Square-root law

For nitrations carried out in nitric acid, the anticatalytic influence of nitrous acid was also demonstrated. The effect was smaller, and consequently its kinetic form was not established with certainty. Further, the more powerful type of anticatalysis did not appear at higher concentrations (up to 0-23 mol 1 ) of nitrous acid. The addition of water (up to 5 % by volume) greatly reduced the range of concentration of nitrous acid which anticatalysed nitration in a manner resembling that required by the inverse square-root law, and more quickly introduced the more powerful type of anticatalytic effect. 

In the classical theory of conductivity of electrolyte solutions, independent ionic migration is assumed. However, in real solutions the mobilities Uj and molar conductivities Xj of the individual ions depend on the total solution concentration, a situation which, for instance, is reflected in Kohhausch s square-root law. The values of said quantities also depend on the identities of the other ions. All these observations point to an influence of ion-ion interaction on the migration of the ions in solution. 

In the case of binary solutions, Eq. (7.49) coincides with the empirical Eq. (7.14), both in form and in the value of the numerical constant k. Therefore, the empirical square-root law can be explained quantitatively on the basis of the theory of ion-ion interaction. 

The initial rate at which the matrix C is formed in these matrix-dependent experiments is related to the initial concentration c by a square-root dependence. This square root law of autocatalysis is found in most self-replicating systems ... 

More information on the square root law and on kinetic and thermodynamic aspects of the minimal replicator theory can be found in the literature (von Kiedrowski, 1999 and 1993). The square root law has its origin in the product inhibition involved in the mechanism of self-replication. The more C units are formed, the greater is the tendency of the molecules of C to dimerize to give C2 this species, however, cannot function as a catalyst. 

In the early days of water radiolysis, it was empirically established in several instances that the reduction of molecular yield by a scavenger was proportional to the cube root of its concentration (Mahlman and Sworski, 1967). Despite attempts by the Russian school to derive the so-called cube root law from the diffusion model (Byakov, 1963 Nichiporov and Byakov, 1975), more rigorous treatments failed to obtain that (Kuppermann, 1961 Mozumder, 1977). In fact, it has been shown that in the limit of small concentration, the reduction of molecular yield by a scavenger should be given by a square root law in the orthodox... 

A special situation arises in the limit of small scavenger concentration. Mozumder (1971) collected evidence from diverse experiments, ranging from thermal to photochemical to radiation-chemical, to show that in all these cases the scavenging probability varied as cs1/2 in the limit of small scavenger concentration. Thus, importantly, the square root law has nothing to do with the specificity of the reaction, but is a general property of diffusion-dominated reaction. For the case of an isolated e-ion pair, comparing the t—°° limit of Eq. (7.28) followed by Laplace transformation with the cs 0 limit of the WAS Eq. (7.26), Mozumder derived... 

This is possible if the equivalent conductivity is proportional to the square root of the concentration Cq, i.e. if the Debye-Hiickel-Onsager law is obeyed. It is known that this square-root law is also obeyed for non-aqueous solvents as a good approximation, as long as the dielectric constant of the solvent is not less than e = 30. Figure 19 shows the equivalent conductivities as a function of Vm for three examples. If one bears in mind that, because of experimental difficulties, the accuracy of measurements in aqueous solutions is not attained, then the square root law is obeyed to a good approximation. 

If the rate is controlled by diffusive mass transfer (Figure 1-1 lb) and if other conditions are kept constant, then (i) the growth (or dissolution) distance is proportional to the square root of time, referred to as the parabolic growth law (an application of the famous square root law for diffusion), (ii) the concentration in the melt is not uniform, (iii) the concentration profile propagates into the melt according to square root of time, and (iv) the interface concentration is near saturation. For the rate to be controlled by diffusion in the fluid, it cannot be stirred. 

Notice the particular features of this kind of ohgonucleotide the hexameric sequence is said to be self-complementary, since two identical molecules can form a duplex via Watson and Crick bases. It may also be noted from Figure 7.5 that two parallel pathways compete for the formation of the template T, namely the template-dependent, autocatalytic pathway, and the template-independent, non-autocatalytic one. This competition is the reason why the initial rate of the autocatalytic synthesis was found to be proportional to the square root of the template concentration -something that von Kiedrowski and colleagues called the square-root law of autocatalysis. As Burmeister (1998) put it ... 

Another way of expressing QD values is to state them as the cube root of the expl wt because certain detonation phenomena scale according to a cube root law. One of these is the instantaneous peak overpressure with distance (Ref 11). Damage can be related to overpressure by the cube root law except with respect to damage within inhabited structures and with respect to flying debris, for both of which a square root law is more nearly correct. 

Experiments in which initiators other than AIBN are used do not indicate any unusual effects. Polymerization rate is nearly always proportional to the square root of the initiator concentration or at least to a value between 0.5 and 0.6. Ulbricht (135) established the square root law for ammonium persulfate, AIBN and benzoyl peroxide in DMF, the rates being fastest with persulfate, slower with AIBN and slowest with the peroxide. One expects that persulfate and peroxide will be more active than AIBN in abstracting hydrogen from other components of the systems. Other initiators have been used, including UV with di-t-butyl... 

For T < 0 a spontaneous polarization exists. It can easily be shown that the Curie temperature 0 is equal to the phase transition temperature Tc. The spontaneous polarization depends on the distance from the phase transition temperature with a square root law. 

See - conductance, - conductivity cell, -> conductometry, - Debye-Falkenhagen effect, -> Debye-Huckel-Onsager theory, - electrolyte, -> ion, -> Kohlrausch square root law, - mass transport. 

Kohlrausch square root law — (1900) A plot of equivalent conductiviy Aeq vs. square root of concentration c according to... 

Ostwald s dilution law — Figure. Plot of Aeq vs. concentration c for aqueous solutions of various electrolytes, (see also - Kohlrausch square root law)... 

Figure 42. Nonergodicity parameter, (effective Debye-Waller factor) of o-terphenyl as obtained from neutron scattering experiments for different values of the momentum transfer Q. (a) Incoherent, (b) coherent full curves below Tc represent fits to the square root law of MCT yielding Tc = 290 K (From Ref. 201.)...

Figure 70 shows that the square-root law is preceded by a linear law for short times, which is indicative of a surface control. In the... 

Nimlos et al. (1993) have foimd that the dependence of the gas-phase oxidation of TCE on the UV intensity, which was linear at high concentrations of TCE, changed its dependence into a square root law when the concentrations were low. This was explained (Upadhya and Ollis, 1998) by a change in the mechanism at high concentrations the mechanism was a chain reaction mechanism induced by chlorine atoms, whereas at low concentrations the mechanism is the common holes /OH attack mechanism. This h)q5othesis correlated well with the fact that the quantum efficiency at high concentrations of TCE was 4-10 times higher than at low concentrations. 

The defect compensation, the square root law, and the low doping efficiency are therefore described by a model of thermal equilibrium bonding configurations. The result is quite surprising because disordered materials cannot usually be described by equilibrium thermodynamics. However, since the model was first proposed, experiments described in the next chapter have confirmed the equilibration of the dopants and defects. 

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