True densities, definition

Although, the true density of solid phase p=m/Vp (e.g., g/cm3) is defined by an atomic-molecular structure (/ ), it has become fundamental to the definition of many texture parameters. In the case of porous solids, the volume of solid phase Vp is equal to the volume of all nonporous components (particles, fibers, etc.) of a PS. That is, Vp excludes all pores that may be present in the particles and the interparticular space. The PS shown in Figure 9.17a is formed from nonporous particles that form porous aggregates, which, in turn, form a macroscopic granule of a catalyst. In this case, the volume Vp is equal to the total volume of all nonporous primary particles, and the free volume between and inside the aggregates (secondary particles) is not included. 

Thus, there are two limitations of the pycnometric technique mentioned possible adsorption of guest molecules and a molecular sieving effect. It is noteworthy that some PSs, e.g., with a core-shell structure, can include some void volume that can be inaccessible to the guest molecules. In this case, the measured excluded volume will be the sum of the true volume of the solid phase and the volume of inaccessible pores. One should not absolutely equalize the true density and the density measured by a pycnometric technique (the pycnometric density) because of the three factors mentioned earlier. Conventionally, presenting the results of measurements one should define the conditions of a pycnometric experiment (at least the type of guest and temperature). For example, the definition p shows that the density was measured at 298 K using helium as a probe gas. Unfortunately, use of He as a pycnometric fluid is not a panacea since adsorption of He cannot be absolutely excluded by some PSs (e.g., carbons) even at 293 K (see van der Plas in Ref. [2]). Nevertheless, in most practically important cases the values of the true and pycnometric densities are very close [2,7],... 

For porous solids such as coal, there are five different density measurements true density, apparent density, particle density, bulk density, and in-place density. The true density of coal is the mass divided by the volume occupied by the actual, pore-free solid in coal. However, determining mass of coal may be deemed as being rather straightforward, but determining volume presents some difficulties. Volume, as the word pertains to a solid, cannot be expressed universally in a simple definition. Indeed, the method used to determine volume experimentally, and subsequently, the density, must be one that applies measurement rules consistent with the adopted definition. 

Based on the definition of density, two new terms are defined. Porosity is defined as the proportion of a powder bed or compact that is occupied by pores and is a measure of the packing efficiency of a powder and relative density is the ratio of the measured bulk density and the true density ... 

The bulk density of a powder is obtained by dividing its mass by the bulk volume it occupies. The volume includes the spaces between particles as well as the envelope volumes of the particles themselves. The true density of a material (i.e., the density of the actual solid material) can be obtained with a gas pycnometer. The bulk density of a powder is not a definite number like true density or specific gravity but an indirect measurement of a number of factors, including particle size and size distribution, particle shape, true density, and especially the method of measurement. Although there is no direct linear relationship between the flowability of a powder and its bulk density, the latter is extremely important in determining the capacity of mixers and hoppers and providing an easily obtained valuable characterization of powders. 

Some important critical remarks to the very definitions of these terms, as well as to their experimental measurements, will be given in Chapter 7, Section 5.1.) The true density of macroporous polymers is usually estimated by helium or nitrogen densitometry, while the apparent density can be determined by measuring the diameters of a sufficient number of spherical beads and weighing them. 

The first tcibles in this book are for the properties of satmated carbon dioxide. Thus the pressures given in these tables are the vapor pressme of pure CO and they end at the critical point One thing that looks imusual is that the heat capacity, C, is infinite at the criticcd point. However, this is true by definition. Subsequent tables cue for the density, enthalpy, entropy and heat capacity for vapor, hquid md supercritical regions. 

This I(S) is always defined when there is at least one distribution for which. S is true but it need not be finite. Thus, if the domain of definition of. S is the set of probability densities p(x) on the whole x axis, a trivial 5 (i.e., true for every p(x)) and also an S which merely gives the value of the first moment, has I(S) = — oo. On the other hand, if S states that the second moment is close to zero, /( S ) is very large, and I(S) -> + oo as this moment approaches zero. Of course there is no p(x) having a zero second moment (only a point distribution, which is not a p(x)). Thus it might seem natural to define I(S) as + oo when S defines an empty set. Then every S without exception has a unique I(S). 

Let us recall that the LRC potential, which is fitted on a panel of atomic IP, leads to good IP for most (if not any) atomic IPs. This is unfortunately not true anymore for the molecular systems for which a shift between experimental IPs and the eigenvalues spectrum remains [76]. This shift is definitely smaller than those of the LDA or other GGAs, but it is significantly larger than for atoms. Indeed, it corresponds to an overcorrection with respect to the LDA, and this could be related to a proportionnally smaller importance of low density domains in molecules than in atoms. This overcorrection was also noticed by Casida with the VLB potential [77]. 

The real key, though, is the definition of the first term of Eq. (1.47). Kohn and Sham defined it as the kinetic energy of noninteracting electrons whose density is the same as the density of the real electrons, the true interacting electrons. The last term is called the exchange-correlation functional, and is a catchaU term to account for all other aspects of the true system. 

---