Square root relation

Furzikov79 proposed a thermal model to describe the etching rate that led to an inverse square root dependence of the threshold fluence on a modified absorption coefficient, aeff, which includes possible changes in the singlephoton absorption coefficient owing to thermal diffusion. This inverse square root relation is given by... 

The speed of carbonation into the depth of the concrete depends strongly on the consistency and porosity of the material and follows a square root relation 403... 

Si and Su are the slopes of the zero-order potential energy surfaces at the intersection (Si = —Sn for an exchange reaction), and v is the velocity with which the point representing the system moves through the intersection region. For typical conditions it is found that p 1 for interactions EIfII of more than 0.5 kcal mol"1 (50). Under these conditions the reactions will be adiabatic, and the square root relation is expected to hold provided EitU is not too large. However, for small EltJ1 ... 

In other words, the square-root relation may also hold for certain classes of nondiabatic reactions (51). Conversely, the fact that a series of reactions obeys the square-root relation does not require that all of the reactions involved be adiabatic. 

Equation (84) follows from (82) since tanh h = 1 when h 1. Thus, a catalyst prepared with very small pores can be expected to have a selectivity factor equal approximately to the square root of the selectivity factor observed for the same catalytic material prepared with very large pores. The physical reason for this is that only a small fraction of catalyst surface is available to the faster of the two reactions while a larger fraction is available to the slower reaction. Thus small pores tend to weaken Type I catalyst selectivity by retarding a fast reaction to a larger extent than a slow reaction. Providing no catalyst poisons are present, it appears that this decrease of selectivity can proceed no further than the square root relation given by eq. (84). For example, if A B + 0 were intrinsically nine times faster than X F + Z, we would observe the first reaction to be only three times faster on a catalyst containing very snaall pores. 

For polymer coils in non-theta systems, this simple square root relation is not valid. In this case, a dependence of the radius from the molar mass similar to the [q]-M-relation is observed ... 

The Fox-Flory equation (Eq. 8.33) in combination with the square root relation of the molar mass (Eq. 8.22) and the calculation of the radius of gyration from the end-to-end distance (Eq. 8.14) allows for a derivation of the [/j]-M-relationship for theta conditions ... 

Hi) A square root relation holds for flow through the valve. 

The involvement of free radicals as part of the reaction sequence in the oxidatic reaction is responsible for some similarities to the non-enzymic autoxidation. Thus, both have OgrDHF stoichiometries that approach 1.0 under some conditions, indicating breakdown beyond diketo succinate (Eq. 7a above). Both are stimulated by cyanide, according to a square root relation, and by manganese. And the DHF oxidase system resembles the ascorbate EDTA-Fe++ system in its initiation of hydroxy-lations. 

This inverse square root relation, suggested in 1829, is Graham s law. 

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