Handbook of Solvents - George Wypych - ChemTech - Ventech!

13.1 Solvent effects on chemical reactivity 747
3
η = πρσ / 6=
ρVHS
[13.1.7]
where:
η packing density
ρ number density N/V= number of particles per unit volume
σ HS diameter
VHS HS volume
For the determination of σ(and henceη), the most direct method is arguably that based
on inert gas solubility data. 39,40 However, in view of the arduousness involved and the uncertainties
in both the extrapolation procedure and the experimental solubilities, it is natural to
look out for alternatives. From the various suggestions, 41,42 a convenient way is to adjust σ
such that the computed value of some selected thermodynamic quantity, related to σ, is consistent
with experiment. The hitherto likely best method 43 is the following: To diminish effects
of attraction, the property chosen should probe primarily repulsive forces rather than
attractions. Since the low compressibility of the condensed phase is due to short-range repulsive
forces, the isothermal compressibilityβT = -(1/V)(∂V/∂P) T might be a suitable candidate,
in the framework of the generalized van der Waals (vdW) equation of state
( )
βT RT V Qr
/ = 1 [13.1.8]
where Qr is the density derivative of the compressibility factor of a suitable reference system.
In the work referred to, the reference system adopted is that of polar-polarizable
spheres in a mean field,
Qr Z
( )
=
⎡ 2 3
5η −2η
⎤
⎢2
−1−2 ⎥
4
μ
⎣
⎢ 1−η
⎦
⎥
[13.1.9]
where Zμ = compressibility factor due to dipole-dipole forces, 43 which is important only for
a few solvents such as MeCN and MeNO2. The HS diameters so determined are found to be
in excellent agreement with those derived from inert gas solubilities. It may be noted that
the method of Ben-Amotz and Willis, 44 also based on βT, uses the nonpolar HS liquid as the
reference and, therefore, is applicable only to liquids of weak dipole-dipole forces. Of
course, as the reference potential approaches that of the real liquid, the HS diameter of the
reference liquid should more closely approximate the actual hard-core length. Finally, because
of its popularity, an older method should be mentioned that relies on the isobaric
expansibility αp as the probe, but this method is inadequate for polar liquids. It turns out that
solvent expansibility is appreciably determined by attractions.
Some values of η and σ are shown in Table 13.1.2 including the two extreme cases.
Actually, water and n-hexadecane have the lowest and highest packing density, respectively,
of the common solvents. As is seen, there is an appreciable free volume, which may
be expressed by the volume fraction η−η0 , where η0 is the maximum value of η calculated
for the face-centered cubic packing of HS molecules where all molecules are in contact with
each other is η0 = π 2/ 6= 074 . . Thus, 1 - η 0 corresponds to the minimum of unoccupied
volume. Since η typically is around 0.5, about a quarter of the total liquid volume is empty
enabling solvent molecules to change their coordinates and hence local density fluctuations
to occur.