for each mass of gas. In this kind of "mixing", the final common volume is equal to each of the respective separate initial volumes, and each gas finally occupies the same volume as it did initially. This constant volume kind of "mixing", in the special case of perfect gases, is referred to in what is sometimes called Gibbs' theorem. It states that the entropy of such "mixing" of perfect gases is zero. Mixing at constant total volume and changing partial volumes, with mechanically controlled varying pressure, and constant temperature An experimental demonstration may be considered. The two distinct gases, in a cylinder of constant total volume, are at first separated by two contiguous pistons made respectively of two suitably specific ideal semipermeable membranes. Ideally slowly and fictively reversibly, at constant temperature, the gases are allowed to mix in the volume between the separating membranes, forcing them apart, thereby supplying work to an external system. The energy for the work comes from the heat reservoir that keeps the temperature constant. Then, by externally forcing ideally slowly the separating membranes together, back to contiguity, work is done on the mixed gases, fictively reversibly separating them again, so that heat is returned to the heat reservoir at constant temperature. Because the mixing and separation are ideally slow and fictively reversible, the work supplied by the gases as they mix is equal to the work done in separating them again. Passing from fictive reversibility to physical reality, some amount of additional work, that remains external to the gases and the heat reservoir, must be provided from an external source for this cycle, as required by the second law of thermodynamics, because this cycle has only one heat reservoir at constant temperature, and the external provision of work cannot be completely efficient. Gibbs' paradox: "mixing" of identical species versus mixing of closely similar but non-identical species For entropy of mixing to exist, the putatively mixing molecular species must be chemically or physically detectably distinct. Thus arises the so-called Gibbs paradox, as follows. If molecular species are identical, there is no entropy change on mixing them, because, defined in thermodynamic terms, there is no mass transfer, and thus no thermodynamically recognized process of mixing. Yet the slightest detectable difference in constitutive properties between the two species yields a thermodynamically recognized process of transfer with mixing, and a possibly considerable entropy change, namely the entropy of mixing.