Roughness equivalent

In 1972 Wegner [25] derived a power-series expansion for the free energy of a spin system represented by a Flamiltonian roughly equivalent to the scaled equation (A2.5.28). and from this he obtained power-series expansions of various themiodynamic quantities around the critical point. For example the compressibility... 

Correlation can be added as a perturbation from the Hartree-Fock wave function. This is called Moller-Plesset perturbation theory. In mapping the HF wave function onto a perturbation theory formulation, HF becomes a hrst-order perturbation. Thus, a minimal amount of correlation is added by using the second-order MP2 method. Third-order (MP3) and fourth-order (MP4) calculations are also common. The accuracy of an MP4 calculation is roughly equivalent to the accuracy of a CISD calculation. MP5 and higher calculations are seldom done due to the high computational cost (A time complexity or worse). 

In providing an isolated molecule description of reactivity, qualitative resonance theory is roughly equivalent to that given above, but is less flexible in neglecting the inductive effect and polarisability. It is most commonly used now as a qualitative transition state theory, taking the... 

The superoxides compare well with a combination of Hquid oxygen or gaseous oxygen systems and a CO2 scmbber, eg, LiOH. Heat release is somewhat less than the combination system, and the volume of superoxides which is required is roughly equivalent to that of lithium hydroxide. 

The price of aniline is also dependent on the cost of benzene, the raw material for nitrobenzene. During the decade of the 1980s, benzene prices ranged from a low of 0.185/L to a high of 0.595/L. A 0.01/L change in price of benzene is roughly equivalent to a 0.01/kg change in the cost of aniline. At times aniline prices are stated on an "ex-benzene" basis to eliminate the effect of volatility in benzene prices on the price of the aniline sold. 

Low-PressureAlulticomponent Mixtures These methods are outlined in Table 5-17. Stefan-MaxweU equations were discussed earlier. Smith-Taylor compared various methods for predicting multi-component diffusion rates and found that Eq. (5-204) was superior among the effective diffusivity approaches, though none is very good. They so found that hnearized and exact solutions are roughly equivalent and accurate. 

Example 1 Calculation of FUG Method A large hutane-pentane splitter is to he shut down for repairs. Some of its feed will he diverted temporarily to an avadahle smaller column, which has only 11 trays plus a partial rehoiler. The feed enters on the middle tray. Past experience on similar feeds indicates that the 11 trays plus the rehoiler are roughly equivalent to 10 equilih-rium stages and that the column has a maximum top-vapor capacity of 1.75 times the feed rate on a mole basis. The column will operate at a condenser pressure of 827.4 kPa (120 psia). The feed will he at its huhhle point (q = 1.0) at the feed-tray conditions and has the following composition on the basis of... 

The concept of ion pairs in nucleophilic substitution is now generally accepted. Presumably, the barriers separating the intimate, solvent-separated, and dissociated ion pairs are quite small. The potential energy diagram in Fig. 5.4 depicts the three ion-pair species as being roughly equivalent in energy and separated by small barriers. 

Most investigators conclude that BS is roughly equivalent to PM,o. However, there... 

Experimental pKa data suggests that simple alkyl groups all affeet acid-base reactivity in roughly the same way. What is more, this universal alkyl effect is roughly equivalent to the effect of a hydrogen atom. For example, the difference in pKa between water and ethanol is approximately the same as that between formic acid and propanoic acid (see table at right). 

Dezocine (30) represents a class of bridged aminotetralins possessing morphine-like analgesic properties. It appears to be roughly equivalent in potency and addiction potential to morphine. The molecule combines molecular features of precedent aminotetralins and benzomor-phans and its structure fits the classical Morphine Rule. The 1-enantiomer is the more active and the p-epimer (equatorial NHj) is the active diastereomer. 

The diffusion coefficients of the constituent ions in ionic liquids have most commonly been measured either by electrochemical or by NMR methods. These two methods in fact measure slightly different diffusional properties. The electrochemical methods measure the diffusion coefficient of an ion in the presence of a concentration gradient (Pick diffusion) [59], while the NMR methods measure the diffusion coefficient of an ion in the absence of any concentration gradients (self-diffusion) [60]. Fortunately, under most circumstances these two types of diffusion coefficients are roughly equivalent. 

It is well established that GGA gives a better description of molecular systems, crystal surfaces and surface-molecule interactions. However, there are cases where the GGA results for solids are in much worse agreement with experiment than the LDA ones (e.g., 3-22 jj has been suggested that the effect of using GGA for solids is roughly equivalent to adding uniform tensile stress, and as a result lattice constants are frequently overestimated. 

An alternative cost also became apparent from the case studies. If the mean hourly steaming rate in kilograms is multiplied by 2.25, it produces a figure roughly equivalent to the annual operating costs in pounds, excluding the cost of fuel. 

Ocean sea water is roughly equivalent in strength to a 3 j % w/v solution of sodium chloride, but it has a much more complex composition, embodying a number of major constituents, and traces at least of almost all naturally occurring elements. For convenience, however, the concentration of salts in any sample of sea water is expressed in terms of the chloride content, either as chlorinity or as salinity. Both these units are again subject to arbitrary definition and do not conform simply to the chemical composition. 

There are three standards controlling surface finish in common use. They are issued by the Steel Structures Painting Council (USA), the Swedish Standards Organisation and the British Standards Institution. They are roughly equivalent. 

Petrin. The earliest reference to its prepn and use appears to be in German patents (Ref 1). Ref 15 gives its heat of explosion as 1204cal/g, and its impact sensitivity as 5 to 10 inches on the PicArsn impact machine (or roughly equivalent to Tetryl)... 

At the highest temps, water will be mostly in the vapor state (eg, at 185° the vapor pressure of water is about 11 atm while the estimated pressure in the bomb of the gaseous products for 50% decompn is about 6 aim) while the PETN is still liq. Thus the hydrolysis reaction does not take place unless there is appreciable soln of water vapor in the liq PETN, At 163° the vapor pressure of water is roughly equivalent to the total pressure in the bomb at 50% decompn. Thus, for the hydrolysis reaction to be significant at lower temps, but not at higher temps, there must be appreciable solubility of water vapor in liq PETN in the 160—70° temp range, and at lower temps, but not at temps above 185°... 

On neutralization an active is obtained with approximately the mole ratio composition 80 20 FAMES/FAS. Sodium methyl sulfate (CH30S03Na) is also produced in a molar quantity roughly equivalent to the FAS. 

A roughly equivalent valence-bond theory would result from allowing the 2s electron of each lithium atom to be involved in the formation of a covalent bond with one of the neighbouring atoms. The wave function for the crystal would be... 

Figure 8. (Continued). As described above, the packing of myosin molecules into the thick filament is such that a layer of heads is seen every 14.3 nm, and this reflection is thought to derive from this packing. Off the meridian the 42.9 nm myosin based layer line is shown. This arises from the helical pitch of the thick filament, due to the way in which the myosin molecules pack into the filament. The helical pitch is 42.9 nm. c) Meridional reflections from actin. Actin based layer lines can be seen at 35.5 nm, 5.9 nm and 5.1 nm (1st, 6th, and 7th layer lines)and they all arise from the various helical repeats along the thin filament. Only the 35.5 nm layer line is shown here.The 5.9 nm and 5.1 nm layer lines arise from the monomeric repeat. The 35.5 nm layer line arises from the long pitch helical repeat and is roughly equivalent to seven actin monomers. A meridional spot at 2.8 nm can also be seen, d) The equatorial reflections, 1,0 and 1,1 which arise from the spacings between crystal planes seen in cross section of muscle.

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