The system shown below has available energy levels of 0, 2, 4, and 8 units. The degeneracyof each of the four levels is 4. The thermodynamic assembly has 7 particles (N = 7) and atotal system energy of 16 units (E = 16).(a) For bosons, what are the available macrostates? How many microstates are associated witheach of these macrostates and what is the most probable macrostate? What is the entropy
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The system shown below has available energy levels of 0, 2, 4, and 8 units. The degeneracy
of each of the four levels is 4. The thermodynamic assembly has 7 particles (N = 7) and a
total system energy of 16 units (E = 16).
(a) For bosons, what are the available macrostates? How many microstates are associated with
each of these macrostates and what is the most probable macrostate? What is the entropy of
the system? For Bose-Einstein statistics, the number of microstates for a given macrostate
N j is given by:
(b) For fermions, what are the available macrostates? How many microstates are associated
with each of these macrostates and what is the most probable macrostate? What is the
entropy of the system? For Fermi-Dirac statistics, the number of microstates for a given
macrostate N j is given by:
(c) What is the entropy of an assembly of fermions that occupies the energy levels shown with
N=16 and E = 56?
2. The system shown below has energy levels of c j B k h 0 K, 5 K, 10 K, 150K, and 200K.
The degeneracy of each of three lower levels is 108 and the two upper levels is 1010. The
system has only these energy levels; there are no translational, rotational, or vibrational
modes. The number of particles in the assembly is 105.
(a) Calculate the energy of the assembly at temperatures of 2, 10, 20, 35, and 50 K,
assuming corrected Maxwell-Boltzmann statistics. Why is CMB statistics a good
assumption?
(b) Plot the energy of the assembly and estimate the heat capacity of the system at 6, 15,
27, and 42 K. Comment on the results.
(c) Calculate the assembly entropy S (in J/K) at temperatures of 2, 10, 20, and 50 K.
Assume corrected-Maxwell-Boltzmann statistics. Comment on the results.
3. The system shown below has available energy levels of 0, kB, 2kB, and 3kB units,
where = 100 K. The degeneracy of each of the four levels is given by gj = 10,000 + 10,000
j. The thermodynamic assembly has 1000 particles (N = 1000) and the temperature of the
assembly is 200 K. For this dilute assembly, the population distribution for the most
probable macrostate is given by the Boltzmann distribution law,
(a) Using the Boltzmann distribution law, calculate the most probable macrostate {N0mp, N1mp,
N2mp, N3mp}. Round the populations to the nearest integer.
(b) What is the entropy (J/K) of the assembly?
(c) What is the energy (J) of the assembly?
(d) Calculate the number of microstates associated with two macrostates that are very similar to
the most probable macrostate. Macrostate A is given by {N0mp, N1mp+5, N2mp-10, N3m+5}.
Macrostate B is given by {N0mp, N1mp-5, N2mp+10, N3m-5}. Comment on the results.
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