# N

for an N-component solution would be given by Pm = ^^ XiPmi, where Xi i is the mole fraction of the i-th component and Pmi is its polarization. By performing an experiment in a stepwise fashion where all the ingredients are added one at a time (e. g. starting with the buffer base added to a nonpolar solvent) one can determine all the Pmis. In the case of a binary solution with nonpolar solvent (i. e. one with only induced polarization) we can approximate the molar polarization of the solvent Pmi as that of the pure solvent and using only the first term of (4.37) arrive at: Pmi = (k^m! where Mi is the mass and pi the density of the solvent. Thus to determine the solute molar polarization at each concentration we can use:

where Pm stands for the measured "bulk" molar polarization of the binary solution. Note here that according to this simplistic formalism one would expect the result of (4.38) to be independent of concentration since it is supposed to be uniquely determined by the molecular structure. However, it is frequently observed that the polarization increases as the concentration decreases due to significant solvent-solute interactions. As a result, it is customary to report the molar polarization extrapolated to an infinite dilution, i. e. Pm2 as lim(X2 ^ 0) . We address further limitations of this approach below.

Hedestrant's procedure  for determining polarizabilities of solutions is based on the above approach plus assumptions that the dielectric constant and the density of the solution are linear in the solute mole fraction, i. e. in our case k = k1 + aX2 and p = p1 + bX2 where a and b are the derivatives of the dielectric constant and density with respect to mole fraction. Substituting these into (4.38) one obtains: