 
SOLUBILITY PARAMETERS AND SPCALC’S USE OF THEM
Screen shot of SPCALC running under Windows Vista®

Theory and Assumptions
Dissolution, as with any chemical process, is spontaneous if the Gibbs free
energy change upon dissolution, ΔG, is negative. The familiar mathematical
definition of the Gibbs free energy of mixing is given as:
ΔG_{m} = ΔH_{m}  TΔS_{m}
where ΔG_{m} is the Gibbs free energy, ΔH_{m} the heat of mixing,
ΔS_{m} the entropy of
mixing, and T the absolute temperature. Since the entropy of mixing is always
strongly positive, the heat (enthalpy) of mixing is the determining factor in
deciding the overall sign of the free energy.
Hildebrand Scott proposed that the enthalpy of mixing could be represented
as:
ΔH_{m} = V_{m}[ (ΔE1/V1)  (ΔE2/V2)]^{2}f1f2
where ΔH_{m} is the overall heat of mixing, V_{m} the total volume of the mixture,
ΔE1
and ΔE2 the energies of vaporization of components 1 and 2, respectively, V1 and
V2 the molar volumes of components 1 and 2, respectively, and f1 and f2 the
volume fractions of the respective components. The expression (ΔE/V), the energy
of vaporization per cc, is usually known as the "cohesive energy density" and
reflects the cavitation energy, i.e. the energy required to create a "hole" in
the liquid to accommodate a molecule of solute.
Rearranging the above equation, we get
ΔH_{m}/V_{m}f1f2 = [ (ΔE1/V1)  (ΔE2/V2)]^{2}
Thus, the heat of mixing at a given concentration is equal to the square of the
difference between the square roots of the cohesive energy density. It therefore
becomes convenient to assign to the square root of the cohesive energy density a
special significance.
This quantity, (ΔE/V), is conventionally given the Greek designation
δ. It is,
in fact, the solubility parameter. Substituting this into the above equation,
ΔH_{m}/V_{m}f1f2 = [δ1δ2]^{2}
we find that the heat of mixing is directly related to the square of the
difference in solubility parameters. If the enthalpy of mixing is to remain
sufficiently small to allow the entropy term in the equation describing the
Gibbs free energy for dissolution to dominate, it follows that the square of the
difference in solubility parameters must remain small. In fact, according to
this treatment, solubility is assured if that difference is zero. Stated another
way, solute solubility increases as the difference in solubility parameters
between solute and solvent decreases.
Note that this approach inherently assumes that the cavitation energy is the
only contributor to the solvation enthalpy. For nonhydrogenbonding solvents
and solutes, this assumption is normally quite good, experimentally. It becomes
increasingly poor with hydrogen bonding solvents and very poor with amphiprotic
solvents. To take these limitations in to account, "threedimensional"
solubility parameters have been devised. Unfortunately, there are very few
solvents for which 3D solubility parameters have been measured. For that
reason, SPCALC currently does not use 3D solubility parameters in its
calculations.
Experimentally, there are a number of ways to measure solubility parameters for
volatile compounds, since the solubility parameter is related to the heat of
vaporization. For nonvolatile compounds like polymers, this approach cannot be
used. While there are methods to determine the solubility parameters of
nonvolatile compounds, they tend to suffer from large errors and are seldom
used experimentally.
Group Molar Attraction Constants  Motivation and Determination
Based on many measurements of vaporization of volatile compounds,
it has proven possible to derive sets of additive constants for various
structural elements, which when properly applied, normally give very good
estimates for the solubility parameters of nonvolatile or otherwise not
previously studied compounds. These are the "group molar attraction constants"
that SPCALC uses to calculate solubility parameters.
Two sets of group molar attraction constants are in common use: those of Small
(derived from heats of evaporation) and those of Hoy (derived from measurements
of vapor pressure). They define structural elements somewhat differently; the
equations describing the relationship between the group molar attraction
constants and the solubility parameter are slightly different in each case. For
that reason, SPCALC allows you to choose which set of group molar attraction
constants you wish to use for a given calculation. One chooses one set or the
other depending on which set provides the most explicit or easiest description
of the solute structure. For the great majority of
materials, the two sets of group molar attraction constants give calculated
solubility parameter values within a few percent of each other.
For reference, the structural features defined in each of the two sets of group
molar attraction constants are listed below. A similar list, including the
actual group molar attraction constants, can be obtained from the Calculations
Menu of SPCALC. These values are embodied in SPCALC. The user can choose which
set of group molar attraction constants he wishes to use.
Note the differences between the two lists. For example, in the
Small group molar attraction constants, one can enter a phenyl group directly.
In the Hoy constants, one must define a phenyl group in terms of its constituent
parts (1 C= (aromatic), 5 CH= (aromatic), and 1 ring, sixmembered).
Small Structural Features
and Group Molar Attraction Constants
Si (in silanes) 87 phenyl 735 ring,sixmembered 100
CH3 214 CH2(singlebonded) 133 CH< 28
>C<93 Si (in silicones)38 naphthyl 1138
phenylene(o,m,p) 658 H (as in SiH) 90 CH2= 190 CH= (doublebonded) 111 >C=
19 CH=C 285 C=C 222 ring, fivemembered 110 O (ethers) 70
CO (ketones) 275 COO (esters) 310 CN 410 Cl
(single) 270 Cl (geminal) 260 Cl (triple as in CCl3) 250 Br 340 I
425 CF2 (nfluorocarbons) 150 CF3
(nfluorocarbons) 274 S (sulfides) 225 SH (thiols) 315 ONO (nitrates) 440 NO2
(aliph. nitro cmpds.) 440 PO4 500 
Hoy Structural Features
and Group Molar Attraction Constants
Si (in silanes) 87 C= (aromatic) 98.12 CH= (aromatic) 117.12 ring,sixmembered23.44 CH3 147.3 CH2(singlebonded) 131.5 CH< 85.99
>C< 32.03 ring,fourmembered 77.76 ring,fivemembered
20.99 Si (in silicones)38 osubstitution 9.69 msubstitution 6.6 psubstitution 40.33 conjugation 23.26 H (as in SiH)50.47 CH2= 126.54 CH= (doublebonded) 121.53 >C= 84.51 cis7.13 trans13.5 O (epoxides)
176.2 bicycloheptyl 22.56 methanoindane 62.5 O (ethers) 114.98 CO (ketones) 262.96 COO (esters) 326.58 CHO 292.64 N 61.08 NH 180.03 NH2 226.56 OH (aromatic) 170.99 OH (Hbonded) 225.84 CN
354.56 Cl (primary) 205.06 Cl (geminal) 342.67 Cl
(secondary) 208.27 Cl (aromatic) 161 Br 257.88 Br (aromatic) 205.6 F 41.33 NCO 358.66 (CO)2O 567.29 S (sulfides) 209.42 
In carrying out mixed solvent calculations, SPCALC assumes that the solvents
contribute to the mixture solubility parameter in a manner directly proportional
to the relative molar volumes of the components. This is equivalent to an
assumption of ideal mixing behavior. While almost never strictly true, this
assumption is usually good to within the error inherent in the assumptions
necessary to derive the solubility parameter in the first place, so long as the
solvents and solutes are not amphiprotic.
