Empty hydrate lattice

Pandit and King (1982) and Bathe et al. (1984) presented measurements using transducer techniques, which are somewhat different from the accepted values of Kiefte et al. (1985). The reason for the discrepancy of the sonic velocity values from those in Table 2.8 and above is not fully understood. It should be noted that compressional velocity values can vary significantly depending on the hydrate composition and occupancy. This has been demonstrated by lattice-dynamics calculations, which showed that the adiabatic elastic moduli of methane hydrate is larger than that of a hypothetical empty hydrate lattice (Shpakov et al., 1998). 

In particular, the extension of the van der Waals and Platteeuw method addresses the first assumption listed at the beginning of Section 5.1.1—namely that encaged molecules do not distort the cavity. In the development of the statistical thermodynamic hydrate model (Equation 5.23), the free energy of water in the standard hydrate (empty hydrate lattice), gt, is assumed to be known at a given temperature (T) and volume (v). Since the model was developed at constant volume, the assumption requires that the volume of the empty hydrate lattice, 7, be equal to the volume of the equilibrium hydrate, v11, so that the only energy change is due to occupation of the hydrate cavities, as shown in Figure 5.3. 

The reader may be confused by the suggestion that the empty hydrate lattice being distorted by the addition of guests. Yet the method is pragmatically justified because it would be impossible to measure the empty lattice energies for all possible combinations of hydrate components. So we simply use methane for si (or propane for sll, or methane + neohexane for sH) as a reference case. With these references, the deviation occurs because an empty methane lattice is not the same as an empty CO2 or xenon lattice, and thus we try to account for that by using this activity term. This point is further discussed in Section 5.1.6. 

The best choice for the standard hydrate is one that is well-characterized and not too different from the real state of the system. If the standard state is well-defined, small perturbations from this standard state can be accounted for correctly. With this in mind, we turn to the three most well-known hydrates of si, sll, and sH, namely methane, propane, and methane+neohexane. Note that the standard states for si, sll, and sH are the empty hydrate lattices of these and not the actual hydrates. Therefore for the reference hydrates, the activity coefficients for methane, propane, and methane + neohexane hydrates will be unity. 

In the event that all are of similar magnitude, equations (1) and (2) and the above values of chemial potentials for equilibrium with ice at 0° lead to (1 - 9)(I)/(1 - 9)(II) = 0.041/0.111 = f(II)/f(I) and structure II first becomes stable at a pressure of the hydrate-forming gas which is only about 40% the pressure at which structure I becomes stable. The assumption of reasonable values for the enthalpy difference between empty hydrate lattices and ice (see section 3) makes structure II relatively even more stable at lower temperatures thus at -100 structure II is stable at a pressure only 20% of that required for structure I stability. 

The Kihara potential function [12] is used as described in McKoy and Sinanoglu [13]. The Kihara potential parameters, a (the radius of the spherical molecular core), a (the collision diameter), and e (the characteristic energy) are taken from Tohidi-Kalorazi [14], The fugacity of water in the empty hydrate lattice, // in Equation , can be calculated by ... 

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. 

The main problem with implementing the cell theory is that it derives the hydrate pressures in terms of quantities that are not well known. In particular, the intermolecular potentials needed to define the cavity potential are not precisely known, and depends on the property of a state that has never been observed (Le, the empty hydrate lattice). Inevitably, this means that a considerable amount of empirical parameterisation is required to implement the theory. 

TABLE IV Differences in thermodynamic properties between ice and the empty hydrate lattice (adapted from Sloan, 1998 )... 

On the whole, the cell theory has proved to be successful. This is particularly true of simple hydrates, for which the calculated hydrate pressures show no significant deviations from experimental values. This might not seem surprising since there are at least 11 empirical parameters that can be chosen to make the theory work however, eight of those parameters describe the water lattice and so should be common to all hydrates with the same structure. One way to test this transferability is to calculate properties of the empty lattice based on parameters derived from different hydrates, and again the cell theory performs well for simple hydrates. For example, calculated vapour pressures for the hypothetical empty hydrate based on CH4, C2H6, and H2S type... 

The lattice of the host in the form it takes in the clathrate is usually thermodynamically unstable by itself—that is, with the holes empty. It is stabilized by inclusion of the guest molecules, and it is of obvious interest in connection with the nonstoichiometry of clathrates to consider the extent to which the cavities in the host lattice must be filled before the system achieves thermodynamic stability. The cavities in the host lattice may all be identical in size and environment, as in the hydroquinone clathrates, or they may be of more than one kind. The gas hydrates, for example, have two possible structures, in each of which there are two sorts of cavity, van der Waals and Platteeuw (15) have developed a general statistical theory of clathrates containing more than one type of cavity. 

Hydration allows water-soluble chemicals to dissolve in water For example, a crystal of table salt (NaCl) is held together by strong ionic interactions. However, when NaCl is dissolved in water, the Na and Cl ions become independent hydrated entities. The energy produced by hydration of the Na and Cl ions more than balances the energy required to remove them from the NaCl crystal lattice. In the Na ion, a lone pair of electrons from a water oxygen atom fills an empty... 

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... 

Hydrates in this class contain water in lattice channels, where the water molecules included lie next to other water molecules of adjoining unit cells along an axis of the lattice, forming channels through the crystal. The empty channels are actually a conceptual construct, since a corresponding low-density structure with empty channels would not be... 

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