Alternatively, in terms of solute concentration in mole fraction (i.e.   moles of solute per total moles of all species), per regular solution theory [53], the single-solute osmotic virial equation for solute i   is [45] and [55] equation(6) π̃=xi+Bii∗xi2+Ciii∗xi3+…,where π̃ is osmole fraction (unitless), xi   is the mole fraction of solute i  , and Bii∗ and Ciii∗ are the second and third mole fraction-based osmotic virial coefficients of solute i, respectively (unitless). Osmole fraction is a rarely-used alternative form of osmolality, defined as [14] equation(7) π̃=-μ1-μ1oRT.Comparing Eqs. (1) and (7), osmolality Sirolimus solubility dmso and osmole fraction

are related by equation(8) π̃=M1π. The osmotic virial coefficients in Eqs. (5) and (6) account for increasing orders of interaction

between molecules of solute i  : the second osmotic virial coefficient represents interactions between two solute i   molecules, the third osmotic virial coefficient represents interactions between three solute i   molecules, and so forth. As such, these coefficients represent the non-ideality of the solute—if they are all zero, solute i   is thermodynamically ideal. For electrolyte solutes, solute concentration must be multiplied by an additional parameter, the dissociation constant [56] equation(9) π=kimi+Bii(kimi)2+Ciii(kimi)3+…,π=kimi+Bii(kimi)2+Ciii(kimi)3+…, Ibrutinib nmr equation(10) π̃=ki∗xi+Bii∗(ki∗xi)2+Ciii∗(ki∗xi)3+…,where ki   is the molality-based dissociation constant of solute i   and ki∗ is the mole fraction-based dissociation constant of solute i. This dissociation constant empirically accounts for ionic dissociation, charge screening, and other additional complexities inherent to electrolytes

[56]; for non-electrolyte solutes, its value is effectively 1. Through a simple, empirical demonstration, Pricket et al. [56] have shown that for applications of interest to cryobiology, this approach for electrolytes is as accurate as the more sophisticated Pitzer–Debye–Huckel approach. To obtain values of the osmotic virial coefficients and (if applicable) the Lepirudin dissociation constant for any solute of interest, Eqs. (5), (6), (9) and (10) can be curve-fit to osmometric (i.e. concentration versus osmolality) data for a binary aqueous solution containing that single solute. The osmotic virial equation can be extended to multi-solute solutions by introducing osmotic virial cross-coefficients, which represent interactions between molecules of different solutes [14] and [45]—for example, for a solution containing (r − 1) solutes, the molality-based osmotic virial equation (i.e. Eq. (5)) can be written as follows equation(11) π=∑i=2rmi+∑i=2r∑j=2rBijmimj+∑i=2r∑j=2r∑k=2rCijkmimjmk+…,where Bij, Ciij, Cijj, Cijk, etc. are cross-coefficients (e.g. Bij accounts for interactions between one molecule of solute i and one of solute j).