VERY LARGE

The method proposed in this monograph has a firm thermodynamic basis. For vapo/-liquid equilibria, the method may be used at low or moderate pressures commonly encountered in separation operations since vapor-phase nonidealities are taken into account. For liquid-liquid equilibria the effect of pressure is usually not important unless the pressure is very large or unless conditions are near the vapor-liquid critical region. 

Large confidence regions are obtained for the parameters because of the random error in the data. For a "correct" model, the regions become vanishingly small as the random error becomes very small or as the number of experimental measurements becomes very large. 

The secondary reactions are series with respect to the chloromethane but parallel with respect to chlorine. A very large excess of methane (mole ratio of methane to chlorine on the order of 10 1) is used to suppress selectivity losses. The excess of methane has two effects. First, because it is only involved in the primary reaction, it encourages the primary reaction. Second, by diluting the product, chloromethane, it discourages the secondary reactions, which prefer a high concentration of chloromethane. 

If F is very large (Douglas has suggested a value greater than 10) in Eq. (4.8), then... 

Dark oil of camphor consists very largely of safrole. 

In the 1970 s, heavy fuel came mainly from atmospheric distillation residue. Nowadays a very large proportion of this product is vacuum distilled and the distillate obtained is fed to conversion units such as catalytic cracking, visbreaking and cokers. These produce lighter products —gas and gasoline— but also very heavy components, that are viscous and have high contaminant levels, that are subsequently incorporated in the fuels. 

Owing to the large number of types of industrial lubricants, the number of constraints, and therefore the number of desired properties, is very large. The main industrial oils are summarized in Tables 6.4 and 6.5, the first giving the constraints common to all applications, and the second addressing the more specific requirements. A few essential properties appear from these tables ... 

The development of a line of lubricant additives is an expensive and slow undertaking the market for these products —on the order of 10 billion dollars in 1992— is very large and is dominated by a few companies. 

For the above reasons, gas Is typically economic to develop only if it can be used locally, i.e. if a local demand exists. The exception to this is where a sufficient quantity of gas exists to provide the economy of scale to make transportation of gas or liquefied gas attractive. As a guide, approximately 10 Tcf of recoverable gas would be required to justify building a liquefied natural gas (LNG) plant. Globally there are few such plants, but an example would be the LNG plant in Malaysia which liquefies gas and transports it by refrigerated tanker to Japan. The investment capital required for an LNG plant Is very large typically in the order of 10 billion. 

Where the distance to the customer is very large, or where a gas pipeline would have to cross too many countries, gas may be shipped as a liquid. Gas has to be chilled to -160°C in a LNG plant to keep it in liquid form, and is shipped in refrigerated tankers. To condition the gas for liquefaction any COj, HjS, water and heavier hydrocarbons must be removed, by the methods already discussed. The choice of how much propane and butane to leave in the LNG depends upon the heating requirements negotiated with the customer. 

The data volume, which can be imported as data block or single slices, can be cutted or rebinned (e.g. if the data set is very large) and interpolated (e.g. interpolating intermediate slices between measured CT cuts in the case of 2D-CT). 

Computed Tomography for very large and heavy objects is a demanding challenge for the manufacturer and his subsuppliers in many respects, especially with regard to the strongest advantages of CT ... 

The case of very large drops or bubbles is easy because only one radius of curvature (that in the plane of the drawings) is considered. Equation 11-12 then becomes... 

For very large lenses, a limiting thickness is reached. Langmuir [59] gave the equation... 

Above 81.5 K the C(2x 1) structure becomes the more stable. Two important points are, first, that a change from one surface structure to another can occur without any bulk phase change being required and, second, that the energy difference between dtemative surface structures may not be very large, and the free energy difference can be quite temperature-dependent. 

There is always some degree of adsorption of a gas or vapor at the solid-gas interface for vapors at pressures approaching the saturation pressure, the amount of adsorption can be quite large and may approach or exceed the point of monolayer formation. This type of adsorption, that of vapors near their saturation pressure, is called physical adsorption-, the forces responsible for it are similar in nature to those acting in condensation processes in general and may be somewhat loosely termed van der Waals forces, discussed in Chapter VII. The very large volume of literature associated with this subject is covered in some detail in Chapter XVII. 

For very clean metal surfaces, m should approach unity, and /t becomes very large, as observed with even a small decrease in m, y, falls to about unity, or to the type of value found for practically clean surfaces. And if a boundary film is present, making m < 0.2, Eq. XII-11 reduces to... 

The rate of physical adsorption may be determined by the gas kinetic surface collision frequency as modified by the variation of sticking probability with surface coverage—as in the kinetic derivation of the Langmuir equation (Section XVII-3A)—and should then be very large unless the gas pressure is small. Alternatively, the rate may be governed by boundary layer diffusion, a slower process in general. Such aspects are mentioned in Ref. 146. 

Periodic boundary conditions force k to be a discrete variable with allowed values occurring at intervals of lull. For very large systems, one can describe the system as continuous in the limit of i qo. Electron states can be defined by a density of states defmed as follows ... 

There are many large molecules whose mteractions we have little hope of detemiining in detail. In these cases we turn to models based on simple mathematical representations of the interaction potential with empirically detemiined parameters. Even for smaller molecules where a detailed interaction potential has been obtained by an ab initio calculation or by a numerical inversion of experimental data, it is usefid to fit the calculated points to a functional fomi which then serves as a computationally inexpensive interpolation and extrapolation tool for use in fiirtlier work such as molecular simulation studies or predictive scattering computations. There are a very large number of such models in use, and only a small sample is considered here. The most frequently used simple spherical models are described in section Al.5.5.1 and some of the more common elaborate models are discussed in section A 1.5.5.2. section Al.5.5.3 and section Al.5.5.4. 

A statistical ensemble can be viewed as a description of how an experiment is repeated. In order to describe a macroscopic system in equilibrium, its thennodynamic state needs to be specified first. From this, one can infer the macroscopic constraints on the system, i.e. which macroscopic (thennodynamic) quantities are held fixed. One can also deduce, from this, what are the corresponding microscopic variables which will be constants of motion. A macroscopic system held in a specific thennodynamic equilibrium state is typically consistent with a very large number (classically infinite) of microstates. Each of the repeated experimental measurements on such a system, under ideal... 

If //is 00 (very large) or T is zero, tire system is in the lowest possible and a non-degenerate energy state and U = -N xH. If eitiier // or (3 is zero, then U= 0, corresponding to an equal number of spins up and down. There is a synnnetry between the positive and negative values of Pp//, but negative p values do not correspond to thennodynamic equilibrium states. The heat capacity is... 

N is very large since the fluctuations around the average behave as 2. A quantum ideal gas with either Fenni or Bose statistics is treated in subsection A2.2.5.4. subsection A2.2.5.5. subsection A2.2.5.6 and subsection A2.2.5.7. 

These results do not agree with experimental results. At room temperature, while the translational motion of diatomic molecules may be treated classically, the rotation and vibration have quantum attributes. In addition, quantum mechanically one should also consider the electronic degrees of freedom. However, typical electronic excitation energies are very large compared to k T (they are of the order of a few electronvolts, and 1 eV corresponds to 10 000 K). Such internal degrees of freedom are considered frozen, and an electronic cloud in a diatomic molecule is assumed to be in its ground state f with degeneracy g. The two nuclei A and... 

Since lis zero when f= 0, the ground state does not contribute to the integral forA. At sufficiently low temperatures, will be very large compared to one, which implies z is very close to one. Then one can approximate z by one in the integrand for N. Then the integral can be evaluated by using the transfonnation v = pe and the known value of the integral... 

This ensures the correct connection between the one-dimensional Kramers model in the regime of large friction and multidimensional imimolecular rate theory in that of low friction, where Kramers model is known to be incorrect as it is restricted to the energy diflfiision limit. For low damping, equation (A3.6.29) reduces to the Lindemann-Flinshelwood expression, while in the case of very large damping, it attains the Smoluchowski limit... 

A very extreme version of surface corrugation has been found in the nonactivated dissociation reactions of Fl2 on W [, ], Pd and Rli systems. In these cases, the very strong chemisorption bond of the FI atoms gives rise to a very large energy release when the molecule dissociates. In consequence, at certain sites on the surface, the molecule accelerates rapidly downliill into the dissociation state. At the unfavourable sites, there... 

---