For an «-component system there are n2 different ay. More than half of all aik for each system can be directly calculated from the other ay, leaving «(« — 1)/2 of the ak to be measured. First, the aik are symmetric (aik = ay), which eliminates «(« — 1)/2 of the alk. An additional n of the alk can be eliminated using the Gibbs-Duhem relation ^; Ni ■ ay = 0. Generally, the availability of experimental methods will dictate which of the alk should be retained. For instance, existence of water vapor pressure data for the given system will make alW the best choice for being retained. However, aww should always be eliminated because of the definition of the molality, which renders a derivative with respect to mw undefined. Furthermore, note that some of the aik can be approximated by ai; = 1/ ml. This is the case for dilute components because the dependence of their chemical potential on their own concentration mi then has the form m = + RT ln(mi).
Take, for example, a dilute protein that is dissolved in a concentrated aqueous osmolyte solution, which contains a dilute buffer component in addition. This is a four-component system (n = 4) with n2 = 16 different aik . Out of these n(n — 1)/2 = 6 are redundant, and n = 4 can be eliminated through the Gibbs-Duhem relation. The aPP and aBB of the dilute components can be equated to 1/mP and 1/mB, respectively, and given that protein and buffer do not interact, we have aPB = 0 in addition. This leaves 16 — 6 — 4 — 3 = 3 of the aik to be measured.
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