A diatomic molecule behaves like a quantum harmonic oscillator with the force constant 12.0 N/m and mass
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- Solid metals can be modeled as a set of uncoupled harmonic oscillators of the same frequency with energy levels given by En = ħwn n = 0, 1, 2,... where the zero-point energy (the lowest energy state) of each oscillator has been adjusted to zero for simplicity. In this model, the harmonic oscillators represent the motions of the metal atoms relative to one another. The frequency of these oscillators is low so that ħw = = 224 KB and the system vibrational partition function is given by 3N Z ² = la₁ - (1 1 e-0/T). (a) If the system contains one mole of atoms, find the average energy (in J) of this system at T= 172 K. (You can use = BkB.) T (b) What is the absolute entropy (in J/K) for this system? You can use either the Gibbs expression for S, or the system partition function to make this evaluation (they are equivalent, as your reading assignment indicates).arrow_forwardA harmonic oscillator consists of a 0.020 kg mass on a spring. The oscillation frequency is 1.50 Hz, and the mass has a speed of 0.480 m/s as it passes the equilibrium position. (a) What is the value of the quantum number n for its energy level? (b) What is the difference in energy between the levels En and En+1? Is this difference detectable?arrow_forwardA nitrogen molecule (N2) vibrates with energy identical to a single particle of mass m = 1.162 x 10-26 kg attached to a spring with a force constant of k = 1500 N/m. Suppose the energy levels of the system are uniformly spaced as shown in the figure below. The lowest energy level is often called the “ground state” and is assigned an integer value n = 1. The next higher energy level is often called the “first excited state” and is assigned an integer value n =2. (1) What is the vibration frequency of this molecule? (2) How much energy is required to excite the molecule from the ground state (n = 1) to the first excited state (n = 2)? (3) How much energy is required to excite the molecule from the first excited state (n = 2) to the state n = 5?arrow_forward
- (2nx sin \1.50. 2nz Consider the case of a 3-dimensional particle-in-a-box. Given: 4 = sin(ny) sin 2.00. What is the energy of the system? O 6h?/8m O 4h²/8m O 3h2/8m O none are correctarrow_forwardRecall from Section 14.3 that the average kinetic energy of an atom in a monatomic ideal gas is given by KE=(3/2)kT, where k = 1.38 x 10-23 J/K and T is the Kelvin temperature of the gas. Determine the de Broglie wavelength of a helium atom (mass = 6.65 x 10-27 kg) that has the average kinetic energy at room temperature (292 K). Number i 7.38E-11 Units marrow_forwardA hypothetical molecule oscillates with a natural frequency of 1.4 × 1013 Hz. Part (a) What is the energy difference, in electron volts, between adjacent harmonic oscillator states of the hypothetical molecule? Part (b) What is the quantum number of the state of the hypothetical molecule that has an energy of 0.75 eV? Round your answer to the nearest integer.arrow_forward
- A collection of atoms has 20% of the sample in a state 7.60 eV above the ground state. If these emit coherent radiation, what is the wavelength of the laser light produced in nanometers? (c = 3.00 × 108 m/s, h = 6.626 × 10-34 J ∙ s, 1 eV = 1.60 × 10-19 J) Give your answer as a whole number.arrow_forwardAn atom of iron has a radius of 156. pm and the average orbital speed of the electrons in it is about ×5.7*10^7 m/s. Calculate the least possible uncertainty in a measurement of the speed of an electron in an atom of iron. Write your answer as a percentage of the average speed, and round it to 2 significant digits.arrow_forwardIn this problem, you will calculate the center of mass of two molecules. For each molecule, draw a picture. Clearly indicate the origin of your coordinate system, the locations of the atoms, and the location of the center of mass. Consult a periodic table to find the masses of the atoms. (a) Electron diffraction experiments reveal that the distance between the centers of the carbon (C) and oxygen (O) atoms in a carbon monoxide molecule is about 1.100 x 10-10 m. Where is the center of mass of a CO molecule? (b) In an ammonia molecule (NH3), the atoms are arranged in a pyramid.* The three hydrogen atoms (H) form an equilateral triangle at the base of a pyramid. The distance between the centers of the hydrogen atoms is 1.628 x 10-10 m; thus, the center of the triangle is 9.399 x 10-"m from each hydrogen atom. The nitrogen atom (N) sits at the apex of the pyramid. The distance between the center of the nitrogen atom and any hydrogen atom is 1.014 x 10-10 m. Where is the center of mass of…arrow_forward
- a)Suppose a hydrogen molecule in its ground state is dissociated by absorbing a photon of ultraviolet light, causing the two hydrogen atoms to fly apart. What photon energy will give each atom a speed of 19 km/s? The mass of a hydrogen atom is 1.7×10^−27 kg Express your answer to two significant figures and include the appropriate units.arrow_forwardA quantum system is described by a wave function (r) being a superposition of two states with different energies E1 and E2: (x) = c191(r)e iEit/h+ c292(x)e¯iE2t/h. where ci = 2icz and the real functions p1(x) and p2(r) have the following properties: vile)dz = ile)dz = 1, "0 = rp(x)T#(x)l& p1(x)92(x)dx% D0. Calculate: 1. Probabilities of measurement of energies E1 and E2 2. Expectation valuc of cnergy (E)arrow_forwardConsider a quantum mechanical ideal harmonic oscillator having a zero point energy of 1.4*10^-20J. how much energy could be released if the oscillator makes a transition from n=4 to n=2 states? a)0.69*10^19J b)2.88*10^-20J c)5.76*10^20J d)none are correctarrow_forward
- University Physics Volume 3PhysicsISBN:9781938168185Author:William Moebs, Jeff SannyPublisher:OpenStax