where Vf and Vi are the final and initial volumes of the system.
The Fermi-Dirac distribution describes the statistical behavior of fermions, such as electrons, in a system:
where P is the pressure, V is the volume, n is the number of moles of gas, R is the gas constant, and T is the temperature. where Vf and Vi are the final and
PV = nRT
The Fermi-Dirac distribution can be derived using the principles of statistical mechanics, specifically the concept of the grand canonical ensemble. By maximizing the entropy of the system, we can show that the probability of occupation of a given state is given by the Fermi-Dirac distribution. By maximizing the entropy of the system, we
where f(E) is the probability that a state with energy E is occupied, EF is the Fermi energy, k is the Boltzmann constant, and T is the temperature.
ΔS = nR ln(Vf / Vi)
The Gibbs paradox can be resolved by recognizing that the entropy change depends on the specific process path. By using the concept of a thermodynamic cycle, we can show that the entropy change is path-independent, resolving the paradox.