and their temperature dependences provide valuable information about the properties of the materials. On a molecular level, the entropy of mixing is of interest because it is a macroscopic variable that provides information about constitutive molecular properties. In ideal materials, intermolecular forces are the same between every pair of molecular kinds, so that a molecule feels no difference between other molecules of its own kind and of those of the other kind. In non-ideal materials, there may be differences of intermolecular forces or specific molecular effects between different species, even though they are chemically non-reacting. The entropy of mixing provides information about constitutive differences of intermolecular forces or specific molecular effects in the materials. The statistical concept of randomness is used for statistical mechanical explanation of the entropy of mixing. Mixing of ideal materials is regarded as random at a molecular level, and, correspondingly, mixing of non-ideal materials may be non-random. Mixing of ideal species at constant temperature and pressure In ideal species, intermolecular forces are the same between every pair of molecular kinds, so that a molecule "feels" no difference between itself and its molecular neighbors. This is the reference case for examining corresponding mixing of non-ideal species. For example, two ideal gases, at the same temperature and pressure, are initially separated by a dividing partition. Upon removal of the dividing partition, they expand into a final common volume (the sum of the two initial volumes), and the entropy of mixing is given by where is the gas constant, the total number of moles and the mole fraction of component , which initially occupies volume . After the removal of the partition, the moles of component may explore the combined volume , which causes an entropy increase equal to for each component gas. In this case, the increase in entropy is entirely due to the irreversible processes of expansion of the two gases, and involves no heat or work flow between the system and its surroundings. Gibbs free energy of mixing The Gibbs free energy change determines whether mixing at constant (absolute) temperature and pressure is a spontaneous process. This quantity combines two physical effects—the enthalpy of mixing, which is a measure of the energy change, and the entropy of mixing considered here. For an ideal gas mixture or an ideal solution, there is no enthalpy of mixing (), so that the Gibbs free energy of mixing is given