Volume increment

There is an alternative traditional recipe which attempts to describe empirically the crystal chemistry of solid-state materials. In contrast to the preceding ionic-radii concept covered in Section 1.1, implicitly based on hard, nonpenetrating spheres, one might simply take the point of view that the total molar volume V of any solid-state material may be approximated by the sum of individual volume increments which are characteristic for the individual particles and, therefore, will contain information about the atom, ion, or ionic groups and also their valence state thus, we can simply write 

It transpires that the above goal has already been accomplished, partly so, in the early 20th century due to the work of Biltz and coworkers [33]. Based on the available experimental data of pyknometric and X-ray-based densities of inorganic (and also organic ) solid-state materials at that time, a tabulation referring to absolute zero temperature and a standard pressure of 1 atm was proposed it is given in Table 1.2. 

In practice, the derivation of these increments was performed in a similar way to Goldschmidt s derivation of ionic radii, namely starting from the alkali halides and thereby deriving basis values, then extending it to other materials. At least two interesting observations can be made from the latter table which, although very dated (1934), is still in use in many solid-state chemical laboratories. 

anionic increments are larger than cationic increments, and cationic increments may adopt a zero value. The latter finding is typical for cations in their highest oxidation states with zero valence electrons, because these highly charged, tiny cations usually rest in the remaining empty sites of the closely packed anions, with no demand for additional space in other words, the whole volume of the solid is fully described by the anions. 

Second, the spatial requirement of inert electron pairs is easily deduced from the table, namely from the entries of In+/In +, T1+/T1 +, Sn +/Sn +, Pb +/Pb +, and Sb +/Sb +, being of the order of 10 cm /mol, about the size of the hydrogen anion, H. Later studies have shown that the latter anion, a strongly polarizable ion depending on the binding partner, also varies greatly in its incremental volume [34]. 

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Liquid Displacement Gas Meter Provers. The Hquid displacement prover is the most prevalent standard for the caHbration of flow meters at low to moderate gas flow rates. The method consists of displacing a known volume of Hquid with gas (Fig. 2). Gas entering the inverted beU causes it to rise and a volume increment can be timed. Typical prover capacities are 1 m or less although capacities as large as 20 m are available. Accuracies can be on the order of 0.5% of actual flow rate. 

Atomic and Structural Diffusion-Volume Increments, ) (cmVmol)... 

Note that the total static pressure curve of Figure 12-145 is limited by the lowest output pressure of the multifan system. The limit curve is established using the fan curve (No. 1 in this example) having the smallest volume increment to the system resistance curve. In this situation fen No. 2 cannot add to the system until its pressure-volume relation reaches the peak point on its curve. 

The optimum volume increment AV depends upon the magnitude of the slope of the titration curve at the equivalence point and this can easily be estimated from a preliminary titration. In general, the greater the slope at the e.p., the smaller should A V be, but it should also be large enough so that the successive values of AE exhibit a significant difference. 

Table 10.4. Atomic and structural diffusion volume increments (mVkmol)1 ...

The nature of the stationary phase in SEC precludes the elution of sample prior to a volume equivalent to that contained in the interstitial spaces, Vj. This is also known as the excluded volume, V. The remainder of the total system void volume, Vj, is made up of the pore volume, Vp. For an eluting species the volume increment at which it appears, V3, will be related to the interstitial and pore volumes as ... 

Another procedure consists in finding the point on the curve where the first derivative, AE/Aml, or ApH/Aml, is maximal and its second derative, A2E/Aml2 or A2pH/Aml2, equals zero, which can be verified graphically or instrumentally for volume increments (ml) around the end-point sought. 

Table 3.6 Additive volume increments for calculating LeBas molar volume, (Cussler,...

In Tables 1.12 and 1.13, the diffusion volumes of some molecules and the volume increments of atoms are given (Perry and Green, 1984, 1999). 

Table II shows the mathematical forms of the normal, log-normal, Rosin-Rammler, and Nukiyama-Tanasawa distributions for an arbitrary ptb-weighted size distribution. As in Table I, the formulas yield the surface increment in the size interval, t to t + dt, for p = 2 number or volume increments are obtained by setting p equal to 0 or 3, respectively. In these expressions 6 and n are constants, and a denotes an appropriate shape factor.

It follows from this equation that/g decreases as the ratio e/kT = c increases. Taking into account the obvious fact that the volume increment per flexed bond should be larger than that for an unflexed one, the above result means that the free-volume... 

This equation can be written in terms of the volume increment dV = At Al as... 

The column effluent was collected in 0.20 pore volume increments by a Spectra/Chrom fraction collector. The column effluent was categorized as one of the following liberated free phase hexadecane, micellarly solubilized hexadecane, or microemulsified middle phase hexadecane. One column effluent sample contained all three phases, where the micellarly solubilized hexadecane was the most dense bottom phase, free phase hexadecane was the least dense top phase, with the microemulsified middle phase in between. Effluent samples of liberated free phase hexadecane were separated from both micellarly solubilized hexadecane and microemulsified middle phase hexadecane, and subsequently quantified with a graduated vial to within 0.01 ml. Micellarly solubilized hexadecane was separated from microemulsified middle phase hexadecane, and both were diluted with methanol to a concentration below 500 mg/L which is in the linear region of the Varian 3300 GC using split injection and FID detection. 

Potentiometric titration curves normally are represented by a plot of the indicator-electrode potential as a function of volume of titrant, as indicated in Fig. 4.2. However, there are some advantages if the data are plotted as the first derivative of the indicator potential with respect to volume of titrant (or even as the second derivative). Such titration curves also are indicated in Figure 4.2, and illustrate that a more definite endpoint indication is provided by both differential curves than by the integrated form of the titration curve. Furthermore, titration by repetitive constant-volume increments allows the endpoint to be determined without a plot of the titration curve the endpoint coincides with the condition when the differential potentiometric response per volume increment is a maximum. Likewise, the endpoint can be determined by using the second derivative the latter has distinct advantages in that there is some indication of the approach of the endpoint as the second derivative approaches a positive maximum just prior to the equivalence point before passing through zero. Such a second-derivative response is particularly attractive for automated titration systems that stop at the equivalence point. 

Parameter N is dimensionless it is shown above as the ratio of two times, each squared. Because of this, the ratio tjr (or fr/r1/2) can be replaced by the ratio of two proportionate distances (the corresponding chart distances for the peak as it unfolds on the chart paper of a recorder) or two proportionate volumes (elution volume and the volume increment observed during the emergence of one standard deviation of the peak). 

Additive-Volume Increments for Atoms and Functional Groups Using Le Bas s Approach... 

TABLE 4.10 Recommended values for molar volume increments at 298 K... 

In most cases the injected concentration is small and A2 can be neglected. Thus, if the optical properties (nQ and Ant Ac) of the polymer solution are known, the molar mass at each elution volume increment can be determined. 

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