In other words, the entropy of mixing is not a continuous function of the degree of difference between the two substances. This is the mixing paradox which states that if the two substances are identical, there will be no entropy change, yet the slightest detectable difference between the two will yield a considerable entropy change, and this is just the entropy of mixing. (In this sense, then the term "entropy of mixing" is a misnomer, since the entropy increase is not due to any "mixing" effect.) Nevertheless, the two substances must be different for the entropy of mixing to exist. The entropy of mixing may be calculated by Gibbs' Theorem which states that when two different substances mix, the entropy increase upon mixing is equal to the entropy increase that would occur if the two substances were to expand alone into the mixing volume. If the substances being mixed are initially at different temperatures and pressures, there will, of course, be an additional entropy increase in the mixed substance due to these differences being equilibrated, but if the substances being mixed are initially at the same temperature and pressure, the entropy increase will be entirely due to the entropy of mixing. We assume that the mixing process has reached thermodynamic equilibrium so that the mixture is uniform and homogeneous. This entropy change must be positive since there is more uncertainty about the spatial locations of the different kinds of molecules when they are interspersed. The entropy of mixing (also known as configurational entropy) is the change in the entropy, an extensive thermodynamic quantity, when two different chemical substances or components are mixed.
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