Show that the kinetic energy (U0) of a 3D gas of N free electrons at 0 K is U0=3/5NEF.
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Show that the kinetic energy (U0) of a 3D gas of N free electrons at 0 K is U0=3/5NEF.
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- The characteristic vibrational temperature for chlorine (Cl₂) gas is 805 K. Calculate the fraction of chlorine molecules in the first excited vibrational state (v-1) when the temperature is 1610 K.p9C.1 Familiarity with the magnitudes of overlap integrals is useful when con- sidering bonding abilities of atoms, and hydrogenic orbitals give an indication of their values. (a) The overlap integral between two hydrogenic 2s orbitals is 1 ( ZR ZR +. 2а, " 12 а, 1 + ZR 240 a, -ZR/Z0 S(2s, 2s)={1+ Plot this expression. (b) For what internuclear distance is S(2s,2s) = 0.50? (c) The side-by-side overlap of two 2p orbitals of atoms of atomic number Z is ZR 1 ( ZR ZR S(2p,2p) ={1+ 10 a, 2a, 120 a. Plot this expression. (d) Evaluate S(2s,2p) at the internuclear distance you calculated in part (b).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
- Grüneisen parameter for silicon The silicon crystal (diamond structure) has the following properties at 500 K. The lattice parameter (a) = 0.5440 nm, thermal expansion coefficient (a) = 3.6x106 K-', and the bulk modulus is about the same as that at room temperature (K = 99 GPa). The Debye temperature of Si is 625 K. At room temperature (300 K) its density is 2.33 g cm³, and its lattice constant is 0.5430 nm. Calculate the Grüneisen parameter at 500 K and compare with its value at room temperature in Table 1. Is 3 constant?Graph below shows the electron occupancy probability P(E) as a function of energy for Bismuth (mBi = 3.47 × 10-25 kg) at the temperature T = 0 K. What is the number of conduction electrons per unit volume for Bismuth? 1 1 2 3 4 5 6 7 8 E (ev) P(E)At what pressure, in atmospheres, would the number of molecules per unit volume in an ideal gas be equal to the number density of the conduction electrons in copper, with both gas and copper at temperature T =300 K?
- Near the surface of a certain kind of star, approximately one hydrogen atom per 10 million is in the first excited level (n = 2). Assume that the other atoms are in the n = 1 level. Use this information to estimate the temperature there, assuming that Maxwell-Boltzmann statistics are valid.Plot the Fermi-Dirac probability of occupation function fFD(E) for T = 0, 10, 100, 200, 300 and 400K.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 → x?