Volume dependence

The second important parameter is the chromatographic peak s width at the baseline, w. As shown in Figure 12.7, baseline width is determined by the intersection with the baseline of tangent lines drawn through the inflection points on either side of the chromatographic peak. Baseline width is measured in units of time or volume, depending on whether the retention time or retention volume is of interest. 

Our primary interest in the Flory-Krigbaum theory is in the conclusion that the second virial coefficient and the excluded volume depend on solvent-solute interactions and not exclusively on the size of the polymer molecule itself. It is entirely reasonable that this should be the case in light of the discussion in Sec. 1.11 on the expansion or contraction of the coil depending on the solvent. The present discussion incorporates these ideas into a consideration of solution nonideality. 

In terms of the derived general relationships (3-1) and (3-2), x, y, and h are independent variables—cost and volume, dependent variables. That is, the cost and volume become fixed with the specification of dimensions. However, corresponding to the given restriedion of the problem, relative to volume, the function g(x, y, z) =xyh becomes a constraint funedion. In place of three independent and two dependent variables the problem reduces to two independent (volume has been constrained) and two dependent as in functions (3-3) and (3-4). Further, the requirement of minimum cost reduces the problem to three dependent variables x, y, h) and no degrees of freedom, that is, freedom of independent selection. 

Both common and systematic names of compounds are used throughout this volume, depending on which the Editor-in-Chief felt was more appropriate The Chemical Abstracts indexing name for each title compound, if it differs from the title name, is given as a subtitle Systematic Chemical Abstracts nomenclature, used in both the 9th and 10th Collective Indexes for the title compound and a selection of other compounds mentioned in the procedure, is provided in an appendix at the end of each preparation. Registry numbers, which are useful in computer searching and identification, are also provided in these appendixes. Whenever two names are concurrently in use and one name is the correct Chemical Abstracts name, that name is adopted. For example, both diethyl ether and ethyl ether are normally used. Since ethyl ether is the established Chemical Abstracts name for the 8lh Collective Index, it has been used in this volume The 9th Collective Index name is 1,1 -oxybisethane, which the Editors consider too cumbersome. 

Here we assume a volume dependence of the Gruneisen parameter... 

Although there have been few data collected, postshock temperatures are very sensitive to the models which specify y and its volume dependence, in the case of the Gruneisen equation of state (Boslough, 1988 Raikes and Ahrens, 1979a Raikes and Ahrens, 1979b). In contrast, the absolute values of shock temperatures are sensitive to the phase transition energy Ejp of Eq. (4.55), whereas the slope of the versus pressure curve is sensitive to the specific heat (see, e.g.. Fig. 4.28). 

Figure 8.1. The cold compression-tension behavior of condensed matter. The volume dependence of energy and pressure or tension are illustrated. The cohesive energy and maximum tension (theoretical spall strength) are properties of the material.

So much for the stress dependence of P. But what of its volume dependence We have already seen that the probability of one sample surviving a stress <7 is Ps(Vq). The probability that a batch of n such samples all survive the stress is just P fl/g) . If these n samples were stuck together to give a single sample of volume 1/ = nVo then its survival probability would still be (P fl/o) . So... 

Since the volume depends on conversion or time in a constant pressure batch reactor, consider the mole balance in relation to the fractional conversion X. From the stoichiometry. 

The intraporous volumes and solid matrix volumes can be expressed as a percentage of the total bed volumes and should be equivalent to 50-55 and 15-18%, respectively. The solid matrix volume depends on the packing density and can be taken as a qualitative number for the control of the reproducibility of repeated packing procedures. 

The only molecular parameter which enters is the total molecular mass M. The volume depends on the number of particles. It is customary to work on a molar scale, in which case V is the volume of one mole of (ideal) gas. 

Figure 1. Volume dependence of the total en gy of rutile (R), anatase (A), brookite (B) and columbite (C) phases. Experimental values of the unit cell volume at ambient conditions are shown with arrows in the following (Bder C-R-B-A (experimental scatter for anatase is illustrated with a box).

These fens usually operate with no piping or duct work on either side and move air or gas from one large open area to another. Pressures are usually very low, and volumes depend on size, blade pitch, the number of blades, and speed. 

The control of these and any other parameters is most usually done in fermenter vessels specifically designed for the purpose and accommodating various working volumes, depending on the yield and production requirements. Laboratory-scale vessels could have a capacity of just 10 litres or less whereas clinical trials and production vessels may be as large as several thousand litres. 

Hie number of Compton scatters occurring in a given volume depends on the number of electrons present and is relatively independent of incident 7-energy. For the lower atomic number elements (excluding hydrogen), the number of electrons present is directly proportional to atomic wt. Thus Compton scattering on a per unit volume basis is a function of density and is independent of chem compn. The density of soils is widely variable and the density of expls falls within the normal range of soil... 

In the past, it has been customary to assume that partial molar volumes depend only on temperature and are independent of composition and pressure (Cl, P13). This assumption is very poor in the critical region. Primarily... 

The expressions in Eq. 1 and Eq. 6 are two different definitions of entropy. The first was established by considerations of the behavior of bulk matter and the second by statistical analysis of molecular behavior. To verify that the two definitions are essentially the same we need to show that the entropy changes predicted by Eq. 6 are the same as those deduced from Eq. 1. To do so, we will show that the Boltzmann formula predicts the correct form of the volume dependence of the entropy of an ideal gas (Eq. 3a). More detailed calculations show that the two definitions are consistent with each other in every respect. In the process of developing these ideas, we shall also deepen our understanding of what we mean by disorder. ... 

In Figures 24.7 through 24.9 are shown the volume dependences of the local and global relaxation times for 1,4-polyisoprene [90], polypropylene glycol [91], and polyoxybutylene [76]. For either mode, volume does not uniquely dehne the relaxation times, as the curves for different... 

Perhaps the best rheological stability criterion is the volume occupied by the sediment per unit volume of original dispersion. A flocculated dispersion settles rapidly to a high sediment volume, while a deflocculated one settles slowly to a low volume, as shown in Figure 8. Of course, the ultimate volume depends on the concentration of the dispersion and in order to give a criterion of more fundamental interest the concept of relative sediment volume RSV may... 

The spatial resolution in quantitative analysis is defined by how large a particle must be to obtain the required analytical accuracy, and this depends upon the spatial distribution of X-ray production in the analysed region. The volume under the incident electron beam which emits characteristic X-rays for analysis is known as the interaction volume. The shape of the interaction volume depends on the energy of the incident electrons and the atomic number of the specimen, it is roughly spherical, as shown in Figure 5.7, with the lateral spread of the electron beam increasing with the depth of penetration. 

When the electron beam enters the sample, it penetrates a small volume, typically about one cubic micron (10-18m3 ). X-rays are emitted from most of this volume, but Auger signals arise from much smaller volumes, down to about 3 x 10 25m3. The Auger analytical volume depends on the beam diameter and on the escape depth of the Auger electrons. The mean free paths of the electrons depend on their energies and on the sample material, with values up to 25 nm under practical analytical conditions. 

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