Thermodynamic Equilibrium Methods

If co-crystals are to solve solubility problems one must assess their true or thermodynamic solubility so that development strategies are guided by the fundamental properties of co-crystals. Measuring the solubility of co-crystals that generate supersaturation of the parent drug is often experimentally impossible due to conversion. Eutectic points, described in Section 11.4, provide a measure of co-crystal solubility under thermodynamic equilibrium conditions. The solution at the eutectic point is saturated with co-crystal and solution concentrations represent experimentally accessible thermodynamic solubility values. Once co-crystal solubility is determined at the eutectic, the solubility under different solution conditions (pH, co-former, micelle concentration) can be obtained from solubility models that consider the appropriate solution phase equilibrium expressions. 

How eutectic point measurement leads to information about co-crystal solubility under stoichiometric solution conditions includes  

[co-former]eu/[drug]eu ratios or values above the co-crystal stoichiometric ratio are associated with increased co-crystal thermodynamic activity of the drug (co-crystal solubility higher than drug solubility on a molar drug basis). 

A higher [co-former]eu/[drug]eu ratio, for a series of co-crystals of the same drug with different co-formers, translates to higher co-crystal solubility. This simple observation will establish a rank order of cocrystal solubilities. 

---

A thermodynamic equilibrium method is used to determine the equilibrium ratio of concentrations of the low-molecular-mass substances in materials in contact. The equilibrium state is usually determined with the help of kinetic curves of mass change in bodies in contact versus time. The analysis of this data permits estimation of distribution coefficients between the studied materials. A long experiment duration (usually a few months at room temperature) is a serious hindrance. If data must be collected at sub-zero temperatures, the experiment may take several years. 

Note that the recommended value for p is not always conservative. In some cases, heat input may be so high that the safety valve cannot vent all the generated vapor. In such cases, the internal pressure will rise until the bursting overpressure is reached, which may be much higher than the vessel s design pressure. For example, Droste and Schoen (1988) describe an experiment in which an LPG tank failed at 39 bar, or 2.5 times the opening pressure of its safety valve. Note also that this method assumes that the fluid is in thermodynamic equilibrium yet, in practice, stratification of liquid and vapor will occur (Moodie et al. 1988). 

The chapter by Gruber deals with thermodynamic equilibrium considerations. It develops a graphical method for presenting these results, and touches on the potentially important problem of carbon-laydown on the catalyst. 

We have noted previously that the forward and reverse rates are equal at equilibrium. It seems, then, that one could use this equality to deduce the form of the rate law for the reverse reactions (by which is meant the concentration dependences), seeing that the form of the equilibrium constant is defined by the condition for thermodynamic equilibrium. By and large, this method works, but it is not rigorously correct, since the coefficients in the equilibrium condition are only relative, whereas those in the rate law are absolute.19 Thus, if we have this net reaction and rate law for the forward direction,... 

We have considered thermodynamic equilibrium in homogeneous systems. When two or more phases exist, it is necessary that the requirements for reaction equilibria (i.e., Equations (7.46)) be satisfied simultaneously with the requirements for phase equilibria (i.e., that the component fugacities be equal in each phase). We leave the treatment of chemical equilibria in multiphase systems to the specialized literature, but note that the method of false transients normally works quite well for multiphase systems. The simulation includes reaction—typically confined to one phase—and mass transfer between the phases. The governing equations are given in Chapter 11. 

Couette Flow Simulation. MD typically simulate systems at thermodynamic equilibrium. For the simulation of systems undergoing flow various methods of nonequilibrium MD have been developed (Ifl iZ.). In all of these methods the viscosity Is calculated directly from the constitutive equation. 

In this work, we developed the safeguard active-set method by modifying the active-set method for thermodynamic equilibrium in order to include the classical nucleation theory. At tn, assume that the partition ( (r ), M(t ), N(t ) and the crystallization time tciyst(t ) forM(t ) are known. For a new feed vector and RH at Vu compute W(tn+i), M(t i), N(t + )) and tciyst(t +i) as follows ... 

In contrast to the ionizing electrode method, the dynamic condenser method is based on a well-understood theory and fulfills the condition of thermodynamic equilibrium. Its practical precision is limited by noise, stray capacitances, and variation of surface potential of the air-electrode surface, i.e., the vibrating plate. At present, the precision of the dynamic condenser method may be limited severely by the nature of the surfaces of the electrode and investigated system. In common use are adsorption-... 

The classical method to study adsorption deals with establishing a correspondence between thermodynamically equilibrium amount of absorbed gas, the temperature and the partial pressure of the gas in space surrounding adsorbent. Usually, a constant temperature is maintained throughout an adsorption experiment. The shape of isotherms obtained gives information on the character of adsorption. 

Headspace methods provide an indirect method of sample analysis suitable for the determination of organic volatiles [11,318-323]. The gas diase in contact with the sample and not the sample matrix itself is taken for analysis. If the sample is in thermodynamic equilibrium with the gas phase in a closed thermostated vessel, then this method of analysis is referred to as static headspace. If a carrier gas is passed over the sample and the sample volatiles accumulated in a cryogenic or sorbent trap, then the method is generally referred to as dynamic headspace. If the carrier gas is introduced below the surface of... 

The EMF of a galvanic cell is a thermodynamic equilibrium quatity. Thus, the potential of a cell must be measured under equilibrium conditions, i.e. without current flow. The measured EMF must be compensated by a known external potential difference. The measurement of the EMF of a cell is thus based on determination of a potential difference that exactly compensates the measured potential difference so that no current passes. This is easily achieved by the Poggendorf compensation method (see Fig. 3.13). 

A rather general method of the calculation of the tunneling taking account of the dissipation was given in Ref. 82. The cases of rather strong dissipation were considered in Refs. 81 and 82, where it was assumed that a thermodynamical equilibrium in the initial potential well exists. The case of extremely weak friction has been considered using the equations for the density matrix in Ref. 83. A quantum analogue of the Focker-Planck equation for the adiabatic and nonadiabatic processes in condensed media was obtained in Refs. 105 and 106. 

McFarland et al. recently [1] published the results of studies carried out on 22 crystalline compounds. Their water solubilities were determined using pSOL [21], an automated instrument employing the pH-metric method described by Avdeef and coworkers [22]. This technique assures that it is the thermodynamic equilibrium solubility that is measured. While only ionizable compounds can be determined by this method, their solubilities are expressed as the molarity of the unionized molecular species, the intrinsic solubility, SQ. This avoids confusion about a compound s overall solubility dependence on pH. Thus, S0, is analogous to P, the octanol/water partition coefficient in both situations, the ionized species are implicitly factored out. In order to use pSOL, one must have knowledge of the various pKas involved therefore, in principle, one can compute the total solubility of a compound over an entire pH range. However, the intrinsic solubility will be our focus here. There was one zwitterionic compound in this dataset. To obtain best results, this compound was formulated as the zwitterion rather than the neutral form in the HYBOT [23] calculations. 

The critical concentration at which the first micelle forms is called the critical micelle concentration, or CMC. As the concentration of block copolymer chains increases in the solution, more micelles are formed while the concentration of nonassociated chains, called unimers, remains constant and is equal to the value of the CMC. This ideal situation corresponds to a system at thermodynamic equilibrium. However, experimental investigations on the CMC have revealed that its value depends on the method used for its determination. Therefore, it seems more reasonable to define phenomenologically the CMC as the concentration at which a sufficient number of micelles is formed to be detected by a given method [16]. In practical terms, the CMC is often determined from plots of the surface tension as a function of the logarithm of the concentration. The CMC is then defined as the concentration at which the surface tension stops decreasing and reaches a plateau value. 

In their subsequent works, the authors treated directly the nonlinear equations of evolution (e.g., the equations of chemical kinetics). Even though these equations cannot be solved explicitly, some powerful mathematical methods can be used to determine the nature of their solutions (rather than their analytical form). In these equations, one can generally identify a certain parameter k, which measures the strength of the external constraints that prevent the system from reaching thermodynamic equilibrium. The system then tends to a nonequilibrium stationary state. Near equilibrium, the latter state is unique and close to the former its characteristics, plotted against k, lie on a continuous curve (the thermodynamic branch). It may happen, however, that on increasing k, one reaches a critical bifurcation value k, beyond which the appearance of the... 

There are numerous advantages to the FAC approach that differentiate it from many forms of bioassay - MS-dependent or otherwise. The FAC method offers thermodynamic and kinetic binding data from the breakthrough curves. As with the classical application of the FA method, the quality of the data is superb relative to other chromatographic or electrophoretic methods [9, 10]. It is an equilibrium method, as opposed to systems that rely upon the separation of bound from unbound, and this forms the basis of its accuracy. 

---