Linear law

The relative weakness of the two effects, and the adoption of the kinetic form of the catalysis to the linear law only when the concentration of the additive was greater than c. 0-2 mol 1 , results from the equilibria existing in anhydrous nitric acid. In the absence of catalyst,... 

The nitronium ions produced in this way tend to repress the selfdehydration of the nitric acid and therefore the net concentration of nitronium ions is not proportional to the concentration of the catalyst. When sufficient sulphuric acid has been added to make the self-ioniza-tion of nitric acid relatively unimportant, the nitronium ions will be produced predominantly from the above ionization, and the acceleration will follow a linear law. 

Although our simple oxide film model explains most of the experimental observations we have mentioned, it does not explain the linear laws. How, for example, can a material lose weight linearly when it oxidises as is sometimes observed (see Fig. 21.2) Well, some oxides (e.g. M0O3, WO3) are very volatile. During oxidation of Mo and W at high temperature, the oxides evaporate as soon as they are formed, and offer no barrier at all to oxidation. Oxidation, therefore, proceeds at a rate that is independent of time, and the material loses weight because the oxide is lost. This behaviour explains the catastrophically rapid section loss of Mo and W shown in Table 21.2. 

A linear law is valid above the roughening temperature and the Wilson-Frenkel rate is an upper hmit to the growth rate which is achieved in the case of very fast surface diffusion. This is illustrated in Fig. 2. 

When a solid metal is attacked by oxygen gas, the product of the reaction is the metal oxide which, if it is not volatile, builds up as a surface layer on the metal. The oxide layer may be protective or non-protective. A non-protective layer does not inhibit the continued access of oxygen to the unchanged metal the rate of growth of such an oxide layer is independent of its thickness X and the law of growth is AX/At =. On integration this gives the linear law... 

Experimental dependences of conductivity cr of the CPCM on conducting filler concentration have, as a rule, the form predicted by the percolation theory (Fig. 2, [24]). With small values of C, a of the composite is close to the conductivity of a pure polymer. In the threshold concentration region when a macroscopic conducting chain appears for the first time, the conductivity of a composite material (CM) drastically rises (resistivity Qv drops sharply) and then slowly increases practically according to the linear law due to an increase in the number of conducting chains. 

The relationship between fluctuation and dissipation is reminiscent of the reciprocal Onsager relations that link affinity to flux. The two relationships become identical under Onsager s regression hypothesis which states that the decay of a spontaneous fluctuation in an equilibrium system is indistinguishable from the approach of an undisturbed non-equilibrium system to equilibrium. The conclusion important for statistics, is that the relaxation of macroscopic non-equilibrium disturbances is governed by the same (linear) laws as the regression of spontaneous microscopic fluctuations of an equilibrium system. In the specific example discussed above, the energy fluctuations of a system in contact with a heat bath at temperature T,... 

Equilibrium fractionation. A simple fractionation law, called the linear law (e.g., Hofmann, 1971), relates the measured and natural isotopic ratios through a function f (Am/) of the mass difference Am/ = m,—m, between the isotopes defining the ratios... 

Figure 3.5 The domain of the linear law for mass-dependent discrimination between two isotopic ratios.

Figure 11 MCT P-scaling for the amplitudes of the von Schweidler laws fitting the plateau decay in the incoherent intermediate scattering function for a R value smaller than the position of the amorphous halo, q = 3.0, at the amorphous halo, q = 6.9, and at the first minimum, q = 9.5. Also shown with filled squares is the P time scale. All quantities are taken to the inverse power of their predicted temperature dependence such that linear laws intersecting the abscissa at Tc should result. 

The linear law. This law (Fig. 1) is probably the most intuitive of all the mass-fractionation laws. Let us expand the transmission T(M) of isotopic beams at mass M as a function of the mass difference with a reference mass M, ... 

This expression shows that the replicates of a sample which fractionates according to the linear law define a linear alignment with slope given by Equation (11). It also explains... 

Unfortunately, the linear law is not consistent if two ratios fractionate according to the linear law, the ratio of these ratios does not. For this reason, two additional mass fractionation laws are usually considered, the power law and the exponential law. It has been shown by Marechal et al. (1999) that these two laws are two members of a broad family of laws, which they refer to as the generalized power law. The mass fractionation law predicted from quantum mechanics for isotopic equilibrium between vibrating molecules (Bigeleisen and Mayer 1947 ... 

Tf transmission for isotope i in cup y m mass bias factor for the linear law... 

Different behaviors and mechanisms were clearly recognized between these resins. Epoxy resin cured with amine showed no degradation during immersion because of its stable crosslinks. Epoxy resin cured with anhydride showed the uniform corrosion with the softening and dissolution of the surface and also behaved similar to the oxidation corrosion of the metal at high temperature obeying linear law. 

Sometimes crystal growth, dissolution, or oxidation is said to follow a linear growth law or a parabolic growth law. The linear law means that the thickness of the crystal depends linearly on time. 

At sufficiently high concentrations of the transported solute particles, the surface coverage becomes important and non-linear laws for the rate of adsorption should be used. 

There are three models for mass fractionation of isotopes in the ionization processes a linear law, a power law and an exponential law (Wasserburg et al., 1981 Lee et al., 2001). 

The measured isotopic ratio is and the corrected isotopic ratio is R. For the linear law, the corrected ratio is given by... 

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