**4. (Kittel 7.14) Two orbital boson system.**Consider a system of \( N \) bosons of spin zero, with orbitals at the single particle energies 0 and \( \varepsilon \). The chemical potential is \( \mu \) and the temperature is \( \tau \). Find \( \tau \) such that the thermal average population of the lowest orbital is twice the population of the orbital at \( \varepsilon \). Assume \( N \gg 1 \) and make what approximations are reasonable.

Answered: 4. (Kittel 7.14) Two orbital boson…

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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G (b) ! (a) X₂ X₂ の ४ P Atmospheric pressure XB → G AG W GB FIGURE 1.38 The effect of interfacial energy on the solubility of small particles. XB We discussed in lecture how in the early phase of a phase transformation, for example as a ß phase forms in a matrix of an a phase as shown in Fig 1.38(a), the interface between the two phases contributes to the Gibbs free energy of the ß phase. (a) In Figure 1.38(b), the Gibbs free energy curves of the ß phase are shown for a case (1) where the 3 phase is of radius r and (2) for the case where the ß phase has a radius of size. Write an expression for the difference in the Gibbs free energy of the two cases. (b) Notice the Gibbs free energy curve for the a phase on the left side of Figure 1.38(b). What can you say is different about the a phase in equilibrium with the ß phase when the ß phase is (1) of size r versus when the ß phase is (2) of size ∞.
3. A block of pure iron needs to be converted to steel. The density of iron is 7.86x10^3kg/m^3, and the atomic weight is 0.0558kg/mol. The concentration of carbon at position A is 4x10^26 at/m^3. The carbon atomic weight is 0.012kg/mol. Carbon must be diffused into the iron to convert it to steel. It is desired to have 1025 atoms of carbon per m' at point B. Position A is 1mm from position B. Assume that you have 1m^3 of iron. You heat the material uniformly to 1200°C. Do, carton = 79x10 * m/sec, Ea = 83kJ/mol, and the density of carbon is 2260kg/m^3. (4pts) 1mm Fe block A What is Dearbon at 1200°C in m?/sec? i. ii. Calculate the FLUX (J) of carbon atoms in atoms per m²-sec at 1200°C? B.
4. Consider a binary alloy of A atoms and B atoms that can exist in a solid phase or a liquid phase. This alloy can also exist in TWO DIFFERENT PHASES SIMULTANEOUSLY: solid and liquid. The system is composed of one mole of atoms, some of which are A atoms and some of which are B atoms. Take GA and GB as the molar Gibbs free energy of pure A and pure B respectively. XA and XB are the molar fractions of A and B atoms respectively. The atoms crystallize in identical crystal structures when they are solid. (a) The starting conditions are such that a partition sits between the A atoms and B atoms so they do NOT mix. (i) Write an expression for the molar Gibbs free energy for the combination of pure components. (ii) Draw schematically how the molar Gibbs free energy of the combination of pure components varies with alloy composition.

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