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The Boltzmann Factor and Partition Functions 85 When the fermion restriction is taken into account (no two identical fermions can occupy the same single-particle energy state), the allowed total energies are ε₁ ε₁ and + and so there are only three terms in Q(2, V, T). 3-25. Redo Problem 3-24 for the case of bosons instead of fermions. In this case there are six allowed terms: the three alloweed in Problem 3-24 and the three in which the are the same ε₂ ε₂, and 3-26. Consider a system of three noninteracting identical fermions, each of which has states with energies ε₁, ε₂, and How many terms are there in the unrestricted evaluation of Q(3, V, T)? Enumerate the allowed total energies in the summation of Equation 3.37 (see Example 3-5). How many terms occur in Q(3, V, T) when the fermion restriction is taken into account? There are 27 terms in the unrestricted evaluation. The allowed total energies given by the summation in Equation 3.37 are 1. + ε₃ + + = = 2. = 3. = = 4. = = 5. = = 6. = = = = 8. 9. 10. When the fermion restriction is taken into account (no two identical fermions can occupy the same single-particle energy state), the only allowed total energy is and so there is only one term in Q(3, V, T). 3-27. Redo Problem 3-26 for the case of bosons instead of fermions. In this case there are ten allowed terms, given by the total energies 3-28. Evaluate (see Table 3.1) for O₂(g) at its normal boiling point, 90.20 K. Use the ideal-gas equation of state to calculate the density of O₂(g) at 90.20 K.