Homework Help
Homework Help
View Details
Assignment Help
Assignment Help
View Details
Online Tutoring
Online Tutoring
View Details
Home » Chemistry Homework Help » Physical Chemistry » Vibrational Motions
Vibrational Motions
A summation can be used to obtain the vibrational partition function and the vibrational contribution to the thermal energy.

With the classical picture, gas molecules fly around, tumble end over end, and vibrate. The atoms of a diatomic molecule are described as vibrating against one another, as do two balls joined by a spring. But the vibrational energy is quantized, and the expression for the allowed vibrational energies was developed. Now we see how the molecules of a macroscopic sample are distributed throughout these allowed vibrational states.

Vibrational partition function: each way in which a molecule can vibrate leads to a set of vibrational states with energies given by



For each vibrational mode we can thus write



Now, in contrast to translational and rotational energy spacings, the vibrational spacings are appreciable compared with kT, and therefore only a few of the terms in the series will contribute appreciably to the partition function. The partition function sum cannot be replaced by an integral, but we can obtain an expression by developing the summation

qvib mode = 1 + e-hv vib/(kT) + e -2hv vib/(kT) + ……

With the introduction of the convenient symbol



This becomes

qvib mode = 1 + e-x + e-2x + …. = 1 + (e-x)1 + (e-x)2 + ……

This series can be recognized as the binomial expansion of (1 – e-x)-1, and thus we have



Example: studies of the absorption of infrared radiation, which is primarily due to vibrational-energy changes, shows that radiation with v⁻ = 1000 cm-1 and a frequency 3 × 1013 s-1 is typical of that required to promote molecules from one vibrational energy to the next higher energy. Calculate the energy, in joules, for this representative vibrational-energy separation. What is the vibrational partition function for such a vibrational mode at (a) 25˚C and (b) 1000˚C?

Solution: the energy of the photons of this infrared radiation is calculated from ε = hv as ε = (6.626 × 10-34 J s) (3.0 × 1013 s-1) = 20 × 10-21 J. now we can proceed to the calculation of x = hv vib/(kT) and then to the first expression
   
At 25˚C = 298 K we have



And



The vibrational energy spacing hvvib is so much greater than kT at this temperature that little more than the lowest-energy vibrational state is available to the molecules.
   
At 1000˚C = 1273 K the results are

X = 1.13 and qvib = 1.48

At this temperature the excited vibrational states are appreciably available to the molecules.

Vibrational energy: the thermal energy for any degree of freedom is obtained from the general expression. The vibrational partition function gives



The thermal energy contributed by a vibrational mode is then obtained as



Alternatively, with x replaced by hv vib/(kT) and R by Nk, this can be written



Plots of (U – U0) vib mode, for various values of T as a function of the vibrational-energy-level spacing are suitable in this amplitude.

Services:- Vibrational Motions Homework | Vibrational Motions Homework Help | Vibrational Motions Homework Help Services | Live Vibrational Motions Homework Help | Vibrational Motions Homework Tutors | Online Vibrational Motions Homework Help | Vibrational Motions Tutors | Online Vibrational Motions Tutors | Vibrational Motions Homework Services | Vibrational Motions

Submit Your Query ???
Topics
Covalent Radii Crystal Shapes, Point Groups Diffraction Pattern Assignments Electron Diffraction Ionic Radii Lattice Energies Diffraction Lattices, Unit Cells Neutron Diffraction Waals Radii X-ray Diffraction Bond Moments Electric Capacitor Atoms, Molecules Properties Paramagnetism Electrolytic Dissociation Solution Ionic Strength Solvent Dielectric Effect Electrolysis Solutions Ionic Mobilities Electrolytes In Solutions Solutions Molar Conductance Solutions Specific Conductance Electrochemical Cell EMF Electrodes Ion Selective Electrodes Junction Potentials Cells Electromotive Force Standard Electrode Potentials Collision Theory Gas Viscosity Theory Elementary Reactions Lasers Molecule-Molecule Collisions Electrochemical Cell Photochemical Quenching Surface Decompositions Atomic Molecular Energies Molecular Energies Particle-in-a-box Particle-on-a-line Rotational Energies Schrodinger Wave Equation De Broglie Wave Length Vibrational Energies Waves And Particles Boltzmann Distribution Gas Heat Capacities Metals Heat Capacities Molecules Collection Energies One Dimensional Motion Partition Function Rotational Motions Thermal Energy Three Dimensional Motion Vibrational Motions Aqueous Ion Energies Bond Energies Chemical Systems Energy Enthalpy, Chemical Reactions Chemical System Enthalpy Thermodynamics First Law Heat Capacities Thermodynamics Molecular Thermal Energy Standard Enthalpy Substance Carnot Cycle Absolute Zero Entropies Entropy Thermodynamics Laws Entropy Molecular Basis Third Law Molecular Basis Rotational Energy Thermodynamics Second Law Thermodynamics Third Law Vapourization Entropy Vibrational Entropy Equilibria And Distributions Real Gases Equilibria Free Energy Equilibrium Constant Free Energy And Pressure Free Energy, Temperature Free Energy Function Free Energy Real Gases Free Energy Fugacity Non-ideal Gases Fugacity Thermodynamic Properties Chemical Equilibria Boyle Gas Pressure Continuity Of States Critical Point Gas Mixtures Kinetic Molecular Theory Gases-Properties, Theories Molecular Energies, Speed Molecular Interactions Real Gas PVT Temperature Volume Waals Gases Behaviour Waals Critical Point Molecular Diameters Virial Equation Diffusion Coefficient Diffusion Molecular View Donnan Membrane Equilibria Electrophoresis Macromolecular Dynamics Average Mass Range Solution Viscosity Sedimentation And Velocity Colloids Macromolecules Micelles Adsorption Isotherm Adsorption Of Gases Boiling Point Diagrams Pressure Temperature Relation Distillation Eutectic Formation Immiscible Liquids Phase Equilibria Liquid Surfaces Phase Rule Pressure Phase Diagrams Solid Compound Foundation Surface Tension Vapour Pressure Three Component System Vapour Pressure Composition Atomic States Bohr Atom Electron Spin Angular Momentum Hydrogen Hydrogen Atom Spectra Hydrogen Radical Factor Quantum Atomic Structure Quantum Mechanical Operators Variation Theorem Enzymes Catalyzed Reactions First Order Rate Equations Flash Photolysis Chemical Reactions Mechanism Enzyme Reactions Mechanism Reactions Mechanisms Photochemical Reactions Rate Equation Second Order Rate Equations Temperatures And Rates Unimolecular Gas Reactions Absorption Coefficient Einstein Coefficient Electromagnetic Induction Electronic Spectra Electron Spin Spectroscopy Infrared Adsorption Spectroscopy Microwave Absorption Nuclear Spin States Nuclear Magnetic Resonance Photoelectron Spectroscopy Polyatomic Vibrational Spectra Rotational Vibrational Spectra Conjugated Systems Spectra Transition Moment Character Tables Symmetry Group Theory Molecular Symmetry Types Orbital Symmetries Point Groups Reducible Representation Symmetry Elements, Operations Molecular Properties Symmetry Transformation Matrices Diatomic Molecule Orbitals Electronegativity Hybridization Hydrogen Molecule Ion Ionic Bond Molecular Orbitals Orbitals Pie Electrons Two Electron Bond Virial Theorem Partial Molal Properties Solute Free Energy Ideal Mixtures Solution Thermodynamic Property Liquid Vapour Free Energies Osmotic Pressure Partial Molal Quantities Solvent Free Energy Vapour Pressure Lowering