1) Take the derivative of this function to find the force function associated with it.2) Demonstrate that for values of r close to re, the potential is close to harmonic: i.e., the forceis proportional to displacement and opposite in direction. (Suggestion: expand the exponentialfunction as a power series.)3) Show that for large amplitudes, the vibrational frequency of the oscillator is less than thefrequency of an equivalent harmonic oscillator. (Suggestion: include higher order terms in theexpansion.) The Morse potentialThe harmonic potential, V(x) = ½kx?, is useful start for modelling molecular vibrations, but it haslimitations. A realistic potential between to atoms should accurately represent the sharpincrease in the potential as two nuclei come in close proximity, and also have the ability for abond to break: that is, an asymptote V →0 as x →00.One option, as shown in the figure, is the Morse potential:V(r) = D(1 – e-«(r=re))21510101520The parameter D is the well depth (or binding energy) of the potential, re is the bond length,and a is the anharmonicity constant.

The harmonic potential, V(x) = ¼kx?, is useful start for modelling molecular vibrations, but it has limitations. A realistic potential between to atoms should accurately represent the sharp increase in the potential as two nuclei come in close proximity, and also have the ability for a bond to break: that is, an asymptote V → 0 as x →o. One option, as shown in the figure, is the Morse potential: V(r) = D(1 – e-a(r=re))2

The harmonic potential, V(x) = ¼kx?, is useful start for modelling molecular vibrations, but it has limitations. A realistic potential between to atoms should accurately represent the sharp increase in the potential as two nuclei come in close proximity, and also have the ability for a bond to break: that is, an asymptote V → 0 as x →o. One option, as shown in the figure, is the Morse potential: V(r) = D(1 – e-a(r=re))2

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

See similar textbooks

Similar questions

The coordinates of unit a, of the atoms in a body-centered cubic lattice are (0,0,0), (0,1,0), (0,0,1), (0,1,1), (1,0,0), (1,1,0), (1,0,1), (1,1,1) and (1/2;1/2,1/2). Calculate the structure factors Fhkl when all the atoms are identical
If one of the two electrons of a H2 molecule is removed, we get a hydrogen molecular ion H+2. In the ground state of an H+2 , the two protons are separated by roughly 1.5 Å, and the electron is roughly 1 Å from each proton. Determine the potential energy of the system. Specify your choice of the zero of potential energy.
Pr. 5. Calculate the force between the two atoms in the diatomic hydrogen molecule that has the following potential: V(x)=(x-xo)²-Vo. Where Vo is the binding energy and ro. 6 are constant parameters.
A one-dimensional square well of infinite depth and 1 Å width contains 3 electrons. The potential well is described by V = 0 for 0 1 Å. For a temperature of T = 0 K, the average energy of the 3 electrons is E = 12.4 cV in the approxination that one neglects the Coulomb interaction between clectrons. In the same approximation and for T = 0 K, what is the average cuergy for 4 electrons in this potential well?
One model for the potential energy of a two-atom molecule, where the atoms are separated by a distance r, is U(r) = Uo[(¹) ¹2 – ( )²] where ro = 0.8 nm and U₁ = 6.1 eV. Note: 1 eV = 1.6 × 10-19 J. Some helpful units: [Force] = eV/nm [Energy] = eV [distance] = nm Equilibrium Distance What is the distance between the atoms when the molecule is in stable equilibrium? Click here for a hint T'eq Hint: Hint: Hint: Hint: Hint: Hint: Force If the distance between the atoms increases from equilibrium by r₁ = 0.35 nm, then what is the force from one atom on the other associated with this potential energy? (Enter your answer as postive if they repel each other, and negative if they attract.) Fr(req+r₁) Hint: Hint: 0.89105934nm Kinetic Energy Hint: The atoms are oscillating back and forth. The maximum separation of the atoms is r₂ = 2 nm. What is the kinetic energy of the atoms when they are separated by the equilibrium distance? Click here for a hint K(req) Hint: Hint: = -1.288eV/nm 3.99eV
Your answer is partially correct. An electron, trapped in a one-dimensional infinite potential well 366 pm wide, is in its ground state. How much energy must it absorb if it is to jump up to the state with n=7? Number 139.6 Units eV
A potential function is shown in the following with incident particles coming from -0 with a total energy E>V2. The constants k are defined as k₁ = 2mE h? h? k₂ = √√2m (E - V₁) h² k3 = √√2m (E - V₂) Assume a special case for which k₂a = 2nπ, n = 1, 2, 3,.... Derive the expression, in terms of the constants, k₁, k2, and k3, for the transmission coefficient. The transmis- sion coefficient is defined as the ratio of the flux of particles in region III to the inci- dent flux in region I. Incident particles E>V₂ I V₁ II V2 III x = 0 x = a
What is the energy level difference between adjacent levels ∆En = En +1 - En for the simple harmonic oscillator? What are ∆E0, ∆E2, and ∆E20? How many possible energy levels are there?
How many electrons can occupy the system with l=0, l=2 and l=4. What is number of possibleorientations of the orbital angular momentum with l=4? What is the smallest z-component of theorbital angular momentum?
Let u,= 0.02 cm 1 and u, = 0.04 cm be the partial linear attenuation coefficients in the slab shown in the figure below. Let L = 5 cm and No = 106 particles. How many particles N, are transmitted, and how %3D many are absorbed by each interaction process in the slab? dt Uncharged partleles No N,
Problem 1 Consider a molecular two-level system with a twofold degenerate ground state (la- beled “0") and a threefold degenerate level (labeled “1") with an energy (expressed in wavenumbers) 150.0 cm-1 above the ground state. (a) What is the proportion N1/N of molecules in the higher level at T = 400 K? (b) What is the molecular average energy (relative to the ground state and expressed in wavenumbers) when the temperature is T = 400 K? (c) What is the proportion No/N of molecules in the ground state when T → 0K? (d) What is the proportion N1/N of molecules in the higher level when T →?
In this and following questions, we develop a model for spontaneous emission of a photon by a diatomic molecule AB (a model molecule), which rotates and vibrates. In intermediate calculations, atomic units (a.u.) will be used: unit of mass = the mass of electron, unit of charge is the proton charge e, (e is a positive constant so that the charge of electron is -e). The initial state of the molecule is an excited rotational (1=1) and excited vibrational state (v=1). We consider a molecule with the reduced mass µ = 10,000 a.u. (it is similar to the mass of CO). After emitting a photon, the molecule will go to the 1=0, v=0 state. The first question is about the model potential of the molecule. It is represented by a potential of the form: V(r) = C6 p12 C12 p6 " where r is the distance between A and B in the molecule, C6 and C12 are positive constants (C6 =2 and C₁2-1). This potential has a well meaning that the molecule is bound. The first thing to do is find vibrational states of the…