Solvatomorphic Systems
Harry G. Brittain
Center for Pharmaceutical Physics, Milford, New Jersey, U.S.A.
INTRODUCTION TO THE PHASE RULE
Bancroft has stated that the two expressions describing in a qualitative manner all states and changes of equilibrium are the Phase Rule and the Theorem of Le Chatelier (1) . One of these principles describes the possibilities that might exist among substances in equilibrium, and the other describes how such equilibrium systems would react to an imposed stress. These changes may entail alterations in chemical composition, but could just as well involve transitions in the physical state. There is no doubt that thermodynamics is the most powerful tool for the char- acterization of such equilibria. Consider the situation presented by elemental sul- fur, which can be obtained in either a rhombic or monoclinic crystalline state. Each of these forms melts at a different temperature, and is stable under certain well- defi ned environmental conditions. An understanding of this system would entail knowing under what conditions these two forms could equilibrate with liquid sul- fur (either singly or together), and what would be the conditions under which the two could equilibrate with each other in the absence of a liquid phase.
These questions can, of course, be answered with the aid of chemical thermo- dynamics, the modern practice of which can be considered as beginning with pub- lication of the seminal papers of J. Willard Gibbs (2) . Almost immediately after the Law of Conservation of Mass was established, Gibbs showed that all cases of equi- libria could be categorized into general class types. His work was perfectly general in that it was free from hypothetical assumptions, and immediately served to show how different types of chemical and physical changes actually could be explained in a similar fashion. Gibbs began with a system that needed only three independent variables for its complete specifi cation, these being temperature, pressure, and the concentration of species in the system. From these considerations, he defi ned a gen- eral theorem known as the Phase Rule, where the conditions of equilibrium could be specifi ed according to the composition of that system.
The following discussion of the Phase Rule, and its application to systems of polymorphic interest, has primarily been distilled from the several classic accounts published in the fi rst half of this century (1,3–10). It may be noted that one of the most fractious disagreements that took place early in the development of physical chemistry took place between the proponents of pure computational thermody- namics and those seeking a more qualitative understanding of physical phenom- ena. The school of exact calculations prevailed (11) , and this view has tended to dominate how workers in the fi eld treat experiment and theory. Nevertheless, hav- ing a qualitative understanding about phase transformation equilibria can provide
2
one with a clearer grasp regarding a particular system, and the Phase Rule is still valuable for its ability to predict what is possible and what is not in a system that exists in a state of equilibrium.
Phases
A heterogeneous system is composed of various distinct portions, each of which is in itself homogenous in composition, but which are separated from each other by distinct boundary surfaces. These physically distinct and mechanically sepa- rable domains are termed phases. A single phase must be chemically and physi- cally homogeneous, and may consist of single chemical substance or a mixture of substances.
Theoretically, an infi nite number of solid or liquid phases may exist side by side, but there can never be more than one vapor phase. This situation arises from the fact that all gases are completely miscible with each other in all proportions, and will therefore never undergo a spontaneous separation into component materials. It is important to remember, however, that equilibrium is independent of the relative amounts of the phases present in a system. For instance, once equilibrium is reached, the vapor pressure of a liquid does not depend on either the volume of the liquid or vapor phases. As another example, the solubility of a substance in equilibrium with its saturated solution does not depend on the quantity of solid material present in the system.
In a discussion of simple polymorphic systems, one would consider the vapor and liquid phases of the compound as being separate phases. In addition, each solid polymorph would constitute a separate phase. Once the general rule is deduced and stated, the Phase Rule can be used to deduce the conditions under which these forms could exist in an equilibrium condition.
Components
A component is defi ned as a species whose concentration can undergo indepen- dent variation in the different phases. Another way to state this defi nition is that a component is a constituent that takes part in the equilibrium processes. For instance, in the phase diagram of pure water, there is only one component (water), despite the fact that this compound is formed by the chemical reaction of hydrogen and oxygen. Because according to the Law of Defi nite Proportions the ratio of hydrogen and oxygen in water is fi xed and invariable, it follows that their concentration can- not be varied independently, and so they cannot be considered as being separate components.
For the specifi c instance of polymorphic systems, the substance itself will be the only component present. The situation complicates for solvatomorphs because the lattice solvent will compromise a second component, and hence, different phases will not have the same composition. The general rule is that the number of compo- nents present in an equilibrium situation is to be chosen as the smallest number of the species necessary to express the concentration of each phase participating in the equilibrium.
Degrees of Freedom
The number of degrees of freedom of a system is defi ned as the number of variable factors that must be arbitrarily fi xed to completely defi ne the condition of the sys- tem at equilibrium. Gibbs (2) demonstrated that the state of a phase is completely
determined if the temperature, pressure, and chemical potentials of its components are known. For a system of one component, there are no chemical potentials involved, so the system becomes specifi ed only through knowledge of the tempera- ture and pressure. One often speaks of the variance of a system, which is defi ned by the number of degrees of freedom required to specify the system.
For example, consider the situation of a substance forming an ideal gas in its vapor phase. The equation of state for ideal gases is given by the familiar equation:
PV nRT= (1)
where P is the pressure, V is the volume, n is the number of moles present, T is the absolute temperature, and R is the gas constant. For a given amount of gas, if two out of the three independent parameters are specifi ed, then the third is determined.
This type of system is then said to be bivariant, or one that exhibits two degrees of freedom. If the gaseous substance is then brought into a state of equilibrium with its condensed phase, then empirically one fi nds that the condition of equilibrium can be specifi ed by only one variable. The system exhibits only one degree of freedom, and is now termed univariant. If this system is cooled down until the solid phase forms, and the liquid and vapor remain in an equilibrium condition, one empiri- cally fi nds that this equilibrium condition can only be attained if all independent parameters are specifi ed. This latter system exhibits no degrees of freedom, and is said to be invariant.
The Phase Rule
For a substance capable of existing in two different phases, the state of equilibrium is such that the relative amounts of substance distributed between the phases in the absence of stress appears to be unchanging over time. This can only occur when the Gibbs chemical potential is the same in each phase, so equilibrium is defi ned as the situation where the chemical potential of each component in a phase is the same as the chemical potential of that component in the other phase.
Consider the system that consists of C components present in P phases. In order to specify the composition of each phase, it is necessary to know the concen- trations of ( C – 1) components in each of the phases. Another way to state this is that each phase possesses ( C – 1) variables. Besides the concentration terms, there are two other variables (temperature and pressure), so that altogether the number of variables existing in a system of C components in P phases is given by:
Variables = ( – 1) + 2VariablesP C (2) In order to completely defi ne the system, one requires as many equations as variables. If for some reason there are fewer equations than variables, then values must be assigned to the variables until the number of unknown variables equals the number of equations. Alternatively, one must assign values to undefi ned variables or else the system will remain unspecifi ed. The number of these variables that must be defi ned or assigned to specify a system is the variability, or the degree of freedom of the system.
The equations by which the system is to be defi ned are obtained from the relationship between the potential of a component and its phase composition, tem- perature, and pressure. If one chooses as a standard state one of the phases in which all of the components are found, then the chemical potential of any component in another phase must equal the chemical potential of that component in the standard
state. It follows that for each phase in equilibrium with the standard phase, there will be a defi nite equation of state for each component in that phase. One concludes that if there are P phases, then each component will be specifi ed by ( P – 1) equa- tions. Then for C components, we deduce that the maximum number of available equations is given by:
Equations = ( – 1)C P (3) The variance (degrees of freedom) in a system is given by the difference between the number of variables and the number of equations available to specify these. Denoting the number of degrees of freedom as F , this can be stated as:
F = Variables – Equations (4) Substituting equations (2) and (3) into equation (4), and simplifying, yields:
F = + 2 – C P (5)
which is often rearranged to yield the popular statement of the Phase Rule:
P F C + = + 2 (6)
One can immediately deduce from equation (5) that for a given number of com- ponents, an increase in the number of phases must lead to a concomitant decrease in the number of degrees of freedom. Another way to state this is that with an increase in the number of phases at equilibrium, the condition of the system must become more defi ned and less variable. Thus, for polymorphic systems where one can encoun- ter additional solid-state phases, the constraints imposed by the Phase Rule can be exploited to obtain a greater understanding of the equilibria involved.
SYSTEMS OF ONE COMPONENT
In the absence of solvatomorphism or chemical reactions, polymorphic systems consist of only one component. The complete phase diagram of a polymorphic sys- tem provides the boundary conditions for the vapor state, the liquid phase, and the boundaries of stability for each and every polymorph. From the Phase Rule, it is concluded that the maximum amount of variance (two degrees of freedom) is only possible when the component is present in a single phase. All systems consisting of one component in one phase can therefore be perfectly defi ned by assigning values to a maximum of two variable factors. However, this bivariant system is not of interest to our discussion.
When a single component is in equilibrium between two phases, the Phase Rule predicts that it must be a univariant system exhibiting only one degree of free- dom. Consequently, it is worthwhile to consider several univariant possibilities, because the most complicated phase diagram of a polymorphic system can be bro- ken down into its component univariant systems. The Phase Rule applies equally to all of these systems, and all need to be understood for the entire phase diagram to be most useful.
Characteristics of Univariant Systems
When a single component exists in a state of equilibrium between two phases, the system is characterized by only one degree of freedom. The types of observable
equilibria can be of the liquid/vapor, solid/vapor, solid/liquid, and (specifi cally for components that exhibit polymorphism) solid/solid types. We will consider the important features of each in turn.
Liquid/Vapor Equilibria
A volatile substance in equilibrium with its vapor constitutes a univariant system, which will be defi ned if one of the variables (pressure or temperature) is fi xed. The implications of this deduction are that the vapor pressure of the substance will have a defi nite value at a given temperature. Alternatively, if a certain vapor pressure is maintained, then equilibrium between the liquid and vapor phase can only exist at a single defi nite temperature. Each temperature point therefore corresponds to a defi nite pressure point, and so a plot of pressure against temperature will yield a continuous line defi ning the position of equilibrium. Relations of this type defi ne the vaporization curve , and are ordinarily plotted to illustrate the trends in vapor pressure as a function of system temperature. It is generally found that vaporiza- tion curves exhibit the same general shape, being upwardly convex when plotted in the usual format of pressure–temperature phase diagrams.
As an example, consider the system formed by liquid water, in equilibrium with its own vapor. The pressure–temperature diagram for this system has been constructed over the range of 1–99°C (12) , and is shown in Figure 1 . The character- istics of a univariant system (one degree of freedom) are evident in that for each defi nite temperature value, water exhibits a fi xed and defi nite vapor pressure.
0 20 40 60 80 100
0 100 200 300 400 500 600 700 800
Vapor Liquid
Temperature (°C)
Vapor pressure (torr)
FIGURE 1 Vapor pressure of water as a function of temperature. The data were plotted from published values (12).
In a closed vessel, the volume becomes fi xed. According to Le Chatelier’s Principle, an input of heat (i.e., an increase in temperature) into a system consisting of liquid and vapor in equilibrium must result in an increase in the vapor pressure.
It must also happen that with the increase of pressure, the density of the vapor must increase, whereas with the corresponding increase in temperature the density of the liquid must decrease. At some temperature value, the densities of the liquid and vapor will become identical, and at that point the heterogeneous system becomes homogeneous. At this critical point (defi ned by a critical temperature and a critical pressure), the entire system passes into one homogeneous phase, and the vaporization curve terminates at this critical point. As evident in Figure 1 , the vapor pressure of a liquid approaches that of the ambient atmospheric pressure as the boiling point is reached.
Continuing with the principle of Le Chatelier, if an equilibrium system is stressed by a force that shifts the position of equilibrium, then a reaction to the stress that opposes the force will take place. Consider, therefore, a liquid/vapor system that is suffi ciently isolated from its surroundings so that heat transfer is prevented (i.e., an adiabatic process). An increase in the volume of this system results in a decrease in the pressure of the system, causing liquid to pass into the vapor state. This process requires the input of heat, but because none is available from the surroundings, it follows that the temperature of the system must fall.
Although qualitative changes in the position of liquid/vapor equilibrium can be predicted by Le Chatelier’s principle, the quantitative specifi cation of the system is given by the Clausius–Clapeyron equation:
2 1
d
d ( – )
q P
T= T v v (7)
where q is the quantity of heat absorbed during the transformation of one phase to the other, v 2 and v 1 are the specifi c volumes of the two phases, and T is the absolute temperature at which the change occurs. Integration of equation (7) leads to useful relations that permit the calculation of individual points along the vaporization curve.
Solid/Vapor Equilibria
As a univariant system, a solid substance in equilibrium with its vapor phase will exhibit a well-defi ned vapor pressure for a given temperature, which will be inde- pendent of the relative amounts of solid and vapor present. The curve represent- ing the solid/vapor equilibrium conditions is termed a sublimation curve , and generally takes a form similar to that of a vaporization curve. Although the subli- mation pressure of a solid is often exceedingly small, for many substances it can be considerable.
One example of a solid that exhibits signifi cant vapor pressure is camphor, for which a portion of its sublimation curve is shown in Figure 2 . This compound exhibits the classic pressure–temperature profi le (13) , fi nally attaining a vapor pres- sure of 422.5 torr at its melting point (179.5°C). When heated above the fusion tem- perature, only a short vaporization curve is possible because the boiling point of camphor is reached at 207.4°C.
The sublimation curve of all substances will have its upper limit at the melting point, and a theoretical lower limit of absolute zero. However, because
low-temperature polymorphic transitions can be encountered, one often encounters considerable complexity in sub-ambient phase diagrams. One need only consider the example of water, where at least seven crystalline forms are known.
If the sublimation pressure of a solid exceeds that of the atmospheric pressure at any temperature below its melting point, then the solid will pass directly into the vapor state (sublime) without melting when that substance is stored in an open ves- sel at that temperature. In such instances, melting of the solid can only take place at pressures exceeding ambient. Carbon dioxide is one of the best known materials that exhibits sublimation behavior. At the usual room temperature conditions, solid
“dry ice” sublimes easily. Liquid carbon dioxide can only be maintained between its critical point (temperature of +31.0°C and pressure of 75.28 atm) and its triple point (temperature of –56.6°C and pressure of 4.97 atm) (14) .
The direction of changes in sublimation pressure with temperature can be qualitatively predicted using Le Chatelier’s principle, and quantitatively calculated by means of the Clausius–Clapeyron equation.
Solid/Liquid Equilibria
When a crystalline solid is heated to the temperature at which it melts and passes into the liquid state, as long as the two phases are in equilibrium, the solid/liquid system is univariant. Consequently, for a given pressure value, there will be a defi - nite temperature (independent on the quantities of the two phases present) at which the equilibrium can exist. As with any univariant system, a curve representing the
0 20 40 60 80 100
0 3 6 9 12 15 18 21
Vapor Solid
Temperature (°C)
Vapor pressure (torr)
FIGURE 2 Vapor pressure of camphor as a function of temperature. The data were plotted from published values (13).
equilibrium temperature and pressure data can be plotted, and this is termed the melting point or fusion curve . Because both phases in a solid/liquid equilibrium are condensed (and diffi cult to compress), the effect of pressure on the melting point of a solid is relatively minor unless the applied pressures are quite large.
Using Le Chatelier’s principle, one can qualitatively predict the effect of pres- sure on an equilibrium melting point. The increase in pressure results in a decrease in the volume of the system. For most materials, the specifi c volume of the liquid phase is less than that of the solid phase, so that an increase in pressure would have the effect of shifting the equilibria to favor the solid phase. This shift will have the observable effect of raising the melting point. For those unusual systems where the specifi c volume of the liquid exceeds that of the solid phase, then the melting point will be decreased by an increase in pressure.
An example of a fusion curve is provided in Figure 3 , which uses benzene as the example (15) . It can be seen that to double the melting point requires an increase in pressure from 1 atm to approximately 250 atm. The fusion curve of Figure 3 is fairly typical in that in the absence of any pressure-induced polymorphic transformations, the curve is essentially a straight line.
The quantitative effect of pressure on the melting point can be calculated using the inverse of the Clausius–Clapeyron equation:
d ( 2 1) d
T v v T
P q
= − (8)
5 7 9 11 13 15 17 19
0 75 150 225 300 375 450
Liquid Solid
Temperature (°C)
Applied pressure (atm)
FIGURE 3 Effect of pressure on the melting point of benzene. The data were plotted from published values (15).