SPCALC Science

SOLUBILITY PARAMETERS AND SPCALC’S USE OF THEM

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 and Scott proposed that the enthalpy of mixing could be represented as:

ΔH_m = V_m[ (ΔE1/V1) - (ΔE2/V2)]²f1f2

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_mf1f2 = [ (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_mf1f2 = [δ1-δ2]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 non-hydrogen-bonding 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, "three-dimensional" solubility parameters have been devised. Unfortunately, there are very few solvents for which 3-D solubility parameters have been measured. For that reason, SPCALC currently does not use 3-D 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, six-membered).

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.

 

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Last modified: 01/01/24