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Question: A system has four energy levels \(\epsilon = 0, 50k, 100k,\) and \(150k,\) where \(k\) is Boltzmann's constant. The degeneracies of the levels are 1, 3, 5, and 9, respectively. The system contains \(10^{10}\) distinguishable particles. 1) At the temperature 25 K, calculate: a) the partition function Z b) the chemical potential \(\mu\) c) the populations \(N_1, N_2, N_3\) and \(N_4\) for each energy level d) the average energy per particle. 2) At what temperature is the populations \(N_1\) and \(N_2\) equal?

          Question:
A system has four energy levels \(\epsilon = 0, 50k, 100k,\) and \(150k,\) where \(k\) is Boltzmann's
constant. The degeneracies of the levels are 1, 3, 5, and 9, respectively. The system
contains \(10^{10}\) distinguishable particles.
1) At the temperature 25 K, calculate:
a) the partition function Z
b) the chemical potential \(\mu\)
c) the populations \(N_1, N_2, N_3\) and \(N_4\) for each energy level
d) the average energy per particle.
2) At what temperature is the populations \(N_1\) and \(N_2\) equal?

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Question:
A system has four energy levels ϵ = 0, 50k, 100k, and 150k, where k is Boltzmann's
constant. The degeneracies of the levels are 1, 3, 5, and 9, respectively. The system
contains 10^10 distinguishable particles.
1) At the temperature 25 K, calculate:
a) the partition function Z
b) the chemical potential μ
c) the populations N1, N2, N3 and N4 for each energy level
d) the average energy per particle.
2) At what temperature is the populations N1 and N2 equal?

Added by Jose M.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Lucas Finney

Step 1

Plugging in the values, we have: Z = e^(-0/kT) + e^(-50k/kT) + e^(-100k/kT) + e^(-150k/kT) Since k is Boltzmann's constant, we can simplify the expression: Z = 1 + e^(-50/T) + e^(-100/T) + e^(-150/T) b) The chemical potential is given by: μ = -kT Show more…

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Thermodynamics & Statistics Question: A system has four energy levels: 0, 50k, 100k, and 150k, where k is Boltzmann's constant. The degeneracies of the levels are 1, 3, 5, and 9, respectively. The system contains 10 distinguishable particles. At the temperature 25 K, calculate a) the partition function Z, b) the chemical potential, c) the populations N1, N2, N3, and N4 for each energy level, and d) the average energy per particle. At what temperature is the populations N1 and N2 equal?

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