19.2.1 General Conditions for Phase and Reaction Equilibrium

We consider a system comprising NC chemical compounds i = 1, …, NC that may be neutral or charged; the charge on compound i is denoted by q_i. The compounds may take part in a number NR of chemical reactions j = 1, …, NR; the stoichiometric coefficient of compound i in reaction j is denoted by ν_ij.

The compounds are distributed over a number NP of phases k = 1, …, NP. If the initial molar amount of compound i in the system is denoted by , then the amounts of each compound i in each phase k at a given temperature T and pressure P are such that the total Gibbs free energy of the system is minimized. This leads to the following equilibrium conditions:

• For each neutral compound i (q_i = 0):
  (19.1)
• For each pair of charged compounds :
  (19.2)
• For each chemical reaction j:
  (19.3)
• For each compound i:
  (19.4)
• For each phase k:
  (19.5)

where

is the chemical potential of compound i in phase k; as indicated above, it is generally a function of temperature, pressure and the vector of molar amounts of the compounds in phase k.

ξ_j is the extent of reaction j = 1, …, NR.

We note that the chemical reaction equilibrium condition (19.3) is written arbitrarily in terms of the chemical potentials in phase 1. It can be shown that, because of the phase equilibrium conditions (19.1) and (19.2), this also holds for all other phases. Similarly, the electroneutrality condition (19.5) is omitted for phase 1 as it is implied by the material balance Eq. (19.4) and the fact that the overall system is neutral.

Conditions (19.1)–(19.5) constitute a system of NC × NP + NR equations that can be solved to determine the phase compositions and reaction extents ξ_j.

The above mathematical formulation assumes that the number NP of phases in the system is known. In reality, this needs to be determined in an iterative manner that postulates a certain number of phases, solves the phase and reaction equilibrium problem, and then considers the stability of each individual phase at the equilibrium point. We will consider the phase stability problem in Section 19.2.5.

19.2.2 Fundamental Conditions for Solid–Liquid Equilibria

The conditions presented above are completely general and apply to any set of phases of any kind, with all compounds under consideration appearing in each and every phase. For solubility calculations in a pharmaceutical context, our interest is focused primarily on systems comprising only solid and liquid phases. Moreover, in most cases of interest, including those of simple APIs, API salts, hydrates and solvates, and co‐crystals, the solid phases have fixed composition, with a subset of the compounds in the system being present in fixed proportions. In fact, for the purposes of solubility calculations, it is convenient to think of these combinations as separate “compounds.” The latter may not exist in the liquid phases; instead, upon dissolution, they dissociate to their constituent compounds in a manner that can be described by an equilibrium chemical reaction. For example, in the case of a (2 : 1) co‐crystal A₂.B between an API A and a coformer B, we have a dissociation reaction of the form

while a sodium salt of an acidic API may dissociate in an aqueous environment according to

We note that, although each solid phase comprises a single compound, the same compound may appear in multiple solid phases corresponding to different polymorphs. In general, at most one of these phases will appear in a nonzero amount at equilibrium, the others corresponding to metastable phases.¹ Furthermore, in addition to the above solid–liquid dissociation reactions, the system may also involve reactions taking place entirely within the liquid phases.

In view of the above discussion, the general phase equilibrium conditions presented in Section 19.2.1 may be amended to take account of the specific features of the solubility calculation problem. The main modification required is to the phase equilibrium condition (19.1). Assuming, for simplicity, that phase 1 is always a liquid:

• For compounds i that exist both in solid and liquid phases (e.g. non‐dissociating or weakly dissociating APIs), Eq. (19.1) is modified to remove the composition dependence of the chemical potential in all solid phases
  (19.6)

  for any solid phase k comprising compound i. Note that Eq. (19.1) remains unchanged for other liquid phases k.

• For compounds i that exist exclusively in solid phases (e.g. API salts, hydrates or solvates, or co‐crystals), Eq. (19.1) is replaced by the equilibrium conditions for the corresponding dissociation reaction. These are a special case of Eq. (19.3), and can be written as
  (19.7)

  for any solid phase k comprising compound i. Here σ_ii′ denotes the stoichiometry of composite compound i in terms of other compounds i^′ in the system (i.e. one mole of compound i comprises moles of compound i^′).

As an illustration, consider again the (2 : 1) co‐crystal and salt examples introduced earlier in this section in a system comprising one solid and one liquid phase. For these cases, Eq. (19.7) will take the form

and

respectively. Here denotes the entire vector of composition including the solvent(s), other ions (e.g. H⁺ and OH⁻ for the salt example), etc.

On the other hand, for an API A that can also exist in the liquid phase, the relevant equation is (19.6), taking the simple form

19.2.3 Liquid‐Phase Chemical Potentials

The discussion in Sections 19.2.1 and 19.2.2 demonstrates that the only thermodynamic property required for solubility calculations is the chemical potential of each compound present in the system. In particular, there is no need for introducing concepts such as “reaction equilibrium constants” (or related quantities such as pK_a) or “solubility products”. In this section, we consider the computation of liquid‐phase chemical potentials, while Section 19.2.4 focuses on solid‐phase chemical potentials.

Since in general we need to be able to handle chemical transformations (e.g. dissociation of composite compounds from solid to liquid phases; liquid‐phase reactions), we need to adopt a reference datum of chemical elements at a reference temperature T^⊖ and pressure P^⊖. Thus, the chemical potential of a compound i in a liquid phase l can be computed by

(19.8)

where is the ideal gas chemical potential of compound i given by

(19.9)

where

• and are, respectively, the enthalpy and entropy of formation of compound i in the ideal gas standard state at temperature T^⊖ and pressure P^⊖.
• is the ideal gas specific heat capacity of compound i.
• x_i is the molar fraction of compound i given by .

The quantity is the residual chemical potential of compound i, representing the difference between the real chemical potential and the ideal gas one under identical conditions of temperature, pressure, and composition.

The liquid‐phase chemical potential may also be expressed in terms of reference states other than the ideal gas, such as the pure liquid state or a variety of infinite dilution states. In such cases, the residual term is replaced by a term based on the corresponding activity coefficient γ_i(T, P, n), which may be computed via a wide range of empirical and semiempirical models [7, 8]. However, as we shall see in Section 19.3, a significant development over the last decade has been the development of EoS that can accurately compute over wide ranges of conditions for general liquid mixtures involving different types of molecules (e.g. neutral compounds, ions, polymers) and complex intermolecular interactions including hydrogen bonding or other forms of association and electrostatic (coulombic) forces. Moreover, the group contribution basis of these EoS implies that they can be applied to systems for which little experimental data are available.

In view of the above, our approach to computing solubilities is based on Eqs. (19.8) and (19.9). It is worth noting that, contrary to what is sometimes stated in the literature, these equations are applicable to any type of compound. In cases where experimentally derived values for the enthalpy and entropy of formation are available for reference states other than the ideal gas, they can easily be converted to the quantities and required by Eq. (19.9) using the EoS itself. For example, the enthalpy and entropy of formation of a pure liquid at temperature T^⊖ and pressure P^⊖ are related to the corresponding ideal gas quantities via

(19.10a)

(19.10b)

where the residual enthalpy and entropy of the pure liquid can be obtained from the corresponding computed by the EoS via the standard thermodynamic relations:

(19.11a)

(19.11b)

Similar relations can be derived for enthalpies and entropies of formation in infinite dilution reference states and are presented in Appendix 19.A.

Figure 19.1 summarizes the different approaches to computing liquid‐phase chemical potentials in the form of thermodynamic paths from chemical elements at standard conditions (T^⊖, P^⊖) to the mixture under the conditions (T, P, n) of interest. Different paths make use of formation properties corresponding to different reference states. All these quantities can be interconverted to each other via the use of an EoS that can accurately compute residual properties of pure compounds. This allows the use of whatever experimentally determined values of formation properties happen to be available, even if these are with respect to different reference states for different compounds in the system under consideration.

All calculations reported in this chapter make use of the thermodynamic path going through the ideal gas reference state as shown by the solid arrows in Figure 19.1. For completeness, this diagram also shows paths based on activity coefficient models as dashed arrows.

19.2.4 Solid‐Phase Chemical Potentials

We now turn our attention to the computation of solid‐state chemical potentials of the kind that appear in the phase equilibrium conditions (19.6) and (19.7).

Below we consider two alternative approaches that differ in their requirements for solid‐state properties. In particular, the approach described in Section 19.2.4.1 makes use of enthalpy and entropy of formation, while that described in Section 19.2.4.2 relies on properties related to the melting of the solid phase.

19.2.4.1 Solid‐Phase Potentials in Terms of Formation Properties

Although, in principle, Eq. (19.8) holds for solid phases too, there is currently no generally applicable way of computing the residual term . In particular, both EoS and activity coefficient models are restricted to phases exhibiting liquid‐like structures only, and cannot describe the behavior of phases where the underlying microscopic structure follows a well‐defined geometric pattern such as crystalline solid phases.²

In view of the above, we express the chemical potential of a compound i in a solid phase corresponding to a particular crystal structure as

(19.12)

where

• and are, respectively, the enthalpy and entropy of formation of the solid phase of the same crystal structure at standard temperature T^⊖ and pressure P^⊖.
• is the specific heat capacity of the solid at temperature T and pressure P^⊖.

The above equation omits the pressure dependence of as this is typically very small at the pressures of interest to pharmaceutical applications.

19.2.4.2 Solid‐Phase Potentials in Terms of Melting Properties

The melting properties approach takes advantage of the fact that at the melting point temperature T_m,i of pure compound i, its liquid‐ and solid‐phase chemical potentials (i.e. free energy) are equal. Neglecting pressure effects and integrating along the thermodynamic path [3]

we obtain the following expression for the solid chemical potential:

(19.13)

where

• is the enthalpy of melting at T_m,i.
• is the difference between liquid‐ and solid‐specific heat capacities.

Since T < T_m,i, the chemical potential corresponds to a metastable (subcooled) liquid phase but can nevertheless be computed from an EoS.

This approach is usually applied to APIs that are stable up to their melting points. In principle, it can be extended to co‐crystals and salts which also have a well‐defined, experimentally measurable melting point. In that case, the liquid‐phase properties (chemical potential and specific heat capacity) used in the expression in the right‐hand side of Eq. (19.13) need to be replaced by those of a liquid‐phase mixture with the fixed composition as dictated by the stoichiometric makeup of compound i (cf. Section 19.2.2).

19.2.5 Determination of Solid Phases Present in the System

The phase equilibrium conditions underpinning solubility calculations (cf. Section 19.2.2) assume that both the number NP of phases that are present in the system and their nature (liquid mixtures; solids of various types) are known. In reality, given the molar amounts of the compounds i = 1, …, NC under consideration, we need to establish the complete set of phases that are present at the temperature T and pressure P of interest. One approach for achieving this aim is to start with a single liquid phase comprising all of the specified amounts and then test its stability with respect to all solid phases that may potentially form given the set of compounds under consideration.³ For the general systems considered here, any nonzero amounts of compounds i that exist only in solid phases (e.g. co‐crystals) can be converted to the equivalent amounts σ_ii′ of the compounds i^′ formed by their dissociation (cf. Eq. 19.7). The resulting liquid phase can then be equilibrated with respect to any chemical reactions taking place in it.

A widely applied stability criterion is that based on the Gibbs surface tangent plane [7]. Given a fluid phase of composition n, it tests whether the formation of an infinitesimally small amount of a new “trial” phase would result in an overall reduction of the Gibbs free energy. If no such trial phase can be found, the original fluid phase is deemed to be stable. Mathematically, this is equivalent to requiring that the quantity

(19.14)

is nonnegative for any trial composition w.

The application of the above criterion in cases where the trial phase is a fluid is a complex problem that has been studied extensively in the literature. The main challenge is to ensure that Ω_G(w) ≥ 0 for all possible nonzero w, something that can be formally guaranteed only via the use of techniques that systematically search over the entire range of trial compositions, e.g. via the use of global optimization algorithms [10–13].

However, testing stability of a liquid phase with respect to solid trial phases comprising a single compound is a significantly simpler problem. Because such phases comprise a single compound i, criterion (19.14) simplifies

• for compounds i that exist in both the solid and the liquid phases:
  (19.15a)
• for compounds i that dissociate into other compounds in the liquid phase (cf. Eq. 19.7):
  (19.15b)

Given a liquid phase of composition n, the above stability criteria can easily be checked for each and every solid phase s under consideration using the values of the liquid‐ and solid‐phase chemical potentials (cf. Sections 19.2.3 and 19.2.4, respectively). If the criterion is found to be violated for any phase s, then that phase may be present in the system and therefore must be added to the set of phases included in the formulation of the phase equilibrium conditions presented in Section 19.2.2. The solution of these equations will then yield a new liquid‐phase composition n, which again can be tested for stability against potential existence of more solid phases. The above procedure is therefore repeated until a stable liquid phase is obtained.

It should be noted that for a given liquid‐phase composition, more than one trial solid phase may be found to violate the stability criterion (19.15). Our approach in such cases is to tentatively add each such solid phase s separately to the system, perform the associated solubility calculation, and then use the resulting equilibrium compositions to compute the total Gibbs free energy of the system:

(19.16)

The trial solid phase s that results in the lowest free energy G_s is selected to be added to the system, and the corresponding liquid‐phase composition is subjected to further stability tests as described above.

19.3.1 Intermolecular Potential Model

Within the SAFT‐γ Mie EoS, each functional group k is characterized by a shape factor parameter S_k, describing the group non‐sphericity as well as a set of parameters that relate to various group‐group interactions, as described below.

Repulsive and dispersive interactions are modeled via the Mie potential and are considered for all group pairs. On the other hand, hydrogen bonding interactions are included only where necessary (i.e. for groups that exhibit association interactions) and are modeled using short‐ranged square‐well sites. It is also possible to formally take into account the presence of permanent charges on functional groups. This allows for the rigorous (and explicit) modeling of the speciation of APIs in aqueous mixtures as will be shown in Section 19.5. This does not require any additional parameters besides the charge itself (positive or negative), which is always known a priori, thus making the extension to charged systems straightforward.

The parameters that describe the Mie intermolecular potential are the diameter σ, the repulsive and attractive exponents λ^rep and λ^att, respectively, and the potential depth ɛ. The association interactions between pair of sites (denoted, for example, as a and b) are, if present, described by two additional parameters, namely, the energy of interaction, , and the volume available for bonding, . Note that no additional parameters are required for describing the formation of the molecular chains; this contribution is inferred by the knowledge of the distinct types of functional groups that comprise each molecule and their multiplicity on the molecular chains. Finally, the type and number of different association sites on a given group need to be defined a priori based on chemical understanding of hydrogen bonding donors and acceptors.

The interaction parameters described above relate to all pair interactions. A schematic representation of the physical meaning of each parameter is given in Figure 19.3. In practice, when developing the group pair interaction parameters, we commonly separate the treatment of the groups that are of the same type (self‐interactions) and the interactions between groups of different types (unlike or cross‐interactions). The self‐interaction parameters are almost always obtained by regression to experimental data, whereas the parameters that describe unlike interaction can be approximated by combining rules. Overall, the application of the SAFT‐γ Mie approach for pharmaceutical system relies on the development of databases of group interaction parameters for relevant chemical groups typically found in APIs. Section 19.3.2 explains the procedure for developing such database based on experimental data.

19.3.2 Estimation of Group Parameters from Experimental Data

As we have seen in Section 19.3.1, most of the parameters in the SAFT‐γ Mie EoS relate to interactions between functional groups. In practice, one can make use of available experimental data on simple systems to estimate these in a sequential manner.

When using a group contribution EoS such as SAFT‐γ Mie, a wide range of different data types can be used, including not only data on the thermodynamic properties and phase behavior of binary and/or multicomponent systems but also pure component data (single phase and equilibrium). This is a significant benefit over activity coefficient models, since only EoS approaches can be applied to the description of pure components and, hence, benefit from the availability of readily available pure component data in the development of group parameters.

In order to obtain reliable parameter estimates, a commonly used strategy is to include various types of experimental data in the regression procedure, including, for instance, densities, vapor pressure, and caloric data (heat capacity and/or vaporization enthalpy). The key aspect of the SAFT‐γ Mie group contribution methodology is that experimental data on any molecule containing the group of interest can be used in the regression procedure. In principle, a database of group parameters relevant to pharmaceutical applications can be developed without the use of API‐specific data.

The details on the exact sequence of development of all the functional group interactions used in the application section (Section 19.5) are beyond the scope of this chapter, but it is worth considering the simple example of CH₃, CH₂, and OH groups to illustrate the general workflow. First the methyl and methylene groups can be obtained by regression to experimental data for a series of (pure) n‐alkanes, from ethane to n‐decane. Having determined the parameters for the CH₃ and CH₂ groups, the OH group can then be obtained using primary alcohol data. Note that the usage of pure component data allows the determination of not only the self‐interactions but also the cross‐interactions between each group pair. A similar sequence can then be used to determine any other group, by selecting either pure component or mixture data.

Finally, it is worth noting that the SAFT‐γ Mie EoS can be used in a “molecular” mode, with an entire compound being represented as a single functional group. This is typically the case for the first member of each chemical family – e.g. methane for the n‐alkanes; methanol for the alcohols; etc. – but can also be employed in the study of more complex molecules. Note that in such cases compound‐specific experimental data are required for the development of the (single) group parameters.

19.4.1 Parameters Involved in Solubility Calculations

Equations (19.8), (19.9), (19.12), and (19.13) involve a number of parameters:

• Pure compound parameters
  • Enthalpies and entropies of formation in the ideal gas reference state at standard temperature T^⊖ and pressure P^⊖.
  • Parameters describing the temperature dependence of specific heat capacities in the ideal gas state.
  • Either
  • • Enthalpies and entropies of formation of crystalline phases at standard temperature T^⊖ and pressure P^⊖.
    • Parameters describing the temperature dependence of solid‐phase specific heat capacities.
  • or
  • • Melting point temperature and enthalpy of melting at that temperature.
    • Parameters describing the temperature dependence of the difference between liquid‐ and solid‐phase specific heat capacities.

• EoS parameters used for the computation of liquid‐phase chemical potentials and related quantities, including:
  • Residual chemical potentials used in Eq. (19.8).
  • Pure compound potential used in Eq. (19.13).
  • (Optionally) the terms required for obtaining the ideal gas formation properties from values of formation properties in other reference states as shown, for example, in Eqs. (19.10) and (19.11).

Some of these parameters can be estimated using group contribution methods, such as the Joback and Reid [22] method for ideal gas specific heat capacities. Others may be determined experimentally either via direct measurement (e.g. melting parameters using DSC techniques) or indirect estimation from experimental data (e.g. formation enthalpies and entropies). As already mentioned in Section 19.3, in the case of the SAFT‐γ Mie EoS parameters for the various functional groups, these experimental data could relate to fluid‐phase measurements (e.g. vapor–liquid or liquid–liquid equilibria) obtained from systems that do not actually involve any of the compounds occurring in the system of interest.

19.4.2 Solid‐Forming Compounds That Also Exist in the Liquid Phase

We start by revisiting the equilibrium condition (19.6) for compounds that exist in both the liquid and solid phases. Using Eqs. (19.8) and (19.9), ignoring the pressure dependence of the solid‐phase potential and applying some minor rearrangement, we obtain an expression of the form

(19.17)

where we have introduced a new quantity ∆g_i defined as

(19.18)

and have retained the phase labeling conventions of the original Eq. (19.6), i.e. k denotes the solid phase formed by compound i and phase “1” is a liquid phase.

We note that the definition (19.18) implies that ∆g_i(T) is purely a characteristic of the solid‐forming compound i and does not depend on any other compound in the system, and that it is solely a function of temperature. Therefore the value of ∆g_i(T) for a given solid API i at a given temperature T will always be the same, irrespective of the system in which this API occurs. Moreover, turning to Eq. (19.17), we can see that we can compute this ∆g_i(T) from any available experimental solubility data for this API by using the measured liquid‐phase composition to replace the quantities and appearing in the right‐hand side – provided, of course, we have an EoS that can compute the residual term .

It is worth noting that the original equilibrium condition (19.6) between solid phase k and liquid phase 1 is valid irrespective of the existence of other solid and/or liquid phases. Therefore, this type of calculation is correct even if the experimental data were derived in systems with multiple solid and/or liquid phases. It is sufficient to have the measured composition of one of the liquid phases.

As an illustration, Figure 19.4 shows the function ∆g_i(T) for ibuprofen. The values shown are computed from the right‐hand side of Eq. (19.17) using published experimental solubility data [23–26] from a range of solvents over a range of temperatures, with the term being computed via the SAFT‐γ Mie EoS. It is clear that all the points define a single function, irrespective of the solvent being considered and the actual solubility values. Moreover, at least in this case, this function can be well approximated by a straight line empirically determined to be of the form ∆g_ibuprofen(T) = 238.78T − 117 460 (∆g_ibuprofen in J/mol, T in K).

Once the function ∆g_i(T) for a given API i has been determined, it can be used directly for the computation of its solubility within any solvent or solvent mixture simply by using Eq. (19.17) to replace the original condition (19.6) in the solubility calculations. Importantly, the values of the various pure compound parameters appearing on the right‐hand side of Eq. (19.18) are not necessary. Effectively, the use of an advanced EoS allows transferability of solubility information from one system (or systems) to another.

As mentioned in Section 19.2.4.2, an alternative to the use of solid‐phase chemical potentials based on formation energies (Eq. 19.12) is to employ an expression based on melting properties as indicated by Eq. (19.13). The latter requires knowledge of the pure liquid chemical potential , which can itself be calculated using Eqs. (19.8) and (19.9):

(19.19)

where the last term on the right‐hand side is the residual chemical potential for the pure liquid that can be calculated using the EoS. It will be noted that the above does not include the +RT ln(x_i) term that appears on the right‐hand side of Eq. (19.9) as x_i = 1 for the case of pure liquid.

The expression for given by Eq. (19.19) can then be substituted in Eq. (19.13) to yield a complete expression for the solid‐phase chemical potential . The latter may then be used on the left‐hand side of equilibrium condition (19.6), with Eqs. (19.8) and (19.9) being used to compute the liquid‐phase chemical potential on the right‐hand side. At this point, all terms arising from the right‐hand side of Eq. (19.19) except the last one will appear identically on both sides and will cancel out, resulting in an equation of the form

(19.20)

where the quantity is defined as

(19.21)

Analogously to the quantity ∆g_i(T) defined by Eq. (19.18), is purely a characteristic of compound i and is solely a function of temperature. It can, therefore, also be described by an empirical relationship derived by fitting values obtained by computing the right‐hand side of Eq. (19.20) at available experimental data points. On the other hand, if the melting properties and of compound i are known, then they can be used directly for the computation of using Eq. (19.21) without the need to resort to experimental solubility data. In either case, there is no need to know the ideal gas properties , and .

19.4.3 Solid‐Forming Compounds That Exist Only in the Solid Phase

As already mentioned in Section 19.2.2, API salts, solvates, hydrates, and co‐crystals are assumed to exist only in the solid phase, dissociating fully upon entering the liquid phase. In such cases, the relevant equilibrium condition is Eq. (19.7). By following an approach similar to that used for the simpler case considered in Section 19.4.2 above, we can derive the equation

(19.22)

where the quantity ∆g_i is now defined as

(19.23)

As in the previous case, Eq. (19.23) shows that ∆g_i(T) is a characteristic of the compound i under consideration (e.g. a particular co‐crystal) and is solely a function of temperature. Inserting any available experimental solubility data for this compound on the right‐hand side of Eq. (19.22) allows us to compute the corresponding ∆g_i(T) and fit an empirical function of temperature to it. The latter can then be used in conjunction with Eq. (19.22) instead of the original condition (19.7) to compute solubilities of compound i in other systems of interest. Once again, we do not actually need to know any of the pure compound quantities appearing on the right‐hand side of Eq. (19.23).

19.4.4 Characterization of Liquid‐Phase Chemical Reactions

Liquid‐phase chemical reactions are described by the chemical equilibrium condition (19.3). Following an approach similar to that presented in Sections 19.4.2 and 19.4.3, the latter can be reformulated to an equation of the form

(19.24)

where the quantity is now defined as

(19.25)

From Eq. (19.25), we note that the free energy quantity is a characteristic of the particular chemical reaction j and depends only on temperature. As before, we can follow a two‐step procedure:

1. Determine the values of this quantity at different temperatures by computing the expression on the right‐hand side of Eq. (19.24) using experimentally measured liquid‐phase compositions for any system⁴ in which this reaction occurs.
2. Use the values computed above to fit an empirical expression (e.g. a linear or quadratic relation) to describe the temperature variation of .

Once we have the empirical expression , we can use it for characterizing reaction j in any system of interest where it may occur by replacing the original equilibrium condition (19.3) by Eq. (19.24).

It is worth noting that it is not always possible to measure directly the complete liquid‐phase composition. For instance,⁵ in aqueous ionic systems what is typically measured is the pH (i.e. the activity of the H⁺ ion) and the API total molar fraction, which comprises contributions from both the undissociated API and its ion(s). In such cases, step 1 of the above procedure needs to be amended as it is no longer possible to explicitly compute at the experimental temperature T simply by evaluating the right‐hand side of Eq. (19.24). Instead, the latter needs to be considered simultaneously with the material balance Eq. (19.4) to determine the value of that leads to the best fit of the available measurements.

19.5.1 The gSAFT Physical Properties Code

All calculations presented in this section were performed using gSAFT [27], a physical properties code developed by Process Systems Enterprise Ltd. for use within its gPROMS® [28] process modeling tool.

Designed to support pharmaceutical applications in the context of drug substance and drug product manufacturing and also oral absorption, gSAFT incorporates several elements of relevance to the topic of this chapter:

• An algorithm for the reliable computation of equilibria in multiple vapor–liquid–solid phases in the presence of chemical reactions, based on the general equilibrium conditions presented in Section 19.2.1 and including a comprehensive phase stability analysis (cf. Section 19.2.5).
• A robust implementation of the SAFT‐γ Mie EoS (cf. Section 19.3) that can be applied efficiently to mixtures with large numbers of neutral and/or charged compounds in the presence of association and/or coulombic interactions.
• A database of SAFT‐γ Mie parameters for functional groups commonly occurring in organic and aqueous systems, including a range of (mainly inorganic) ions.
• A tool for the reliable⁶ estimation of SAFT‐γ Mie parameters from multiple data sets derived from a range of standardized experiments such as pure compound vapor pressures and saturated liquid densities, vapor–liquid or liquid–liquid phase equilibria for binary mixtures, solubility measurements in single solvents or solvent mixtures, titration curves, etc.
• A tool for the derivation of empirical models for the temperature dependence of quantities:
  • ∆g_i(T) and from experimental solubility measurements (cf. Sections 19.4.2 and 19.4.3).
  • from experimental liquid‐phase composition measurements (cf. Section 19.4.4).

In addition to SAFT‐γ Mie, gSAFT also incorporates implementations of the SAFT‐VR SW [29, 30] and PC‐SAFT [31, 32] EoS and associated databanks. In principle, these EoS can also be used to perform solid–liquid equilibrium calculations within the framework described in this chapter (see, for example, [33–36]). However, they require parameters to be estimated specifically for each individual API, typically using experimental solubility data. This is in contrast to SAFT‐γ Mie EoS, which, by virtue of its functional group basis, is capable of making use of experimental information from systems involving different compounds and/or phases. This advantage of SAFT‐γ Mie EoS is particularly pronounced in systems involving complex molecules (such as pharmaceutical APIs) for which relatively little experimental data may be available.

19.5.2 Example 1: Solubility of Fenofibrate in Alcohols

The molecular structure of fenofibrate and its decomposition into functional groups is presented in Figure 19.5. The parameters for all the functional groups are already available within the gSAFT databank, having been estimated entirely from published experimental data derived from fluid‐phase experiments; no solubility data of any kind were used for this purpose.

The solid‐phase chemical potential is computed using the melting point approach described in Section 19.2.4.2, with the experimentally measured values of melting temperature and melting enthalpy reported in [37]. The specific heat capacity difference was set to zero.

The approach described in this chapter is now used to predict the solubility of fenofibrate in ethanol and 1‐propanol.⁷ A comparison between the predicted values and experimental data reported in the literature is shown in Figure 19.6. The solubility is presented as the liquid‐phase molar fraction of fenofibrate, with a logarithmic scale being employed to allow a clearer inspection of the low solubility values. It can be seen that the predictions are in good agreement with the experimental data, the slight deviations being attributed to the fact that some of the unlike interactions between functional groups of the solute and the solvents are approximated by means of combining rules, and have not been regressed using experimental data.

In order to assess further the predictive capability of SAFT‐γ Mie, we present in Figure 19.7 the computed solubilities in a binary mixture of water + ethanol. In general the computation of solubilities of APIs in a mixture of solvents is straightforward in gSAFT, the accuracy of the prediction depending on whether the majority of the group cross‐interactions available in the gSAFT database have been regressed using experimental data, instead of being approximated by combining rules.

19.5.3 Example 2: Solubility of Ibuprofen in PEG Polymers

This example considers the solubility of ibuprofen in polyethylene glycol polymer of given molecular weight (MW = 10 000 g/mol), an application of relevance to the solubilization of APIs with poor aqueous solubility via the preparation of solid dispersions. The molecular structure of ibuprofen and its decomposition into functional groups in gSAFT are presented in Figure 19.8.

The functional group decomposition of the PEG polymer is shown in Figure 19.9. The group contribution basis of SAFT‐γ Mie provides a natural mechanism for modeling complex polymers of arbitrary molecular weight. For example, PEG of molecular weight 10 000 g/mol is described as 2 hydroxyl groups combined with 454 methylene groups and 226 ether groups.

The melting point approach was used for the characterization of the solid phases of both ibuprofen and PEG, with the corresponding melting temperatures and enthalpies shown in Table 19.1. The specific heat capacity differences were set to zero.

Experimental data for this particular system [39] suggest the potential appearance of a co‐crystal with a 5 : 1 ibuprofen‐to‐PEG ratio, reflecting the experimental composition at which the solid‐liquid equilibrium temperature exhibits a maximum. Within our general framework for solid–liquid equilibria, the co‐crystal is considered as a compound that appears only in a solid phase, dissociating to its constituents in the liquid phase:

Moreover, in view of the lack of the necessary solid‐phase data, we characterize this compound using the approach described in Section 19.4.3 to derive an empirical representation for the function Δg_i(T). In this case, we assume that the latter has a simple linear form obtained by drawing a straight line through two experimental solubility data points (shown by black circles in Figure 19.10). The resulting empirical model is also shown in Table 19.1.

Based on this information, we can now trace the entire solid–liquid phase diagram for this system over a range of temperatures T and ibuprofen mass fractions w. This is shown in Figure 19.10, where the lines represent calculations obtained using SAFT‐γ Mie and the symbols the experimental solubility points. It can be seen that starting off with a liquid solution at a given temperature, a range of different solid forms can be obtained upon cooling depending on the initial composition.

19.5.4 Example 3: Solubility of Ketoprofen in PEG Polymers

This example examines the solubility of ketoprofen in PEG of molecular weight 6000 g/mol. The molecular structure and functional group decomposition of ketoprofen are shown in Figure 19.11.

Experimental data [40] suggest the potential formation of a co‐crystal with a 10 : 1 ratio of ketoprofen to PEG. As in the previous example, the co‐crystal is considered as a compound that appears only in a solid phase, dissociating to its constituents in the liquid phase:

The approach used for the characterization of the solid‐phase behavior of ketoprofen, PEG 6000, and the KT₁₀.PEG co‐crystal is identical to that adopted for the ibuprofen/PEG 10000 system presented in the previous Section 19.5.3. The parameters used for the solid–liquid equilibrium calculations are summarized in Table 19.2.

A comparison between the predictions obtained using the SAFT‐γ Mie approach and the experimental solubility data is presented in Figure 19.12. The latter illustrates both the complexity of the phase diagram and the accuracy of the proposed approach to the computation of solid–liquid equilibria.

19.5.5 Example 4: Aqueous Solubility of Ibuprofen as Function of pH

An application of great theoretical and practical interest is the study of the solubility of an ionizable API at different values of pH. Here we consider the aqueous solubility of ibuprofen in the presence of the pH‐modifying agents HCl and NaOH at a constant temperature of 298.15 K.

Figure 19.13 shows the results of experimental investigations [42, 43] conducted on this system. As expected for a weakly acidic compound such as ibuprofen, an increase in the pH generally results in increased solubility. However, above a pH of about 9, the observed solubility remains essentially constant. This is because further addition of NaOH causes precipitation of ibuprofen in the form of sodium ibuprofen dihydrate [44].

19.5.5.1 System Modeling and Characterization

Both water and ibuprofen dissociate partially in the liquid phase according to the reactions

and

On the other hand, the pH‐modifying agents are assumed to dissociate fully in the aqueous environment.

Ibuprofen may form a crystalline solid form. Sodium ibuprofen dihydrate is assumed to exist only in the solid phase, dissociating fully into its constituents in the liquid phase according to the reaction

The functional group decomposition of ibuprofen employed by gSAFT has already been shown in Figure 19.8. The gSAFT databank also includes groups corresponding to the water molecule (see Appendix 19.B) and to the ionic species H⁺, OH⁻, Na⁺, and Cl⁻. The SAFT‐γ Mie parameters for all these compounds were estimated from experimental data sets for simple inorganic systems.

The ideal gas formation enthalpy and entropy at standard temperature T^⊖ and pressure P^⊖ for water were obtained directly from Ref. [45]. The enthalpies and entropies of ion formation are tabulated in Ref. [45] for the infinite dilution reference state under the molality scale and were converted to the corresponding ideal gas formation properties via Eqs. (19.A.3b) and (19.A.4) in Appendix 19.A; the results are shown in Table 19.3.

Ion	Enthalpy of Formation (kJ/mol)		Entropy of Formation (J/mol·K)
	Infinite Dilution Reference State	Ideal Gas Reference State	Infinite Dilution Reference State (Molality Scale)	Ideal Gas Reference State
H+	0.0	855.091	0.0	421.584
OH−	−229.994	437.505	−244.006	794.196
Na+	−240.120	82.051	73.067	409.285
Cl−	−167.159	269.024	−120.513	111.431

The values for the infinite dilution reference state were obtained from Ref. [45] and were converted to the corresponding ideal gas properties via Eqs. (19.A.4) and (19.A.3b).

The ibuprofen anion, IB⁻, produced by the dissociation reaction of ibuprofen, is modeled using the same functional groups as for ibuprofen, simply replacing the carboxylic acid group (COOH) with a carboxylate group (COO⁻), with the latter featuring a negative charge. As its enthalpy and entropy of formation are unknown, we employ the procedure described in Section 19.4.4 to estimate the free energy quantity Δg^R associated with the liquid‐phase dissociation of ibuprofen using a single experimental data point, namely, the one shaded in black in Figure 19.13. In this context, we need to take account of the fact that the reported experimental ibuprofen solubility is actually the combined molar concentration of ibuprofen and ibuprofen ion and that the pH value is a measurement of the activity (and not the molarity) of the H⁺ ion. The value of was estimated to be 29.231 kJ/mol.

The solid‐phase characterization of crystalline pure ibuprofen has already been shown in Table 19.1. For the characterization of sodium ibuprofen dihydrate, we apply the procedure described in Section 19.4.3 for compounds that exist only in the solid phase. A single experimental data point, that is, the one shaded in gray in Figure 19.13, is used to estimate the free energy quantity , yielding a value of −698.18 kJ/mol.

19.5.5.2 Results

Starting with a system containing water, an excess of ibuprofen, and a given quantity of the pH‐modifying agent, the combined phase and reaction equilibrium calculation is performed at T = 298.15 K to determine the liquid‐phase composition in terms of the molar fractions of water, ibuprofen, and the ions IB⁻, H⁺, OH⁻, Na⁺, and Cl⁻. From these, other derived quantities can be computed, including the apparent solubility of ibuprofen (i.e. the combined molar concentrations of ibuprofen and ibuprofen anion) and the H⁺ activity (and hence the pH) illustrated in Figure 19.14a and b, respectively.

As can be seen from the left parts of Figure 19.14a and b, the solubility of ibuprofen in acidic environments, induced by the addition of HCl, is very low. On the other hand, the addition of NaOH causes a significant increase in the solubility of ibuprofen until solid sodium ibuprofen dihydrate starts forming. Thereafter, both crystalline molecular ibuprofen and sodium ibuprofen dihydrate coexist with the liquid solution. The solubility remains constant until the initial amount of ibuprofen has been fully converted to ions , after which point the solubility decreases as expected for a basic salt in a basic solution. The pH increases upon initial addition of NaOH, remains constant while the two solid phases coexist, and then rises sharply.

Combining the results of Figure 19.14a and b, the solubility of ibuprofen can be plotted against pH, as shown in Figure 19.14c, which employs a logarithmic solubility scale in order to allow the very low solubility values to be seen more clearly. We note that in this figure, the region of coexistence of the two solids is reduced to a single point corresponding to the constant solubility and constant pH regions in Figure 19.14a and b, respectively. In the acidic region, ibuprofen solubility remains constant until the pH drops below 1, beyond which point it decreases rapidly.

Figure 19.14c also shows the experimental data of Figure 19.13 for comparison. Agreement between predictions and experiments is generally good despite the fact that, as described in Section 19.5.5.1, only two experimental data points were used to characterize this system.

Figure 19.14d shows the concentrations of ibuprofen, ibuprofen ion, H⁺, and OH⁻ as functions of pH along the solubility boundary. It can be seen that the phase and reaction equilibrium algorithms implemented in gSAFT are capable of resolving reliably very small ionic concentrations across the entire pH spectrum.

Finally, it is common practice in the pharmaceutical industry to describe the behavior of weakly dissociating acids in aqueous environments in terms of pK_a constants. The values of the latter are published in the literature and are used widely for the calculation of reaction equilibria in important applications such as the determination of bioavailability in oral absorption [1]. In the specific case of ibuprofen, pK_a corresponds to the quantity with the molar concentrations C_i being expressed in mol/dm³. Figure 19.15 shows the variation of this quantity over the pH range based on C_i values computed from the liquid‐phase compositions established by our solid–liquid phase equilibrium calculations. For comparison, the dashed line in Figure 19.15 indicates a published value [25] of the pK_a. It is clear that, especially in the alkaline region, the quantity is neither constant nor even approximately equal to the published pK_a values of around 5; in fact, the latter may overestimate the value of the concentration product by 3–4 orders of magnitude.

19.5.6 Example 5: Polymorphic Transition and Hydrate Formation of Caffeine in Water

Caffeine (CFN) is known to exhibit a polymorphic transition between two different crystalline forms (here denoted as forms I and II). Moreover, a separate solid phase consisting of a hydrate CFN₅∙(H₂O)₄ with a ratio of caffeine to water of 5 : 4 may also form [45, 46]. These factors lead to a complex phase diagram describing the behavior of caffeine in aqueous solutions.

19.5.6.1 System Modeling and Characterization

The molecular structure of caffeine is shown in Figure 19.16. For the purposes of describing the liquid‐phase behavior of this compound in the SAFT‐γ Mie framework, we choose to represent the entire molecule as a single functional group.

The necessary SAFT parameters were estimated from published solubility data [47, 48] for caffeine in simple organic solvents. This type of parameter estimation also requires a description of the solid‐phase behavior of caffeine. This was based on the melting point approach described in Section 19.2.4.2 using melting properties reported in the literature [35, 49–51] and summarized in Table 19.4.

		Form I	Form II
Tm	K	509.15	500.39
Δh[s → l](Tm)	kJ/mol	21.60	22.45
	J/mol·K	101.52	101.52

The specific heat capacity difference is assumed to be constant over the temperature range of interest.

Figure 19.17 compares the solubilities computed by gSAFT using the estimated SAFT‐γ Mie parameters with the experimental solubility data used for the estimation. As can be seen, a single set of SAFT‐γ Mie parameters leads to a good fit for all four organic solvents, which provides a degree of confidence in employing this molecular model for describing the behavior of caffeine in other systems, such as the aqueous one of interest to this example.

The standard SAFT‐γ Mie representation of the water molecule as a single functional group with two electron donor and two electron acceptor association sites was employed. The SAFT‐γ Mie parameters describing the cross‐interactions between the caffeine and water functional groups were estimated from the experimental solubility data for the high temperature polymorph (form I) shown in Figure 19.18a.

Since caffeine is a non‐ionizable compound, no liquid‐phase reactions need to be considered in this system.

The caffeine hydrate CFN₅∙(H₂O)₄ is assumed to exist only in the solid phase, dissociating into its constituents in the liquid phase:

The solid‐phase free energy of the hydrate was characterized via the procedure described in Section 19.4.3 for compounds that exist only in the solid phase. Two experimental data points (shown as filled black circles in Figure 19.18b) were used to determine the value of the free energy quantity at two different temperatures, which allows the derivation of the empirical model: (in kJ/mol).

19.5.7 Example 6: Co‐Crystal Solubility of Nicotinamide/Succinic Acid in Mixed Solvents

The system of nicotinamide and succinic acid is an example of the general category of API + coformer systems, where alongside the crystalline forms of the two pure components, a co‐crystal can be formed. In this specific case, the co‐crystal, denoted as NIC₂∙SUCC, involves a ratio of nicotinamide to succinic acid of 2 : 1.

The study presented here focuses on the solid–liquid phase diagram of nicotinamide and succinic acid in a 50 : 50 (by weight) mixture of ethanol and water at 298.15 K.

19.5.7.1 System Modeling and Characterization

The molecular structures of nicotinamide and succinic acid are shown in Figure 19.19.

As in the previous example (cf. Section 19.5.6.1), each solute molecule is modeled as a single functional group in the SAFT‐γ Mie framework, and the corresponding parameters are estimated from published experimental solubility data in pure organic solvents and in pure water. The solid‐phase characterization necessary for this estimation was based on the melting point approach described in Section 19.2.4.2 using melting properties reported in the literature [53–55] and summarized in Table 19.5.

		Nicotinamide	Succinic Acid
Tm	K	401.15	461.15
Δh[s → l](Tm)	kJ/mol	28.0	38.91
	J/mol·K	78.12	69.78

A comparison between the computed and the experimental solubility values is presented in Figure 19.20a for nicotinamide and Figure 19.20b for succinic acid.

The solid‐phase free energy of the co‐crystal was characterized via the procedure described in Section 19.4.3 for compounds that exist only in the solid phase. Since our present study is performed entirely at a single temperature of 298.15 K, the free energy quantity is required only at this temperature and can be determined from a single experimental data point relating to the solubility of the co‐crystal in any solvent that can be described by SAFT‐γ Mie. Here we used the experimentally determined solubility of the co‐crystal in pure ethanol at a temperature of 298.15 K and ambient pressure reported in Ref. [46], obtaining the value 36.404 kJ/mol.

No liquid‐phase reactions were taken into account in this study.

19.5.7.2 Calculation of Solid–Liquid Phase Diagrams

Figure 19.21 shows solid–liquid phase diagrams of nicotinamide and succinic acid in pure ethanol, pure water, and a 50 : 50 (by weight) mixture of water with ethanol, all computed by gSAFT at a temperature of 298.15 K. Experimental data reported in Ref. [46] are also shown on these diagrams. It is noted that only the single data point shown as a filled black square in Figure 19.21a was used for the characterization of this system (cf. Section 19.5.7.1); all other gSAFT results reported represent predictions.

Overall, there is good agreement between gSAFT predictions and experimental data. In particular, the results in Figure 19.21c confirm the co‐crystal's significantly increased solubility in the mixed solvent compared to that in pure water or pure ethanol.

Ionic compounds in aqueous systems cannot exist in pure form. Accordingly, from the perspective of relating thermodynamic theory to experimentally observable quantities, it is preferable to consider their properties with respect to an “infinite dilution” reference state corresponding to the limit of their presence tending to zero (or, equivalently, the molar fraction of water, x_w, tending to 1) in an aqueous solution at standard T^⊖, P^⊖.

The chemical potential of ionic compound i generally depends on the logarithm of the corresponding molar fraction x_i (cf. the term +RT ln(x_i) in Eq. 19.9). This dependence needs to be removed in order to ensure that the partial molar Gibbs free energy at the reference state has a finite value. Historically, three different approaches have been proposed to achieve this, each making use of a different measure of ionic concentration:

• Molar fraction‐based reference state:
  (19.A.1a)
• Molality‐based reference state:
  (19.A.1b)

  where m_i ≡ x_i/x_wM_w denotes the molality, with M_w being the molecular weight of water and m₀ = 1 mol/kg.

• Molarity‐based reference state:
  (19.A.1c)

  where C_i ≡ x_iρ_w/M_w denotes the molarity (molar concentration), with ρ_w being the mass density (in kg/dm³) of pure water at T^⊖, P^⊖ and C₀= 1 mol/dm³.

The free energy and enthalpy of formation in the infinite dilution reference state that are typically reported in the literature are related to via the standard thermodynamic relations:

(19.A.2)

It can be shown that, while each of the three infinite dilution reference states leads to a different value for , they all result in the same value for .

Using Eqs. (19.8) and (19.9) for the chemical potential appearing in Eqs. (19.A.1a)–(19.A.1c), we can derive the following relations between the ideal gas formation entropy and the formation entropies in the various infinite dilution reference states:

• Molar fraction‐based reference state:
  (19.A.3a)
• Molality‐based reference state:
  (19.A.3b)
• Molarity‐based reference state:
  (19.A.3c)

  where is the residual chemical potential of compound i in an infinitely diluted composition. In practical terms, this can be computed by applying an EoS such as SAFT‐γ Mie to a binary mixture comprising water and compound i and setting the corresponding molar fractions to x_w = 1 and x_i = 0, respectively.

In all three cases, the ideal gas enthalpy of formation can be obtained from the corresponding infinite dilution values via the equation

(19.A.4)

In summary, Eqs. (19.A.3a)–(19.A.3c) and (19.A.4), coupled with an EoS to compute the residual term , allow the determination of the ideal gas formation properties and from experimentally derived values and in any one of the commonly used infinite dilution reference states.

The functional group decomposition of the compounds involved in the examples of Section 19.5 within the SAFT‐γ Mie framework is shown in the table below.