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by Bartleby Expert- Problem 1: Consider a classical ideal gas in three dimensions, with N indistinguishable atoms confined in a box of volume N³. Assume the atoms have zero spin and neglect any internal degrees of freedom. Starting from the energy levels of a single atom in a box, find: (a) The Helmholtz free energy F' Hint: ſ. -ax² d.x e Va (b) The entropy o (c) The pressure pFor 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 VThe value of a partition function roughly represents the maximum energy of the states at a given temperature. O True False
- Please help to prove this to be trueConsider a classical ideal gas of N diatomic heterogeneous molecules at temperature T. The charac- teristic rotational energy parameter is € = 1 and the natural frequency of vibrations is wo. Consider the temperature region where T≫er/kB, but T is of the order of ħwo/kB. Ignore contributions from all other internal modes. Calculate the canonical partition function, the average energy, and the heat capacity at constant volume, Cv.Calculate the partition function of a two-level system at 25 °C with an energy gap of 10-2¹ J, assuming: a) Both states are non-degenerate. b) The ground state is non-degenerate, and the excited state is 3-fold degenerate.
- For a system of particles at room temperature (300K), what value must & be before the Fermi-Dirac, Bose-Einstein, and Maxwell-Boltzmann distributions agree within 0.1% ? Justify your answer.A diatomic gas molecule can be in one of two vibrational energy levels, separated by 0.1 eV. Give the probabilities to be in either state and use these to calculate their relative populations at room temperature, T≈ 300 K. [You may use that kB ≈ 8.6 × 10−5 K eV−1]Consider a three-dimensional infinite-well modeled as a cube of dimensions L x L x L. The length L is such that the ground state energy of one electron confined to this box is 0.50eV. (a) Write down the four lowest energy states and evaluate their corresponding degeneracy. (b) If 15 (total) electrons are placed in the box, find the Fermi energy of the system. (c) What is the total energy of the 15-electron system? (d) How much energy would be required to lift an electron from Fermi energy of part (b) to the first excited state? Need full detailed answers and explanations to understand the concept.