(11), one must use multi-solute osmometric data. Alternatively, it is possible to develop mixing rules to avoid this requirement. Thermodynamic mixing rules are theoretical relations that predict the values of

cross-coefficients using the values of individual solute coefficients. Elliott et al. [14] and [15] have proposed the following second and third order mixing rules for the molality- and mole fraction-based osmotic virial equations equation(12) Bij=Bii+Bjj2, equation(13) Cijk=(CiiiCjjjCkkk)1/3,Cijk=(CiiiCjjjCkkk)1/3, equation(14) Bij∗=Bii∗+Bjj∗2, equation(15) Cijk∗=(Ciii∗Cjjj∗Ckkk∗)1/3. Applying these mixing rules yields the molality- and mole fraction-based Elliott et al. multi-solute osmotic virial equations equation(16) π=∑i=2rmi+∑i=2r∑j=2r(Bii+Bjj)2mimj+∑i=2r∑j=2r∑k=2r(CiiiCjjjCkkk)1/3mimjmk+…, ABT-737 clinical trial equation(17) π̃=∑i=2rxi+∑i=2r∑j=2r(Bii∗+Bjj∗)2xixj+∑i=2r∑j=2r∑k=2r(Ciii∗Cjjj∗Ckkk∗)1/3xixjxk+…,or, in the presence of electrolytes equation(18) π=∑i=2rkimi+∑i=2r∑j=2r(Bii+Bjj)2kimikjmj+∑i=2r∑j=2r∑k=2r(CiiiCjjjCkkk)1/3kimikjmjkkmk+…,

equation(19) π̃=∑i=2rki∗xi+∑i=2r∑j=2r(Bii∗+Bjj∗)2ki∗xikj∗xj+∑i=2r∑j=2r∑k=2r(Ciii∗Cjjj∗Ckkk∗)1/3ki∗xikj∗xjkk∗xk+…,where r is the number of solutes present. These equations have been found to provide accurate predictions of osmolality in a wide variety of non-ideal multi-solute solutions [3], [7], [14], [43], [54], [55] and [56]. It should, however, be noted that although Eqs. (16) (or (18)) and (17) (or (19)) are similar in form and were derived using similar methods, they were obtained SGI-1776 solubility dmso using different Clomifene starting assumptions (regarding concentration units i.e. Landau and Lifshitz solution theory versus regular solution theory). They are not equivalent, will not necessarily yield the same predictions for a given solution, and it is not possible to directly convert the coefficients of one to those of the other. That is, Eqs. (16) and (17) are effectively separate and distinct solution theories. The Kleinhans and Mazur

freezing point summation model is similar to the osmotic virial equation in that it also models the osmolality (or, in this case, freezing point depression directly) as being a polynomial function in terms of solute concentration [38]. For a binary aqueous solution containing a single solute i, this model represents the freezing point depression as [38] equation(20) ΔTm=Tmo-Tm=-(C1imi+C2imi2+C3imi3),where C1i, C2i, and C3i are empirical solute-specific coefficients. Like the osmotic virial coefficients, the coefficients in Eq. (20) can be obtained by fitting to single-solute solution osmometric data. For multi-solute solutions, Kleinhans and Mazur proposed summing the freezing point depression equations of all solutes present, i.e. [38] equation(21) ΔTm=Tmo-Tm=-∑i=2r(C1imi+C2imi2+C3imi3),where the number of solutes present is (r − 1).