In order to understand the thermodynamics of solubility, it is appropriate to begin with a simplified model of solution, namely that of an ideal solution. An ideal solution is defined as one where the activity coefficient of all components in the solution equals one. Under these stipulations, the activity of the dissolved solute, the activity of the solid, and the molar solubility of the dissolved solute would be equal.
As discussed above, the absolute activity of the solid depends on the chosen reference or standard state, and the usual practice is to take the supercooled liquid state of the pure solute at the temperature of solution as the standard state of unit activity. At temperatures lower than the melting point, the liquid state of the solute is less stable than its solid state, making the activity of the corresponding solid less than one.
An ideal solution requires that the scope of solute-solute, solvent-solvent, and solute-solvent intermolecular forces be all the same. Thus, the net energy change associated with breaking bonds between two solute molecules and two solvent molecules, and then forming new bonds between solute and solvent molecules must be zero. Moreover, the mixing process is ideal as well, so that the total volume of the solute/solvent system does not change during the mixing process.
where AUmx is the energy of mixing, A Hmx is the enthalpy of mixing, and AV^ix is the volume change of mixing. The ideal entropy of mixing, A Smix, can be derived from pure statistical substitution
where nA and nB are the number of moles of the solvent (A) and the solute (B), respectively. Because the mole fractions of the solvent and the solute, XA and XB, are less than unity, it follows that A Smix is always positive. From this analysis, one can conclude that the mixing processes associated with an ideal solution would be thermodynamically favored.
The dissolution of a solid in a solvent can be considered as consisting of two steps. The first step would be, in effect, a melting of the solid at the absolute temperature (T) of the solution, and the second step would entail mixing of the liquidized solute with the solvent. The enthalpy of solution (A H) is therefore equal to the sum of the enthalpy of fusion (A HfT) and the enthalpy of mixing (A Hmx). However, since the enthalpy of mixing must equal zero for an ideal solution, it follows that the enthalpy of solution must equal the enthalpy of fusion of the solid at the given temperature, T:
For those situations where the temperature of study is not the same as the melting point, then A HfT = A Hfm, where now A Hfm is the enthalpy of fusion at the melting point( Tm). If one makes the approximation that the enthalpy of fusion is constant over the temperature range in the vicinity of the melting point, then:
Applying the Clausius-Clapeyron equation to the solubility calculation yields:
d T Jp RT2
Integration of equation (16) provides the relationship known as the van't Hoff equation, which expresses the temperature dependence of the solubility of a solid solute (identified as species B) in an ideal solution:
By combining equations (15) and (17), one finds that the molar solubility of the solute in an ideal solution (expressed in natural logarithmic form) is given by:
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