Probability volume dependence

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... 

Low cross-linked polystyrene resins (1% divinylbenzene) is probably the most popular solid support. These resins swell to 2-6 times their original volume depending on the solvent used. Swollen resin, after removal of solvent and without excessive drying, remains in a rubbery state and can be easily flattened for FTIR study in the transmission mode. The support-bound compound should be washed free of reagent and solvent. 

Much interest remains, however, in the question of the long time limiting density because Fig. 4 indicated that the deviation from the Batchinski equation seen in all laboratory studies at low D becomes greater than the uncertainty in the simulation diffusion coefficients (only) at Z)<5x 10" cm sec. At this diffusivity, relations discussed in Section II. B imply that the time needed for full structural equilibration would exceed 10 psec. The dependence of D on the degree of equilibration is not known at this time, but it seems probable that a major computing effort will be needed to determine whether the volume dependence of the dense atomic liquid diffusivity is as simple and significant as (13) implies or is more complex, as for the known behavior of low-diffusivity laboratory liquids (Fig. 4). 

It appears that the lower the temperature the higher the nu-eleation rate and eonsequently it is very difficult to get the solidification of a pure liquid near its bulk solid-liquid equilibrium temperature. Another less obvious consequence is that the temperature at which a given sample will solidify is not unique. Therefore, only a most probable temperature ean be given and this temperature T appears to be volume dependent and of eourse it also depends on the sample composition as it is shown thereafter. 

Identically this hold to be true for Ti (15 for ZSM-5), Fe (11 for ZSM5, 12 for ZSM-23) Zr (23 for ZSM-5 with u.c. volume varying from 5.345 to 5.380 nn with 0.5 Zr per u.c., etc. However it appears that this does not actually held true for Al in ZSM-5 (Fig. 1 left). This is probably due to the fact that water content was depending in Al content and that u.c. volume depends on water content as it will be described below. This makes that many data in the litterature are questionable when ambient atmosphere has not been well defined as outgassing at a given temperature or under a given water pressure. This holds particularly true since substitution may lead to more or less hydrophilic materials. Moreover when isomorphous substitution corresponds to a maximum of a few percents as for ti or Fe in MFI structure, the u.c. volume changes are within experimental error. 

Fig. 5.12 The volume-dependence of the exact stationary probability distribution function of system / and comparison with its Euler-McLaurin approximation (after Ebeling and Schimansky-Geier).

A definite size effect exists in brittle materials, such as glass, and their strength depends on the volume of the material. This volume-dependence is simply explained by the fact that the probability of finding proper-sized cracks increases with volume. 

Fig. 7.11. Demonstration of the volume dependence of the survival probability. The circled symbols denote the chances for survival and failure of the component, respectively ( survival, failure)...

In writing the above equatiorrs, we have explicitly indicated the volume dependence of the probability density P(P C) and classical Hamiltonian H(p, qiV) to reflect the fact that the system voltrme V is variable. 

As in the case of biphenyl, current worldwide production figures for terphenyls are not readily obtainable, but the volume is probably around 6.8—8.2 million kg/yr. Currently, most of the terphenyl produced is converted to a partially hydrogenated form. U.S. production of terphenyls has remained steady at several thousand metric tons per year over the past decade. The 1991 small lot price for mixed terphenyls was about 3.89/kg whereas the specially fractionated heat-transfer-grade terphenyl—quaterphenyl mixture sold as Therminol 75 heat-transfer fluid was priced around 6.93/kg. Partially hydrogenated mixed terphenyls were priced in the 6.05—7.48/kg range depending on quantity and grade. 

This, then, is our final design equation. It shows how the survival probability depends on both the stress (rand the volume V of the component. In using it, the first step is to fix on an acceptable failure probability, Pp 0.3 for chalk, 10 for the cutting tool, 10 for the vacuum-chamber window. The survival probability is then given by P = 1 -. ... 

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