Logarithmic relationships

A broad range of solvents can be used in solvent-borne BR and PIB adhesives. Hydrocarbon (hexane, heptane, naphtha) and chlorinated solvents (perchloroethy-lene) can be used they provide higher viscosities. The presence of small amounts of stabilizer may cause cloudy solutions and settling is rarely produced. There is a logarithmic relationship between viscosity and solids content. In fact, a small... 

The oxidation of aluminium at room temperature is reported to conform to an inverse logarithmic equation for growth periods up to 5 years duration. At elevated temperatures, oxidation studies over shorter periods illustrate conformity to parabolic, linear and logarithmic relationships according to time and temperature. These kinetic variations are attributed to different mechanisms of film formation . ... 

At 180-290°C it was found that the parabolic law first applied but subsequently changed to a logarithmic relationship of the type... 

Owing to the logarithmic relationship between E and AAG, a small increase in AAG produces a dramatic change in E. For instance, when AAG is increased by only 1 kcal mol approximately, the enantiomeric excess of the product is enhanced from 80 to 95% [2]. 

The kinetic method, developed by Cooks et al., is commonly nsed to measnre acidities for systems that readily form proton-bound clnster ions. For example, acidities are determined from the branching ratios for dissociation of proton-bonnd cluster ions of anions and reference. In the simplest form, there is assnmed to be a semi-logarithmic relationship between the CID branching ratio, r, and the difference in the free energies for the two processes (Eq. 5.2). 

The logarithmic relationship above indicates that if we halve the length of a PCB trace, we can make its inductance halve, too. But we have to increase its width... 

Equations (65) and (66) show that a linear logarithmic relationship involving the effect of a change in chain length on the kinetic EM and the equilibrium EM of the same cyclisation reaction is impossible, unless 0AH° is a constant throughout the series. 

There is a strong limitation in the concentration range due to the logarithmic relationship between transmission and concentration (optical densities reasonably to measure range from 0.1 to 1.5). Nevertheless, protein quantification by direct UV-measurement or after staining with dyes in the visible range is a very robust method and can be found, e.g., as a common detection mode in HPLC or other chromatographic techniques. 

Pullman GS, DeVay JE, Garber RH (1981a) Soil solarization and thermal death a logarithmic relationship between time and temperature for four soilbome plant pathogens. Phytopathology... 

Pauling has found empirically that there was a simple logarithmic relationship between the bond length R and the bond order 0, namely ... 

Supernatant BOD5 and TOA concentrations of piggery slurry provide useful indicators of potential odour nuisance. They have very highly significant logarithmic relationships with the odour offensiveness as assessed by a panel of people. Specific odorants including VFA, indoles and phenols are shown to be significantly less reliable. 

We recall that both of the currents, i.e. lanodk and Icathodic (see equations (7.14) and (7.15)), follow a logarithmic relationship with q. Next, we recall from the relationship in equation (7.13) that l e, = forward + hack< SO all the values of the current are net currents, thus explaining the observed decrease at the centre of the graph. The two constituent currents are comparable in magnitude, but of opposite signs, and so one current partially cancels out the other, and l ei is small. 

Whereas the Langmuir equation assumes a uniform energy of adsorption, the Temkin equation uses a logarithmic relationship... 

The graphs given in Fig. 1.38 show the logarithmic relationship between the water-cement ratio and the permeability coefficient of hardened cement paste. Thus concrete with a paste water-cement ratio of 0.4 will be almost impermeable. Water-reducing agents can be used to reduce the water- cement ratio, so ensuring that the permeability is kept to a minimum. 

Many factors act together to determine the optimum scale of a process. These include the demand for the product, competitors share of the market, any technical limitations on the size of operation and also economies of scale effects. There is an approximate logarithmic relationship between the unit production costs for a product and the volume of production, whereby considerable economies of scale can be achieved. If the costs of a process of one size (C ) is known then the costs of larger or smaller factories (C ) can be approximately obtained from the relationship C = Cx (or n° ), where n is the scale-up ratio, i.e. n=l for a plant that is twice as big. Alternatively, a graph of log capital costs vs. log of plant capacity gives a straight line with a slope equal to the scale-up factor (n). The power term varies from case to case, but is invariably less than one. This scale effect is one reason why unit production costs are inversely proportional to the scale of manufacture. For example, most amino acids are expensive and can only be used in... 

Values of (s ) for reaction of the more stable tertiary carbocations X-2+ with a solvent of 50 50 (v/v) water/trifluoroethanol, determined using the azide ion clock, were used to establish a good linear logarithmic relationship between k (s ) for reaction of X-2+ and the first-order rate constants fcobsd (s ) for their formation as intermediates in the stepwise reaction of X-2-C1 in the same solvent (Eq. 2 and Scheme 2.5)." ... 

Note T Urbanski ( ). Rychter, CR 208, 900(1939) found for eleven expls tested by them that logarithmic relationship between the time lag prior to expln and the reciprocal of absolute temperature gives a straight line. Graphs for NC s, MF, ErTeN and EDNA did not give straight lines... 

It is possible to display the relationship between size and number of particles graphically. Groves [79] reported a logarithmic relationship between the number of... 

In practice, the values of v necessary to hold Rx or Rj at a constant value, most commonly equal to 0.500, are determined and defined as vc, i.e. f o.s when the peak current ratio is held constant at 0.500. Under these conditions, z can conveniently be determined by the logarithmic relationship... 

Regardless of the type of lethality induced by a sterilization process—whether it be heat, chemical, or radiation—micro-organisms, upon exposure to adequate levels of such treatments, will die according to a logarithmic relationship between the concentration or population of living cells and the time exposure or radiation dose to the treatment. This relationship between the microbial population and time may be linear or nonlinear, as seen in Figure 1. The D value, or the time or dose required for a one-log reduction in the microbial population, may be calculated from these plots. 

Logarithmic relationship between pseudo-limiting-current of the prewave at E = 0.45 V vs. SCE and the hydrogen peroxide concentration for various pH values, recorded at a glassy-carbon electrode rotating at 16.67 Hz. The numbered curves correspond to pH values of (1) 10, (2) 11, (3) 12, (4) 13 and (5) 14 other curves correspond to each increment of 0.2 pH units. 

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