(a) An atom has a non-degenerate electronic state 1000 cm-1 above a non- degenerate electronic ground state. Determine: (i) the electronic partition function qE at 1500 K; (ii) the relative population of the upper state at 1500 K; (ii) the mean electronic energy (ɛ) at 1500 K;
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- (1) A single particla quantum mechanical oscillator has energy levels (n + 1/2) hw, where n = 0, 1, 2, .. and w is the natural frequency of the oscillator. This oscillator is in thermal equi- librium with a reservoir at temperature T. (a) Find the ratio of probability of the oscillator being in the first excited state (n = 1) to the probability of being in the ground state. (b) Assuming that only the two states in Part la are occupied, find the average energy as a function of T. (c) Calculate the heat capacity at a constant volume. Does it depend on temperature?True or false? (a) The photon emitted in an n = 3 to n = 2 transition in the H atom has a lowerfrequency than the photon for an n = 2 to n = 1 H-atom transition. (b) The groundstate energy ofHe is about 4 times the ground-state energy of H. (c) yis zero at the nucleus for all H-atomstationary states. (d) For the ground state of the H atom, y2 is a maximum at the nucleus. (e) Themost probable value of the electron–nucleus distance in a ground-state H atom is zero. ( f ) Thesmallest allowed value of the atomic quantum number n is 0. (g) For H atom stationary stateswith l = 0, yis independent of qand f. (h) For the H-atom ground state, the electron is confinedto move on the surface of a sphere centered around the nucleus. (i) For the H-atom ground state, the electron is confined to move within a sphere of fixed radius.For 3D free electron gas, the density of states counts the number of degenerate electron states dn per energy interval dE around a given energy E as g(E): = dn dE 3 (2m₂)2V 1 E2 2π²ħ³ At absolute zero temperature, N electrons can fill up all low lying energy levels (following Pauli exclusion principle) up to a given energy level E called Fermi energy. From the density of states, what is the relation between the total electron states N below a given energy E? Use this result to show that the Fermi energy EF is given by - - 2010 (307² M)³ ħ² 3π²N\3 EF 2me V
- (4) Electronic energy level of a hydrogen atom is given by R ; п %3D 1,2, 3,... n2 E = - and R = 13.6 eV. Each energy level has degeneracy 2n2 (degeneracy is the number of equivalent configurations associated with the energy level). (a) Derive the partition function for a hydrogen atom at a constant temperature. (b) Consider that the energy level of a hydrogen atom is approximated by a two level system, n = 1,2. Estimate the mean energy at 300 K.A 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)(a)How many sets of quantum numbers are possible for a hydrogen atom for n = 4,? ). (b) Write out a set of possible values for the quantum numbers n, ℓ, mℓ, and ms for each electron if all states are occupied including n=4. (c)Write table of occupancy of quantum numbers: n, ℓ, mℓ, and ms for Arsenic As, including spin orientations.
- View a system of two particles that do not interact with each other, where each particle can occupy three possible states, each with energy &, 2ɛ, 3E (i) Marwell-Boltzmann: Na. Configuration 1 2 3 4 S 6 7 2 9 Nader Conligation 1 2 2 + 1 2 5 6 (1) Fermi-Dirac 3 No. Configuration E AG A A B B AA A A Distinguishable 28 E AB (6) Boson: Base- Ginstein: Indistinguishable င် A A B A A 8 28 AA A A 38 A 2 A AB B A CO B A JE AA A A Energy system A A wwwww Indistinguishable We know formion follow exclusion principle. 28 38 Smarty tem 28 32 38 4E SE SE Energy system 22 48 68 38 48 SE 38 4€ SE Calculate the average energy of the system as a temperature function for the three statistics above.A diatomic molecule is modeled as a Morse oscillator, and one finds that its energy level differences decrease from E2 - E₁ = 1374.2 cm-¹ to E7 - E6 = 1139.6 cm ¹. (a) Use this information and the quantum energy levels for the Morse oscillator to find the harmonic angular frequency, w, in cm-¹. (b) What is the dissociation energy, D (in kJ/mole) for this Morse oscillator? (Note that the energy units, 11.9627 J/mole = 1 cm-¹.)- Calculate the average number of phonons occupying a vibrational mode with angular fre- quency w = 4.0 x 10¹2 s-1 at T = 300 K. - Calculate the total energy of the mode at this temperature, expressing your answer in meV.