Note that (4.35a) and (4.35b) can give a and p in terms of macroscopic, measurable parameters such as the dielectric constant k and volume and number of molecules in the sample, while (4.34) is not easily applicable to an experimental determination of p as it contains the difficult-to-measure P. However, (4.35) is not applicable to cases where the local field cannot a _
k be approximated by the simple field assumed by (4.33). These cases include water, where, if the measured permanent dipole moment (6.1 x 10-30 Cm) is inserted into the Clausius-Mossotti equation one arrives at a negative value for k.
Noting that antiparallel orientations between the local field and the dipole moment of a molecule in a sample will have higher interaction energies U (where U = —p • E = —\p\\E\ cos 9) than parallel ones, we can find the contribution of the permanent electric dipole of a molecule to the volume electric dipole moment of a bulk sample as follows: consider an equilibrium situation where the thermal energy kBT ^ U and we can expand the probability of each dipole's orientation (determined by Boltzmann statistics a: exp-U/ksT) keeping only the zeroth and first-order terms. This gives the average dipole moment as pave = \p\2S'/(3kBT), (4.36)
and the volume polarizability ignoring high frequencies can be written as a = « + «i + «d = «i + \p\2/(3kBT), (4.37)
so volume polarizability varies inversely with temperature as expected (at lower temperatures it is easier to reorient dipoles as kBT is closer to p • E). In the case of a dilute liquid, for instance when a protein is present at low concentration inside a nonpolar buffer, and assuming the properties of the solution are the sum of the properties of the components, we can write the effective molar polarizations of each component in terms of their mole fractions. For instance, by replacing V/N in (4.35a) by Vm the molar volume of the material, we can write the molar polarizability am = 3e0()Vm and also the molar polarization Pm = ^ = P0 + Pp where P0 and Pp are the induced and permanent dipole moments respectively. P0 = a0/3e0 and Pp = \p\2/(9e0kBT). In this dilute approximation, the molar polarization
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