Method points

instead of considering shell elements, we sample the spherical space at the spherical surfaces in Fig. 3.8 at r, r2. .. We then 

This is again seen to be almost identical with the box result (Eq. 3.35) which has, in effect, a backward difference for 3c/3r. As suggested previously (Britz 1980), the above comparison provides rather strong argument in favour of the point method - it is faster to use and leaves no doubts. The argument will be further strengthened when we come to convection tetms (Chapt. 8). 

The corresponding cylindrical development starts with the cylindrical diffusion equation (Eq, 2.7 the 2/r term in Eq. 3.42 is replaced with 1/r). Similar development to the above easily leads to the discrete form 

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Evaporation (NF M 41-012) < rC Final boiling point (so-called 95% point method) < - 15°C ... 

The existence of this situation (for nonporous solids) explains why the ratio test discussed above and exemplified by the data in Table XVII-3 works so well. Essentially, any isotherm fitting data in the multilayer region must contain a parameter that will be found to be proportional to surface area. In fact, this observation explains the success of Ae point B method (as in Fig. XVII-7) and other single-point methods, since for any P/P value in the characteristic isotherm region, the measured n is related to the surface area of the solid by a proportionality constant that is independent of the nature of the solid. 

Molecular Weight Determinatioos. Details of the determination of molecular weights on a semi-micro scale by the Freezing-point method are given on p. 436, and by the Boiling-point method on p. 440. 

The method is used to determine the molecular weight primarily of those compounds which are too slightly soluble in the usual cold solvents for the more efficient freezing-point method to be nployed. 

The same considerations with regard to association in solution apply here as in the freezing-point method (p. 435). 

Table II, 9, A.—Reference Substances for the Calibration of Thermometers BY THE Melting Point Method...

Assays using equiUbrium (end point) methods are easy to do but the time requited to reach the end point must be considered. Substrate(s) to be measured reacts with co-enzyme or co-reactant (C) to produce products (P and Q) in an enzyme-catalyzed reaction. The greater the consumption of S, the more accurate the results. The consumption of S depends on the initial concentration of C relative to S and the equiUbrium constant of the reaction. A change in absorbance is usually monitored. Changes in pH and temperature may alter the equiUbrium constant but no serious errors are introduced unless the equihbrium constant is small. In order to complete an assay in a reasonable time, for example several minutes, the amount and therefore the cost of the enzyme and co-factor maybe relatively high. Sophisticated equipment is not requited, however. 

The general vaUdity of single-point methods has been questioned, however (28). An even simpler but equally useful method is to approximate the limiting... 

Ring-and-ball method (ASTM D36-26). In the United States, two other softening point methods ate employed cube-in-air (35) and cube-in-water (36). Cube-in-air softening point = R-and-B softening point +4 C. Cube-in-water softening point = R-and-B softening point +10 C. 

Dew-Point Method For many applications, the dew point is the desired moisture measurement. VHien concentration is desired, the relation between water content and dew point is well-known and available. The dew-point method requires an inert surface whose temperature can be adjusted and measured, a sample gas stream flowing past the surface, a manipulated variable for adjusting the surface temperature to the dew point, and a means of detecting the onset of con-densation. 

Dew-Point Method The dew point of wet air is measured directly by observing the temperature at which moisture begins to form on an artificially cooled polished surface. The polished surface is usually cooled by evaporation of a low-boihng solvent such as ether, by vaporization of a condensed permanent gas such as carbon dioxide or liquid air, or by a temperature-regulated stream of water. 

Although the dew-point method may be considered a fundamental technique for determining humidity several uncertainties occur in its use. It is not always possible to measure precisely the temperature of the polished surface or to eliminate gradients across the surface. It is also difficult to detect the appearance or disappearance of fog the usual practice is to take the dew point as the average of the temperatures when fog first appears on cooling and disappears on heating. 

BP Method for Distillation The bubble-point method for distillation, particularly when the components involved cover a relatively narrow range of volatihty, proceeds iteratively by the following steps, where k is the iteration index for the entire distillation column. 

Mettler softening point method — ASTM D 3461-76. This is the most recent method. This automatic method measures the temperature at which the resin flows out of a sample cup under its own weight. The temperature is recorded when the first drop crosses the light path of a photocell (Fig. 21). This method is quite accurate and reproducible. 

Steveirs, R. K., R. J. Drago, and Y. Mamane. A Long Path Differential Optical Absorpnon Speerrometer and EPA-Approved Fixed-Point Methods Inrercomparison. Atmospheric Enro vanincnt 27B (1993), pp. 231-236. 

As in the freezing-point method, the molecular weight is calculated from the weight of substance lequired to raise the boiling-point of too grams of solvent i°, and the result multiplied by a coefficient which depends upon the nature of the solvent. The following is a list of solvents commonly employed and their coefficients and boiling-points —... 

Although the boiling-point method is able to dispose of a greater numbei of convenient solvents than are suitable for freezing-point determinations, it is never so accurate, mainly on account of the difficulty of avoiding fluctuations in the boiling-point, due to radiation, to the dripping of cold liquid from the condenser, to impure solvent, and to barometric fluctuations. 

Boiling-point method, 37 determination of, 58 correction for, 59 / 1 iroinacetanilicle, 152 Bromacetic acid, 8g Broiiiaceiyl bromide, 89 Bromuhenzene, 140 y /-Bromobenz.oic acid, 201 Brouiocresol, 165 / Bromoioluene, 167 Butyric acid, 99... 

Filter-pump, 44 Filtration through cloth, 131 under reduced pressure, 43 with fluted filter, 53 "Fischer s ester method, 133 Fluorescein, 187 Fluted filter, 53 Formic acid, 106 Fractional distillation, 136 Fractionating columns, 137 Freezing-point method, 32 I -riedel-Crafis reaction, 210 furnace, combustion, 4 tube, 23 ... 

Another obvious advantage that the LG method typically enjoys over floating-point methods is one of programming efficiency. Because LG dynamics are a natural transcription of the kinetic processes taking place on the micro-lattice, the actual program code is generally concise and straightforward, and often amounts to little more than a few dozen lines of FORTRAN. 

Finally, we should mention that the stability of a running program itself is never really an issue for LG simulations, unlike that for floating-point methods for which certain (sometimes unforeseen) conditions can lead to serious software crashes. 

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