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  • https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/25%3A_Bose-Einstein_and_Fermi-Dirac_Statistics
    In developing the theory of statistical thermodynamics and the Boltzmann distribution function, we assume that molecules are distinguishable and that any number of molecules in a system can have the s...In developing the theory of statistical thermodynamics and the Boltzmann distribution function, we assume that molecules are distinguishable and that any number of molecules in a system can have the same quantum mechanical description. Fortunately, it turns out that more rigorous treatment of the conditions imposed by quantum mechanics usually leads to the same conclusions as the Boltzmann treatment.
  • https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Statistical_Thermodynamics_(Jeschke)/05%3A_Quantum_Ensembles/5.02%3A_Quantum_and_Classical_Statistics
    The next part of the derivation is the same as for the Boltzmann distribution in Section [subsection:Boltzmann], i.e., it relies on maximization of \ln \Omega using the Stirling formula and consid...The next part of the derivation is the same as for the Boltzmann distribution in Section [subsection:Boltzmann], i.e., it relies on maximization of \ln \Omega using the Stirling formula and considering the constraints of conserved total particle number N = \sum_i N_i and conserved total energy of the system . The initial result is of the form
  • https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/25%3A_Bose-Einstein_and_Fermi-Dirac_Statistics/25.01%3A_Quantum_Statistics
    We often assume that molecules are distinguishable and that any number of molecules in a system can have the same quantum mechanical description. These assumptions are not valid for many chemical syst...We often assume that molecules are distinguishable and that any number of molecules in a system can have the same quantum mechanical description. These assumptions are not valid for many chemical systems. Fortunately, it turns out that more rigorous treatment of the conditions imposed by quantum mechanics usually leads to the same conclusions as the Boltzmann treatment. The Boltzmann treatment can become inadequate when the system consists of low-mass particles or low temperatures.
  • https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/18%3A_Quantum_Mechanics_and_Molecular_Energy_Levels/18.07%3A_Particle_Spins_and_Statistics-_Bose-Einstein_and_Fermi-Dirac_Statistics
    The spin of a particle is an important quantum mechanical property. It turns out that quantum mechanical solutions depend on the spin of the particle being described. Particles with integral spins beh...The spin of a particle is an important quantum mechanical property. It turns out that quantum mechanical solutions depend on the spin of the particle being described. Particles with integral spins behave differently from particles with half-integral spins. When we treat the statistical distribution of these particles, we need to treat particles with integral spins differently from particles with half-integral spins.
  • https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Thermodynamics_and_Chemical_Equilibrium_(Ellgen)/25%3A_Bose-Einstein_and_Fermi-Dirac_Statistics/25.02%3A_Fermi-Dirac_Statistics_and_the_Fermi-Dirac_Distribution_Function
    Let us consider the total probability sum for a system of particles that follows Fermi-Dirac statistics.

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