The Chemical Potential of Water in Hydrates

Our aim is to derive the chemical potential of water to enable phase equilibria calculations. Note that, while Equation 5.7 is the grand canonical partition function (2guest) an(j with respect to the solute (guest) molecule, it is the canonical partition function ( 2host) with respect to the host (water) because k = 1, so that we have 

using the letter h to denote the host, and g to denote the guest, each of the partition functions in Equation 5.8 can be related to their macroscopic thermodynamic properties in the usual way (see McQuarrie, 1976, p. 58) as 

Since entropy and volume are extensive properties, they can be combined, 

By dropping the superscript combined, taking the left-most derivative, and using d[ij = k l Mnkj on the right, we get  

With the development of Equation 5.12 relating the partition function and the macroscopic properties, all of the macroscopic thermodynamic properties may be derived from Equation 5.7. For example, differentiating In E with respect to the absolute activity (A.) of./, provides the total number of guest molecules J over all the cavities i 

---

Equation 5.23 may be used with Equation 5.22a to determine the chemical potential of water in hydrate /z, which is one of the major contributions of the model. The combination of these two equations is of vital importance to phase equilibrium calculations, since the method equates the chemical potential of a component in different phases, at constant temperature and pressure. 

Equation 5.23 is considered to be an ideal solid solution model. If we choose to extend our equations from one hydrate crystal to a large number Na (Avogadro s number) of crystals, we must replace the Boltzmann constant k with the universal gas constant R (=kN ). Ballard (2002) defined the chemical potential of water in hydrates as... 

When it comes to the equilibration of water concentration gradients, the relevant transport coefficient is the chemical diffusion coefficient, Dwp. This parameter is related to the self-diffusion coefficient by the thermodynamic factor (see above) if the elementary transport mechanism is assumed to be the same. The hydration isotherm (see Figure 8) directly provides the driving force for chemical water diffusion. Under fuel-cell conditions, i.e., high degrees of hydration, the concentration of water in the membrane may change with only a small variation of the chemical potential of water. In the two-phase region (i.e., water contents of >14 water molecules... 

Equation 5.22a enables the calculation of the chemical potential of water in the hydrate as a function of the fractional occupation in the cavities. Equation 5.18 provides the chemical potential of water in terms of the chemical potential in the empty hydrate, as well as the product of the individual cavity partition function and the absolute activity ... 

Figure 5.4 illustrates the processes needed to determine the chemical potential of water in the hydrate, as given by Equation 5.28... 

The major advance in hydrate thermodynamics was the generation of the van der Waals and Platteeuw model bridging the normal macroscopic and microscopic domains. Only a brief overview is given here to provide a basis for model improvements the reader interested in more details should refer to another source. The essence of the van der Waals and Platteeuw model is the equation for the chemical potential of water in the hydrate phase ... 

Equation (1) indicates that when a guest fills the cavities of a hydrate, the chemical potential of water in the cage is lowered, thereby stabilizing the hydrate phase. In principle. Equation (1) solves the hydrate prediction dilemma - that is, the hydrate formation conditions are determined by the pressures and temperatures that cause equality between the hydrate chemical potential of water in Equation (1), and the chemical potential of water in the other phase(s), as determined by separate equations of state. There are two important terms on the right the molar Gibbs free... 

If jul is the chemical potential of a hypothetical empty hydrate, each of the chemical potentials of water in Eq. (1), can be calculated relatively to. For, the hydrate phase. Van der Waals and Platteeuw have derived the following expression by calculating the great canonical partition function of the enclathrated gas, based on the principles of Statistical Mechanics ... 

For calculating the chemical potential of water in the liquid solution, or ice phase. Holder, Corbin, Papadopoulos generated chemical potential, enthalpy, and heat capacity functions for gas hydrates at temperatures between 150 and 300 K and derived... 

Another aspect of current interest associated with the lipid-water system is the hydration force problem.i -20 When certain lipid bilayers are brought closer than 20-30 A in water or other dipolar solvents, they experience large repulsive forces. This force is called solvation pressure and when the solvent is water, it is called hydration pressure. Experimentally, hydration forces are measured in an osmotic stress (OS) apparatus or surface force apparatus (SFA)2o at different hydration levels. In OS, the water in a multilamellar system is brought to thermodynamic equilibrium with water in a polymer solution of known osmotic pressure. The chemical potential of water in the polymer solution with which the water in the interlamellar water is equilibrated gives the net repulsive pressure between the bilayers. In the SEA, one measures the force between two crossed cylinders of mica coated with lipid bilayers and immersed in solvent. 

The chemical potential of water in the clathrate hydrate is also calculated from the thermodynamic potential as... 

Here i refers to the chemical potential of water in the phase a, with a = H being the hydrate at the appropriate composition and a = p being the hypothetical empty hydrate lattice. Other symbols in (1) are v- is the number of cavities of type i in the hydrate lattice, k is Boltzmann s constant, and T is the temperature. 

Equation (1) is readily recast in a form that is suitable for calculating the conditions for hydrate stability. Note that for a true three-phase equilibrium between water, hydrate and guest phases, the chemical potential of each individual species must be the same in all three phases. Thus must equal the chemical potential of water in its stable pure phased and we have that... 

Stresses caused by chemical forces, such as hydration stress, can have a considerable influence on the stability of a wellbore [364]. When the total pressure and the chemical potential of water increase, water is absorbed into the clay platelets, which results either in the platelets moving farther apart (swelling) if they are free to move or in generation of hydrational stress if swelling is constrained [1715]. Hydrational stress results in an increase in pore pressure and a subsequent reduction in effective mud support, which leads to a less stable wellbore condition. 

In order to assign the Raman bands and determine the absolute oceupancies of O2 and N2 molecules in the small and large cavities, we use the statistical thermodynamic expression derived by van der Waals and Platteeuw. " Let us consider an equilibrium state of the ice-hydrate-gas system. Then, the difference between the chemical potential of water molecules in ice, //h 0), and that in a hypothetical empty lattice of structure II hydrate, ju (h ), is given by... 

In this part of the paper we examine the thermodynamic properties of hydrated ionomers. By strongly hydrated we mean that we are beyond the state of solvation shells, where V, the number of water molecules per cation, is a small number ("v A to 6). In a strongly hydrated sample, the water molecules are considered to be free, and make a concentrated solution with the cations (the counter ions) and eventually with some mobile anions (the coions). This subject has already been extensively studied because of its practical importance (1-4). From the following discussion, we shall see that some of the usual classical laws are no longer valid. For instance, the variation of the chemical potential of water with the concentration of cations may no longer hold. 

We further impose an experimentally accessible condition that the clathrate hydrate is in equilibrium with a fluid mixture of guest and water. This is realized by requiring that the chemical potentials of water and guest in the fluid phase are equal to those in the clathrate and the pressure in the fluid is equal to p in Eq. (7). Then, A fig in Eq. (8) is replaced in terms of dp and dJfrom the Gibbs-Duhem equations as... 

Even when the number of molecules of type a is fixed to a constant, the chemical potential of water can be expressed apparently by the same equation [40]. In case of an equilibrium between clathrate hydrate and ice, there are three phases (/Jp = 3 clathrate, ice, and guest fluid) with the components of two species (tic = 2 water and one guest type, /). From the phase rule, n[ = iic + 2 — rip=l, and Tcan be an independent thermodynamic variable while the occupancy Xa (= nalNj) is kept fixed and is regarded as a parameter. The equilibrium between ice and clathrate for a given is obtained by solving... 

It is highly desirable to establish a relationship between reduction of the dissociation pressure and the efficiency as hydrogen storage in a wide range of pressure with varying composition of a promoter species. To this end, we should find a simple way to evaluate the thermodynamic stability of clathrate hydrates and its cage occupancy from intermolecular interactions currently available under a fixed occupancy of a promoter guest elaborated in Section III.D. Since the chemical potential of water is written as... 

Of course, it is impractical to calculate the free energies up to quadruple occupation mi = A. However, the above equation provides a way to evaluate the chemical potential of water once a set of jris known. This is achieved by GC/NPT simulations with a fixed number of promoter species. The mean hydrogen number per water 2h is listed for various and p in Table V for stable clathrate hydrate. It should be noted this table is valid for any promoter species. 

Owing to the assumptions in the vdWP theory, the chemical potential of water is divided into two parts one part is the chemical potential of empty lattice, /cj , which depends only on the crystal structure of the clathrate hydrate, and the other part is the chemical potential of cage occupancy, A/Cc. which depends on the properties of the guest molecule and the cage size. 

The other part of the chemical potential of water /Xc is contributed from the chemical potential of empty lattice, which is independent from the choice of the guest molecule. The chemical potential of the hypothetical empty clathrate hydrate in the limit of low pressure is decomposed into two terms ... 

A remarkable success of the vdW-P model is the prediction that type II structure should be more stable than type I if the guest is very small.According to Eq. [8], the chemical potentials of water molecules in structures I and II hydrate are... 

In the case of the retro Diels-Alder reaction, the nature of the activated complex plays a key role. In the activation process of this transformation, the reaction centre undergoes changes, mainly in the electron distributions, that cause a lowering of the chemical potential of the surrounding water molecules. Most likely, the latter is a consequence of an increased interaction between the reaction centre and the water molecules. Since the enforced hydrophobic effect is entropic in origin, this implies that the orientational constraints of the water molecules in the hydrophobic hydration shell are relieved in the activation process. Hence, it almost seems as if in the activated complex, the hydrocarbon part of the reaction centre is involved in hydrogen bonding interactions. Note that the... 

Hydration of polymeric membranes may be influenced by the chemical identity of the polymers. A hydrophilic polymer has a higher potential to hydrate than a hydrophobic one. Sefton and Nishimura [56] studied the diffusive permeability of insulin in polyhydroxyethyl methacrylate (37.1% water), polyhydroxy-ethyl acrylate (51.8% water), polymethacrylic acid (67.5% water), and cupro-phane PT-150 membranes. They found that insulin diffusivity through polyacrylate membrane was directly related to the weight fraction of water in the membrane system under investigation (Fig. 17). 

Having obtained the elastic equations in terms of shifted entities, and reverting to total entities, the constitutive equations express the total stress cr, the chemical potentials of the extrafibrillar water p,wE and of the salt psE, and the hydration potential of the intrafibrillar water //hydl . in terms of the generalized strains, namely the strain of the porous medium e, the mass-contents of the extrafibrillar water mWE and of the cations sodium mNae, and the mass-content of intrafibrillar water mwi. The interested reader is directed to [3]. 

---