to a single homogeneous phase as the amount of acetic acid is increased.
This document discusses electronic spectra of metal complexes. It begins by defining quantum numbers related to electron configuration, such as L (total orbital angular momentum) and l (secondary quantum number). It then describes two main types of electronic transitions in coordination compounds: d-d transitions specific to metals, and charge-transfer transitions. The remainder of the document discusses charge-transfer transitions in more detail, defining ligand-to-metal and metal-to-ligand charge transfer, and how solvent polarity affects these transitions.
Transition metal carbonyls form when carbon monoxide bonds to a transition metal through both sigma and pi bonding. This synergistic metal-ligand bonding strengthens the metal-carbon bond. Metal carbonyls can be classified based on the ligands present and the number/structure of metal atoms. They exhibit a variety of reactions including substitution, reactions with halogens, and disproportionation. Metal carbonyls display properties related to their toxicity, magnetic behavior, thermal stability, and thermodynamic instability.
Arenium Ion Mechanism in Aromatic Electrophilic Substitution SPCGC AJMER
This document describes the arenium ion mechanism for aromatic electrophilic substitution reactions. The mechanism involves two steps: 1) the rate-determining attack of an electrophile on the aromatic ring to form a resonance-stabilized arenium ion intermediate, and 2) rapid departure of the leaving group to regenerate aromaticity. Evidence for this mechanism includes isolation of arenium ion intermediates and a lack of isotope effects. The first step is highly endothermic due to loss of aromaticity, making it rate-determining, while the second step regains aromatic stabilization in an exothermic process.
Electron Spin Resonance (ESR) SpectroscopyHaris Saleem
Electron Spin Resonance Spectroscopy
Also called EPR Spectroscopy
Electron Paramagnetic Resonance Spectroscopy
Non-destructive technique
Applications
Extensively used in transition metal complexes
Deviated geometries in crystals
This document provides an overview of metal carbonyls. It discusses how metal carbonyls are formed from transition metals and carbon monoxide, and examples like nickel tetracarbonyl and iron pentacarbonyl. The molecular orbital diagram of carbon monoxide is shown, explaining why it can participate in pi-backbonding. Infrared spectroscopy is described as a useful technique for analyzing metal carbonyls, as it can distinguish terminal from bridging carbonyl ligands based on the infrared absorption frequency. Factors like metal charge and other ligands that affect the carbonyl stretching frequency are also outlined. Finally, some applications of infrared spectra of metal carbonyls are mentioned.
Acid Base Hydrolysis in Octahedral ComplexesSPCGC AJMER
This document discusses acid and base hydrolysis in octahedral complexes. It covers factors that affect the rate of acid hydrolysis, including the charge on the complex, steric hindrance effects, and the strength of the leaving group. A higher positive charge, more steric hindrance, or stronger metal-leaving group bond each decrease the rate of acid hydrolysis according to first-order kinetics through a dissociative SN1 mechanism. Base hydrolysis of octahedral complexes can proceed by either associative SN2 or dissociative SN1 pathways depending on conditions.
This Presentation describes about the evidence of metal ligand bonding in a molecule. In this presentation various evidences are explained. Learn and grow.
1. Reaction mechanisms can be determined through various methods like identifying products, detecting intermediates through isolation, trapping or labeling studies, studying the effects of catalysts and acids, and performing kinetic studies.
2. Isotope labeling and crossover experiments involve using isotopically labeled reactants to determine whether reaction pathways are intra- or intermolecular. Kinetic isotope effects also provide information about which bonds are broken or formed in the rate-determining step.
3. Acid and base catalysis can indicate whether proton transfer is involved in the rate-determining step. General acid catalysis means proton transfer is rate-determining while specific catalysis means it is not.
Non-heme oxygen carrier proteins, Hemocyanin, Copper containing metalloprotein, Active site of deoxyhemocyanin and oxyhemocyanin, Oxidative addition of dioxygen, peroxide bridging, antiferromagnetic, Hemerythrin, Active site structure of deoxyhemerythrin and oxyhemerythrin, Comparison between hemoglobin, hemerythrin and hemocyanin
Metal nitrosyl compounds contain nitric oxide bonded as an NO+ ion, NO- ion, or neutral NO molecule. They can be classified into three classes based on the nitric oxide group present. Metal nitrosyls are coordination compounds where an NO molecule is attached as an NO+ ion to a metal atom or ion. Examples include metal nitrosyl carbonyls such as Co(NO+)(CO)3, metal nitrosyl halides such as Fe(NO+)2I, and metal nitrosyl thio-complexes involving Fe, Co, and Ni metals. These compounds can be prepared through the reaction of nitric oxide with metal compounds like carbonyls, halides, or ferrocyanides. Metal
Rotational spectroscopy measures the energies of rotational states of molecules. It can observe the rotation of polar molecules using microwave or infrared spectroscopy, and of non-polar molecules using Raman spectroscopy. Molecules can be modeled as rigid or non-rigid rotors. Diatomic and linear molecules can be modeled as rigid rotors, while distortions are accounted for in non-rigid rotor models. Vibrational states are modeled as harmonic oscillators, though anharmonicity is considered. Rotational and vibrational states are quantized, and selection rules apply to rotational-vibrational transitions.
Mössbauer spectroscopy involves the interaction of gamma rays with atoms and molecules. It provides information about the chemical environment and oxidation states of atoms based on how they absorb gamma rays. For the Mössbauer effect to occur, the emitting and absorbing atoms must be embedded in a solid crystal lattice to minimize recoil effects. This allows the resonant absorption of gamma rays. Analysis of parameters like isomer shift, electric quadrupole interactions, and magnetic interactions in the Mossbauer spectrum provide details about the chemical environment and oxidation state of atoms in a sample.
This document provides an overview of statistical mechanics. It defines microstates and macrostates, and explains that statistical mechanics studies systems with many microstates corresponding to a given macrostate. The Boltzmann distribution is derived, which gives the probability of finding a system in a particular microstate as being proportional to the exponential of the negative of the energy of that microstate divided by the temperature. Maxwell-Boltzmann statistics are described as applying to classical distinguishable particles, yielding the Maxwell-Boltzmann distribution. References for further reading are also included.
The document discusses a lecture on statistical thermodynamics. It introduces the concept of a partition function, which describes the possible energy states of a system and the probability of occupying those states. It provides examples of using the Boltzmann distribution and Lagrange multipliers to determine the most probable distribution of molecules among energy levels for different systems. The summary focuses on key statistical thermodynamics concepts introduced in the document.
For UG students of All Engineering Branches (Mechanical Engg., Chemical Engg., Instrumentation Engg., Food Technology) and PG students of Chemistry, Physics, Biochemistry, Pharmacy
The link of the video lecture at YouTube is
https://www.youtube.com/watch?v=t3QDG8ZIX-8
Nuclear Quadrupole Resonance Spectroscopy (NQR) is a chemical analysis technique that detects nuclear energy level transitions in the absence of a magnetic field through the absorption of radio frequency radiation. NQR is applicable to solids due to the quadrupole moment averaging to zero in liquids and gases. The interaction between a nucleus's quadrupole moment and the electric field gradient of its surroundings results in quantized energy levels. Transitions between these levels are detected as NQR spectra and provide information about electronic structure, hybridization, and charge distribution. NQR finds applications in studying charge transfer complexes, detecting crystal imperfections, and locating land mines.
Electron spin resonance (ESR) spectroscopy is a technique used to study compounds with unpaired electrons. In ESR, a sample is placed in a static magnetic field and irradiated with microwaves. This causes transitions between the electron spin energy levels. The absorption of microwave energy is detected to obtain an ESR spectrum. ESR spectra provide information about electron environments through parameters like g-values and hyperfine splitting patterns. ESR finds applications in studying transition metal complexes and unstable free radicals.
CONTENTS
INTRODUCTION
CONCEPTS OF WALSH DIAGRAM
APPLICATION IN TRIATOMIC MOLECULES
[IN AH₂ TYPE OF MOLECULES(BeH₂,BH₂,H₂O)]
INTRODUCTION
Arthur Donald Walsh FRS The introducer of walsh diagram (8 August 1916-23 April 1977) was a British chemist, professor of chemistry at the University of Dundee . He was elected FRS in 1964. He was educated at Loughborough Grammar School.
Walsh diagrams were first introduced in a series of ten papers in one issue of the Journal of the Chemical Society . Here, he aimed to rationalize the shapes adopted by polyatomic molecules in the ground state as well as in excited states, by applying theoretical contributions made by Mulliken .
It contains what are the shift reagents, and how they will use in NMR spectroscopy. It includes lanthanide shift reagents and their effect using NMR spectroscopy. It has mostly used shift reagents like Europium and their importance. paramagnetic species that affect the NMR spectra are also explained in detail. What are contact shift and pseudo-contact shift also explained. It contains what are the chiral shift reagent, and the advantages, and disadvantages of lanthanide shift reagents. Reference books are also included.
Hyperfine splitting occurs due to the interaction between an electron's spin and the nucleus' spin. This interaction causes each electron spin state to split into 2I+1 levels, where I is the nuclear spin quantum number. As examples, the document discusses the hyperfine splitting in hydrogen, where the nuclear spin is 1/2, and deuterium, where the nuclear spin is 1. Hyperfine splitting has applications in radio astronomy, nuclear technology such as laser isotope separation, and atomic clocks.
This document provides an overview of the application of phase rule to a three component system of acetic acid, chloroform, and water. It defines key terms like phases, components, and degrees of freedom. It explains Gibbs phase rule and how it applies to a three component system. Specifically, it discusses how the water-acetic acid-chloroform system can be represented on a triangular phase diagram, with acetic acid enhancing the miscibility of water and chloroform. The document outlines how the system transitions from two heterogeneous phases to a single homogeneous phase as the amount of acetic acid is increased.
This document discusses electronic spectra of metal complexes. It begins by defining quantum numbers related to electron configuration, such as L (total orbital angular momentum) and l (secondary quantum number). It then describes two main types of electronic transitions in coordination compounds: d-d transitions specific to metals, and charge-transfer transitions. The remainder of the document discusses charge-transfer transitions in more detail, defining ligand-to-metal and metal-to-ligand charge transfer, and how solvent polarity affects these transitions.
Transition metal carbonyls form when carbon monoxide bonds to a transition metal through both sigma and pi bonding. This synergistic metal-ligand bonding strengthens the metal-carbon bond. Metal carbonyls can be classified based on the ligands present and the number/structure of metal atoms. They exhibit a variety of reactions including substitution, reactions with halogens, and disproportionation. Metal carbonyls display properties related to their toxicity, magnetic behavior, thermal stability, and thermodynamic instability.
Arenium Ion Mechanism in Aromatic Electrophilic Substitution SPCGC AJMER
This document describes the arenium ion mechanism for aromatic electrophilic substitution reactions. The mechanism involves two steps: 1) the rate-determining attack of an electrophile on the aromatic ring to form a resonance-stabilized arenium ion intermediate, and 2) rapid departure of the leaving group to regenerate aromaticity. Evidence for this mechanism includes isolation of arenium ion intermediates and a lack of isotope effects. The first step is highly endothermic due to loss of aromaticity, making it rate-determining, while the second step regains aromatic stabilization in an exothermic process.
Electron Spin Resonance (ESR) SpectroscopyHaris Saleem
Electron Spin Resonance Spectroscopy
Also called EPR Spectroscopy
Electron Paramagnetic Resonance Spectroscopy
Non-destructive technique
Applications
Extensively used in transition metal complexes
Deviated geometries in crystals
This document provides an overview of metal carbonyls. It discusses how metal carbonyls are formed from transition metals and carbon monoxide, and examples like nickel tetracarbonyl and iron pentacarbonyl. The molecular orbital diagram of carbon monoxide is shown, explaining why it can participate in pi-backbonding. Infrared spectroscopy is described as a useful technique for analyzing metal carbonyls, as it can distinguish terminal from bridging carbonyl ligands based on the infrared absorption frequency. Factors like metal charge and other ligands that affect the carbonyl stretching frequency are also outlined. Finally, some applications of infrared spectra of metal carbonyls are mentioned.
Acid Base Hydrolysis in Octahedral ComplexesSPCGC AJMER
This document discusses acid and base hydrolysis in octahedral complexes. It covers factors that affect the rate of acid hydrolysis, including the charge on the complex, steric hindrance effects, and the strength of the leaving group. A higher positive charge, more steric hindrance, or stronger metal-leaving group bond each decrease the rate of acid hydrolysis according to first-order kinetics through a dissociative SN1 mechanism. Base hydrolysis of octahedral complexes can proceed by either associative SN2 or dissociative SN1 pathways depending on conditions.
This Presentation describes about the evidence of metal ligand bonding in a molecule. In this presentation various evidences are explained. Learn and grow.
1. Reaction mechanisms can be determined through various methods like identifying products, detecting intermediates through isolation, trapping or labeling studies, studying the effects of catalysts and acids, and performing kinetic studies.
2. Isotope labeling and crossover experiments involve using isotopically labeled reactants to determine whether reaction pathways are intra- or intermolecular. Kinetic isotope effects also provide information about which bonds are broken or formed in the rate-determining step.
3. Acid and base catalysis can indicate whether proton transfer is involved in the rate-determining step. General acid catalysis means proton transfer is rate-determining while specific catalysis means it is not.
Non-heme oxygen carrier proteins, Hemocyanin, Copper containing metalloprotein, Active site of deoxyhemocyanin and oxyhemocyanin, Oxidative addition of dioxygen, peroxide bridging, antiferromagnetic, Hemerythrin, Active site structure of deoxyhemerythrin and oxyhemerythrin, Comparison between hemoglobin, hemerythrin and hemocyanin
Metal nitrosyl compounds contain nitric oxide bonded as an NO+ ion, NO- ion, or neutral NO molecule. They can be classified into three classes based on the nitric oxide group present. Metal nitrosyls are coordination compounds where an NO molecule is attached as an NO+ ion to a metal atom or ion. Examples include metal nitrosyl carbonyls such as Co(NO+)(CO)3, metal nitrosyl halides such as Fe(NO+)2I, and metal nitrosyl thio-complexes involving Fe, Co, and Ni metals. These compounds can be prepared through the reaction of nitric oxide with metal compounds like carbonyls, halides, or ferrocyanides. Metal
Rotational spectroscopy measures the energies of rotational states of molecules. It can observe the rotation of polar molecules using microwave or infrared spectroscopy, and of non-polar molecules using Raman spectroscopy. Molecules can be modeled as rigid or non-rigid rotors. Diatomic and linear molecules can be modeled as rigid rotors, while distortions are accounted for in non-rigid rotor models. Vibrational states are modeled as harmonic oscillators, though anharmonicity is considered. Rotational and vibrational states are quantized, and selection rules apply to rotational-vibrational transitions.
Mössbauer spectroscopy involves the interaction of gamma rays with atoms and molecules. It provides information about the chemical environment and oxidation states of atoms based on how they absorb gamma rays. For the Mössbauer effect to occur, the emitting and absorbing atoms must be embedded in a solid crystal lattice to minimize recoil effects. This allows the resonant absorption of gamma rays. Analysis of parameters like isomer shift, electric quadrupole interactions, and magnetic interactions in the Mossbauer spectrum provide details about the chemical environment and oxidation state of atoms in a sample.
This document provides an overview of statistical mechanics. It defines microstates and macrostates, and explains that statistical mechanics studies systems with many microstates corresponding to a given macrostate. The Boltzmann distribution is derived, which gives the probability of finding a system in a particular microstate as being proportional to the exponential of the negative of the energy of that microstate divided by the temperature. Maxwell-Boltzmann statistics are described as applying to classical distinguishable particles, yielding the Maxwell-Boltzmann distribution. References for further reading are also included.
The document discusses a lecture on statistical thermodynamics. It introduces the concept of a partition function, which describes the possible energy states of a system and the probability of occupying those states. It provides examples of using the Boltzmann distribution and Lagrange multipliers to determine the most probable distribution of molecules among energy levels for different systems. The summary focuses on key statistical thermodynamics concepts introduced in the document.
The document summarizes key concepts from a lecture on combinatorics, probability, and multiplicity as they relate to statistical mechanics. It discusses two model systems - a two-state paramagnet and the Einstein solid. For the paramagnet, it defines the energy levels and multiplicity of macrostates. The multiplicity represents the number of configurations or arrangements of particles between energy levels. It also discusses the probability distribution of particles between energy levels for large systems. For the Einstein solid model, it describes a solid as 3N independent quantum harmonic oscillators, each with discrete energy levels defined by integers. It provides expressions for the total internal energy of the system.
This document discusses phase space and the statistical mechanics of classical particles. It can be summarized as:
1. The state of a classical particle is defined by its position and momentum coordinates, which together form a point in the particle's 6D phase space. For a system of N particles, the full 6N-dimensional phase space is called the Γ-space.
2. The minimum volume element in phase space is called the unit cell, with volume h^3 according to Heisenberg's uncertainty principle.
3. The number of quantum states available to particles with energies between E and E+dE is given by the ratio of the volume of phase space to the volume of a unit cell.
Helium gas with Lennard-Jones potential in MC&MDTzu-Ping Chen
Helium gas with Lennard-Jones potential in MC&MD
- The document discusses using Monte Carlo and Molecular Dynamics simulations to model helium gas atoms interacting via the Lennard-Jones potential.
- Initial MC simulations of many helium atoms did not converge well. Simulations were then reduced to just four atoms, allowing analysis of compressibility and heat capacity as functions of volume and temperature.
- For four atoms in a tetrahedral configuration, equations were derived relating potential energy, pressure, compressibility, and heat capacity to volume and temperature in the low-temperature regime.
This document discusses quantum theory and the electronic structure of atoms. It introduces quantum numbers like principal, angular momentum, and electron spin quantum numbers used to describe atomic orbitals. Atomic orbitals like s, p, and d orbitals are described along with their shapes and orientations. Electron configurations follow rules like the Aufbau principle, Pauli exclusion principle, and Hund's rule. The document shows how electrons fill atomic orbitals in order of increasing energy to write electron configurations of elements, which are represented using noble gas cores. Exceptions to electron filling order are noted for some transition metals.
Partition function indicates the mode of distribution of particles in various energy states. It plays a role similar to the wave function of the quantum mechanics,which contains all the dynamical information about the system.
1) The document discusses molecular partition functions, which represent the sum of probability factors for how the energy of an assembly of molecules is partitioned among different energy levels.
2) It provides equations for calculating translational, rotational, and vibrational partition functions for diatomic molecules based on principles of quantum mechanics and by modeling molecular motion as simple harmonic oscillators.
3) The key factors that determine each type of partition function are discussed, such as temperature, molecular mass, moment of inertia, and vibrational frequency.
This document discusses a computational study of MAX phases using density functional theory. MAX phases are a group of materials that exhibit both metallic and ceramic properties. The study uses the WIEN2k software to calculate the electronic structure and properties of MAX phases like Cr2AlC and Cr2GaC from their density of states and band structure plots. Manganese is incorporated into the structures at varying concentrations to study their magnetic properties.
1) The document discusses the Maxwell-Boltzmann distribution, which describes the distribution of velocities or energies of particles in a gas. Maxwell and Boltzmann developed this distribution based on assumptions about molecular motion in gases.
2) The Maxwell-Boltzmann distribution can be derived using statistical mechanics and considering the multiplicity, or number of arrangements, of particles into different energy states. Maximizing the multiplicity subject to conservation constraints leads to the Maxwell-Boltzmann distribution.
3) The derivation utilizes concepts such as the density of states function, integrals over energy states, and results in identifying temperature as proportional to the average kinetic energy per particle divided by the Boltzmann constant.
This document discusses the methodology of thermodynamics and statistical mechanics. It explains that thermodynamics studies the relationships between macroscopic properties like volume and pressure, while statistical mechanics links these macroscopic properties to the microscopic properties of individual molecules through analysis of their positions and momenta. It introduces the key ensembles used in statistical mechanics - the canonical ensemble, which models systems in thermal equilibrium with a heat bath at fixed temperature; the grand canonical ensemble, which models open systems that can exchange both energy and particles with a reservoir; and the microcanonical ensemble, which models isolated systems with a fixed total energy.
Struggling with your statistical physics exam? Live Exam Helper offers expert exam help services for all levels. Our experienced tutors and study aids can help you master the concepts and formulas of statistical physics, so you can ace your exam with confidence. Visit our website https://www.liveexamhelper.com/physics-exam-help.html to learn more about our services!
1) Statistical mechanics deals with relating the macroscopic behavior of a system to the microscopic properties of its particles. A system's macrostate is defined by the distribution of particles among compartments, while each distinct microscopic arrangement is a microstate.
2) Phase space combines position and momentum space, specifying the complete state of a system. For classical particles, the Maxwell-Boltzmann distribution describes average particle numbers. Quantum statistics include Bose-Einstein and Fermi-Dirac distributions.
3) A photon gas in an enclosure reaches thermal equilibrium where the Bose-Einstein distribution applies. The number of photon energy states is calculated from phase space considerations.
This document summarizes research exploring different two-body interactions and their effects on nuclear spectra. Key findings include:
1) Using only T=1 two-body matrix elements, it is possible to generate near-equally spaced spectra for even spin states in 44Ti, similar to vibrational spectra.
2) A J(J+1) interaction produces perfect rotational spectra, while a scaled T=1 interaction still yields good rotational behavior.
3) A "123" interaction with specific T=1 matrix elements produces a spectrum with equally spaced levels from I=6-12, resembling vibrational behavior.
4) Wavefunctions for higher spin states under this interaction show a separation based on the
Different physical situation encountered in nature are described by three types of statistics-Maxwell-Boltzmann Statistics, Bose-Einstein Statistics and Fermi-Dirac Statistics
Nuclear Decay - A Mathematical PerspectiveErik Faust
Radioactivity as a phenomenon is often misunderstood: if one says ‘Radioactive’, most people will think about disastrous electrical plants, dangerous bombs and other forms of life-threatening details. In my native Germany, members of the Green party have been campaigning for a decade to put an end to nuclear energy. Only few think of the useful aspects of this unique actuality, although radiotherapy is most promising of tools in the fight against cancer, and radioactive dating allows us to identify the age of any historical item. But even fewer people see radioactivity as the natural process that it actually is: A spontaneous mechanism, in which one nucleus decays into another. As an aspiring Physicist and Engineer, Radioactivity is one my favourite topics in the realm of science. I am fascinated at how we are able to predict exactly how many Nuclei will decay in a certain amount of time, but not say for certain which Nuclei exactly will do so.
This document discusses statistical thermodynamics and key concepts such as:
- Relating microscopic properties like molecular energies to macroscopic bulk properties using statistics.
- Developing the Maxwell-Boltzmann distribution of molecular velocities based on a statistical analysis of energy levels.
- Using statistical mechanics and undetermined multipliers to calculate thermodynamic properties like heat capacity from microscopic properties.
The document discusses the four quantum numbers used to describe electrons in atoms:
1. The principal quantum number (n) designates the principal electron shell. n is a positive integer.
2. The azimuthal quantum number (l) describes the orbital shape and ranges from 0 to n-1.
3. The magnetic quantum number (ml) determines the number and orientation of orbitals in a subshell.
4. The electron spin quantum number (ms) gives the direction of electron spin.
Together, the four quantum numbers uniquely describe each electron in an atom. Electrons fill atomic orbitals according to the Aufbau principle, occupying lower energy orbitals first.
CBSE Class XI Chemistry Quantum mechanical model of atomPranav Ghildiyal
Classical mechanics successfully describes macroscopic objects but fails for microscopic objects like atoms and molecules. Quantum mechanics was developed to account for these microscopic objects, which exhibit both wave-like and particle-like properties. It describes that electrons can have distinct quantum states defined by quantum numbers like principal (n), azimuthal (l), magnetic (m), and spin (s). The values of these quantum numbers determine properties of atomic orbitals like shape, size, and energy.
Quantum numbers describe the distribution of electrons in an atom and include:
1) The principal quantum number (n) which designates the size of an orbital and possible energy level. Larger n corresponds to larger orbitals.
2) The angular momentum quantum number (l) which describes the shape of an orbital as s, p, d, or f.
3) The magnetic quantum number (ml) which specifies the orientation of an orbital.
Together these quantum numbers fully describe an atomic orbital and the electron it contains. Quantum numbers are necessary to apply the Pauli exclusion principle, which states that no two electrons in an atom can have the same set of quantum numbers.
Similar to Introduction statistical thermodynamics.pptx (20)
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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2. Introduction
The limitation of the classical treatment is that we are not certain about the
rate and mechanism of the process under investigation. The relevant
information can be had provided we know the link between the
macroscopic properties and the microscopic properties of the system.
Such types of correlation are studied by statistical mechanics.
The nature and details of the microscopic properties (like speed,
momentum, energy of each particle) are provided by Quantum
mechanics. The macroscopic properties of the system are determined
by thermodynamic principles. Statistical mechanics acts as a bridge
between thermodynamics and quantum mechanics.
3. Thus the study of link between thermodynamics and quantum mechanics
is called statistical thermodynamics.
Summarizing ,we can state that quantum mechanics provides information
about the energy of the molecular system, statistical mechanics tells us
about the possible arrangement of the energy among various molecules
of the system, and introduces the concept of probability & partition functions.
Statistical thermodynamics deals with the relationships between the
probability, partition functions and the thermodynamic properties.
4. The Role of Statistical Mechanics
The energy expressions suggest that the energy of the molecules present
in a system will depend on quantum numbers. For a lower
quantum number, the energy of the state is lower than that for a higher
quantum number.
Now the question is that how the molecules are distributed amongst the
possible energy levels?
In other words, we want to know that how many molecules are present
in the lowest energy level, highest energy level, and how many are
present in various intermediate energy levels.
That is to say, we wish to have information about the best possible
arrangements of the molecules in various quantum or energy levels.
Such a description is given by the term probability denoted by W.
5. Common Terms
Assembly A number N of identical entities is called an assembly. If the entities
are single particles then the assembly is called simply as a system.
If the entities are assemblies of particles then we call the number N as assembly
of assemblies or an ensemble.
An ensemble consists of a large number of replicas of the system under
consideration. Each member of the ensemble has same number of molecules (N),
same volume V, and same energy E. It is called a micro canonical ensemble.
Canonical ensemble The ensemble consisting of a large number of
assemblies (systems) each having the same value of N, V, and T is called a
canonical ensemble. At equilibrium each assembly (member) of the canonical
ensemble has the same temperature, but not necessarily the same energy E.
The value of E will fluctuate about the ensemble average value.
Occupation number It is the number of system in that particular state. The
set of occupation number is called a distribution.
6. Suppose we have to distribute 100 distinguishable balls into five boxes in such
a way that each box contains 20 balls. This over all distribution is called a
macrostate. The detailed description of the distribution that balls numbered 1
to 20 can appear in one box, 21 to 40 in the second, 41 to 60 in the third etc, is
called the microstate. The number of microstates which correspond to a
macrostate is referred to as thermodynamic probability.
7. Statistical weight factor g It is the degree of degeneracy of a particular energy
level, and is equal to the energy states of any energy level. For example, the energy
level of a particle in the three dimensional box is given by 1 = (nx
2 + ny
2 + nz
2 )
h2/8mV. The energy of the quantum states 211, 121, and 112 is the same, but the
three states are distinct. Hence the degree of degeneracy is 3, and the statistical
weight factor g is also 3.
Configuration Various equivalent ways of achieving a state is called a
configuration of the system. For example, consider two coins a and b. The state of
showing 1 head (H) and 1 tail (T) can be achieved in two ways:
•Coin a shows a head and coin b tail.
•Coin a shows a tail and coin b head.
There are two ways to arrive at the same state of 1 H and 1T, hence the
number of configuration is 2.
8. Probability The probability of a state of a system is defined as the
number of configurations leading to that particular state divided by the
total number of configurations possibly available to the system. For
instance consider the tossing of a coin. It can either show head or tail.
Thus, total number of possible configurations of the state of the coin is
two (1 head + 1 tail), and the probability of showing head is one out of
two configurations i.e. ½, similarly the probability of showing a tail is
½.
9. Thermodynamic Probability
The thermodynamic probability of system is equal to the number of
ways of realizing the distribution. The symbol for probability is W.
Expression of Thermodynamic Probability
Consider an assembly of N identical particles of a gas at a temperature T,
volume V and total energy E. Let N0, N1, N2, … etc., group of molecules be palced in
energy levels 0, 1, 2, … etc. in such a way that the total number of molecules and
total energy are constant.
The number of ways in which the molecules can be distributed into
different energy level is calculated by the principle of permutation and
combination. Thus, the probability W is expressed as
… (6.5).
Here N! is N factorial and is written as
N! = N x (N- 1) (N- 2) x … x 4 x 3 x 2 x 1
Similarly, the terms in the denominators are different.
10. Permutation and combination. Suppose there are 25 particles which
have to be put in groups of 12, 6, 4, 2, 1.
The number of ways of arranging there are
This can be calculated as follows. If there is no restriction on
group placement, the number of ways would be 25 x 24 x 23 ….. x 1
which is denoted by 25! Since one has to place the particles in groups, the
number has to be divided b y 12 ! 6 ! 4 ! 2 ! 1 !.
The methods of evaluating the configurations for systems of different
nature are summarized as follows.
i) Number of ways of selecting n distinguishable objects from the N
distinguishable objects (n < N) is
ii)The number of ways of arranging N indistinguishable objects into l
distinguishable locations with only one object in a location is (l ≥ N) is
11. iii) The number of configurations for N indistinguishable objects into l
distinguishable locations without any restriction on occupation is
iv) The number of ways of putting N distinguishable objects into l locations
with no restriction is
v) In writing the factorials we keep in mind that
12. Example 6.1. Calculate the number of ways of distributing distinguishable
molecules a, b, c, between three energy levels so as to obtain the following set
of occupation number.
N0 = 1, N1 = 1, N2 = 1, that is each energy level is occupied by one molecule.
Solution: The probability W is given by
Here N = 3; N0 = N1 = N2 = 1,
There are six ways of distributing the three molecules as required in the
problem. The same result may be obtained by considering the following
treatment:
13. Energy state
N = 1, N1 = 1,
N2 = 1
Configurations
I II III IV V VI
0 a a b b c c
1 b c c a a b
2 c b a c b a
There are six possible configuration
14. Example 6.2. Calculate the number of ways of distributing four
molecules in four energy levels so as there are 2 molecules in the level
0, 1 molecule in the 1 energy level, 1 molecule in the 2 energy level,
and zero in the 3 energy level i.e.
N0 = 2, N1 = 1, N2 = 1, N3 = 0,
Solution : The probability equation gives
Energy state
N0 = 2, N1 = 1,
N2 = 1, N3 = 0
Configuration
I II III IV V VI VII VIII IX X XI XII
0 ab ab ac ac ad ad bc bc dc dc bd bd
1 c d b d c b a d a b a c
2 d c d b b c d a b a c a
3 0 0 0 0 0 0 0 0 0 0 0 0
15. Example 6.3. Calculate the number of ways to distribute
•Two distinguishable objects in two boxes
•Two distinguishable objects in three boxes
•Two indistinguishable objects in two boxes
•Two indistinguishable objects in three boxes
Solution :
•Number of objects = N = 2
Number of boxes (location) = l = 2
Number of configurations = W = lN = 22 = 4
• N = 2; l = 3;
W = lN = 32 = 3 x 3 = 9