Solids equations

Knopoff, L., Solids Equations of State of Solids at Moderately High Pressures, in High Pressure Physics and Chemistry, Vol. 1 (edited by Bradley, R.S.), Academic Press, New York, 1963, pp. 227-245. 

Once equation (10.158) has been obtained for relating Cy. m to T for a Debye solid, equations relating (Um - U0 m), (Hm - Uo.m), and Sm to T can be derived. Tables of values, expressed in terms of 9d/T. can be found in Table A4.7, Appendix 4, with more extensive tables found in the literature1 to calculate these thermodynamic properties. 

The diffusion layer model satisfactorily accounts for the dissolution rates of most pharmaceutical solids. Equation (43) has even been used to predict the dissolution rates of drugs in powder form by assuming approximate values of D (e.g., 10 5 cm2/sec), and h (e.g., 50 pm) and by deriving a mean value of A from the mean particle size of the powder [107,108]. However, as the particles dissolve, the wetted surface area, A, decreases in proportion to the 2/3 power of the volume of the powder. With this assumption, integration of Eq. (38) leads to the following relation, known as the Hixon-Crowell [109] cube root law ... 

In the reaction of chlorocarbonylphenylketene 283 and A -phenylcinnamoyl amide 284, the mesoionic 1,3-oxazi-nium derivative 95 was obtained as a metastable, but isolable, red solid (Equation 27) <2005JOC5859>. 

Combining with equation 2 for solids, equation 3 gives equation 7 ... 

Certain special cases are of interest here. When one ion is present at concentrations low enough for loading of the solid phase to be only a small fraction of ion exchange capacity, C, expressed in equivalents/kg of solid. Equation 3 may be written... 

Compared to other members of Group 5 antimony displays a marked preference for polymeric structures over cyclic structures (equations 38 and 39112). Alkyl primary stibines break down at -78 °C to afford brown/black solids (equation 40). 

The van der Waals force exists not only between individual atoms and molecules but also between solids. Equations for the van der Waals force between solids were deduced... 

Equation (3.6) illustrates that the solubility of a solid in a liquid depends on the enthalpy change at Tm and the melting temperature of the solid. Equation (3.6) is a valid one when T > Tm because the liquid solute in an ideal solution is completely miscible in all proportions. Table 3.1 shows the ideal solubilities of compounds and their heat of fusion. Equation (3.6) is the equation for ideal solubility. The relationship of In x2 (ideal or nonideal solubility) vs. 1/T is shown in Figure 3.1. 

Sublimation of the dithiadiazolyl radical 139 in a partial atmosphere of oxygen resulted in the formation of dithiatetrazocine 140, as a yellow solid (Equation 43) <2003CC2774, 2004PS981>. Earlier, it was shown that dithiadiazolyl radicals reacted with dioxygen in MeCN to generate dithiatetrazocines <1996CHEC-II(9)705>. 

A particularly intriguing arylhthium reagent is the bis(etherate) of 2,6-diphenylphenyllithium, (2,6-Pli2C6H3) Li(Et20)2. This reagent, readily prepared by reaction of (2,6-diphenyl)-1-iodobenzene with n-butyllithium, is isolated as a white crystalline solid (equation 6). ... 

Gas and solids equations of state, and a reactor pressure equation, are needed to complete the definition of the dynamic model. Pressure drop as a linear function of coal bed height is used for the gasifier pressure equation and the ideal gas law is used for the gas equation of state. The solids equation of state is expressed in terms of both the bulk and raw densities. These equations are given in Stillman (7, 8). 

However, it is important to note that there is an inherent numerical stiffness in the coupled gas-solids equations because the gas stream moves through the reactor much more rapidly than the solids stream. In a typical example, while it only takes the gas about 7 seconds to move through the reactor, it takes the solids stream about a 1000 times longer. 

The continuity equations for mass and energy were used to derive an adiabatic dynamic plug flow simulation model for a moving bed coal gasifier. The resulting set of hyperbolic partial differential equations represented a split boundary-value problem. The inherent numerical stiffness of the coupled gas-solids equations was handled by removing the time derivative from the gas stream equations. This converted the dynamic model to a set of partial differential equations for the solids stream coupled to a set of ordinary differential equations for the gas stream. 

Methanolysis of the N-Si bond in diphosphirene complex 87d gave a 48% yield of complex 130 as a yellow solid (Equation 22) <2001AGE2471>. 

The speed and efficiency of chain growth relative to termination of the metathesis catalysts provides high molecular weight polymers. Different catalyst mixtures produce different double bond geometries and tacticities. These factors have been studied in detail since one of the first promising polymers produced by ROMP was polypentenamer. The all-trans isomer is an elastomer and was studied as an excellent material for tire manufacture, while the cis isomer is a nonelastomeric solid (equation 11). ... 

Exploration of the chemistry of cyclopropanimines has been carried out by Quast and coworkers during extensive studies on a-bromoketimines. 1,3-Elimination of hydrogen bromide from 14 by excess potassium t-butoxide in THF yields the imines as colorless oils or low melting solids (equation 6). As shown by their H- and C-NMR spectra, these... 

A related zirconium complex with a [N2 ]-type of ligation is tetrakis(l,3-diphenyltriazenido)zirconium(lv) 74,85 which is obtained from the amine elimination approach. Thus, treatment of Zr(NMe2)4 with 4equiv. of 1,3-diphenyltriazene in hexanes at ambient temperature affords 74 in 79% yield as a deep red crystalline solid (Equation (6)). This Zr complex is eight coordinate with four 2-l,3-diphenyltriazenido ligands. Complex 74 crystallizes with a distorted dodecahedral geometry about Zr consistent with this solid-state structure, the solution molecular weight measurements indicate a monomeric structure in benzene. 

D is replaced by DeB for porous solids. Equation (2.369) agrees in terms of its form with (2.53) for heat conduction in a rod. In this case the temperature profile was calculated with the boundary conditions of constant temperature at the beginning of the rod and vanishing heat flow at the end of the rod. In the current problem these correspond to the boundary conditions (2.370) and (2.371). The solution of (2.369) to (2.371) therefore corresponds to the relationship (2.59) found earlier for heat conduction in rods. The solution is... 

The values of H listed in Table 6.1 for the physical hardness of solids raise an interesting question. Should there not be a corresponding number, H for the physical hardness of molecules After all, there are force constants in molecules as well as in solids. Equation (6.13) might serve for a diatomic molecule, if n were simply set equal to one. 

The first arsaethyne results from the reaction of (Me3Si)2AsLi with 2,4,6-(t-Bu)3C6H2COC1 (ArCOCl). The intermediate aroyl derivative decomposes through loss of siloxane to give the arsaethyne 90 as a pale yellow crystalline solid (equation 84)107. 

Both Fade approximant " and the more powerful Tova approximant" predictions of the tails of the virial series are consistent with the location of the first-order pole at the crystal close-packed density, as required by the closure for the virial series. In fact, Baram and Luban" give this as a conclusion of their work. The known virial coefficients in the soft-sphere, inverse twelfth power models also imply that the virial series contains information on the crystalline phase at very high pressure, but is unrelated to the freezing transition, the glass transition, or the amorphous solid equation of state. °... 

The value of R can be calculated from the measured value of the refractive index of the liquid or solid. Equation (26.24) can be combined with Eq. (26.21) to express the dielectric constant at low frequencies ... 

The differential mass balances for the pores (Equation 6.5) and the solid (Equation 6.6) are formulated together with Equations 6.23 and 6.24 as well as Equations 6.30 and 6.31 as... 

Treatment of thiuram disulfides 190 with CuClz leads to a two-electron oxidation of the ligand, reduction of the Cu(ll) to Cu(l), and formation of a chlorocuprate(l) salt. The resulting Cu(l) complexes 191 are diamagnetic red solids (Equation 15) <1995JCR19>. 

Ash et al. (1973) extended the preceding theory to the adsorption from liquid mixtures. It follows very similar lines to the gas/solid situation and so, in the interests of concision, it will not be reproduced here. For a binary mixture of solute 2 in solvent 1, the analogue of the gas/solid equation (17.17) is... 

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