Carter equation

For a solid-state diffusion mechanism, the growth of the reaction product in powder systems occurs at the contact points and for nearly equal-sized spheres, the number of contact points is small. Nevertheless, for many systems, the Jander equation and the Carter equation give a good description of the reaction kinetics for at least the initial stages of the reaction. It appears that rapid surface diffusion provides a uniform supply of one of the reactants over the other. Alternatively, if the vapor pressure of one of the reactants is high enough [e.g., ZnO in Eq. (2.3)], condensation on the surface of the other reactant can also provide a uniform supply of the other reactant. In this case, the powder reaction can be better described as a gas-solid reaction rather than a solid-state reaction (32). 

Figure 2.16 Kinetics of reaction between spherical particles of ZnO and AI2O3 to form ZnAl204 at 1400°C in air, showing the validity of the Carter equation. (From Ref. 32.)...

Outline the derivation of the lander equation and discuss its limitations. How are the limitations corrected in the Carter equation One-micron spheres of AI2O3 are surrounded by excess ZnO powder to observe the formation of zinc aluminate spinel ZnAl204. It is found that 25% of the AI2O3 is reacted to form ZnAl204 during the first 30 min of an isothermal experiment. Determine how long it will take for aU... 

Valensi C ) earlier developed the same solid-state reaction model mathematically from a different starting point. Thus Eq. (9) is referred to as the Valensi-Carter equation. 

The initial morphology of the reactants exerts considerable influence on the reaction kinetics, even in purely diffusion-controlled reactions. The oxydation of metal spheres of radius rM [496], which are not in contact with each other, is a comparatively simple case. The solution is the Carter equation [497] ... 

Fig. 6.70 Description of the course of reaction for the oxydation of nickel spheres for differing sphere diameters and temperatures according to the Carter equation. The continuous lines represent the theoretical course. The left scale is linear and ranges from 1.32 to 1.52. Prom Ref. [497].

The most consequent and the most straightforwaid realization of such a concept has been carried out by Handy, Carter, and Rosmus (HCR) and their coworkers. The final form of the vibration-rotation Hamiltonian and the handling of the corresponding Schrddinger equation in the absence of the vibronic... 

McQueen, R.G., S.P. Marsh, J.W. Taylor, J.N. Fritz, and W.J. Carter (1970), The Equation of State of Solids from Shock Wave Studies, in High Velocity Impact Phenomena (edited by R. Kinslow), Academic Press, New York, pp. 293-299. 

As can be seen from the above equations, the standard deviation of the strength increases significantly with the number of processes used in manufacture that are adding the residual stresses. This may be the reason for the apparent reluctance of suppliers to give precise statistical data about their product (Carter, 1997). 

Equation 4.34 represents probably one of the most important theories in reliability (Carter, 1986). The number of load applieations defines the useful life of the eompo-nent and is of appropriate eoneern to the designer (Bury, 1974). The number of times a load is applied has an effeet on the failure rate of the equipment due to the faet that the probability of experieneing higher loads from the distribution population has inereased. Eaeh load applieation in sequenee is independent and belongs to the same load distribution and it is assumed that the material suffers no strength... 

The approaeh taken by Carter (1986, 1997) to determine the reliability when multiple load applieations are experieneed (equation 4.34) is first to present a Safety Margin, SM, a non-dimensional quantity to indieate the separation of the stress and strength distributions as given by ... 

Using Carter s approach first, from equation 4.47 we can calculate LR to be ... 

In estimating the value of Ed by means of the transcendental equations (28), the circumstance utilized is that the variation of em for a given change in Tm is much less than the variation of exp(em) (31). Until now, only particular solutions have been available for the hyperbolic and linear heating schedules and for the first-order and second-order desorptions. They can be found for example in the fundamental papers by Redhead (31) and Carter (32) or in the review by Contour and Proud homme (106), and therefore will not be repeated here. Recently, a universal procedure for the... 

An analysis of the rate of release of adsorbed atoms from sites with a continuous energy spectrum for the case of an arbitrary distribution function of initial site populations was given by Carter (32). The rate equation for the t th desorption process with x = 1 and negligible readsorption is... 

Hulbert [77] discusses the consequences of the relatively large concentrations of lattice imperfections, including, perhaps, metastable phases and structural deformations, which may be present at the commencement of reaction but later diminish in concentration and importance. If it is assumed [475] that the rate of defect removal is inversely proportional to time (the Tammann treatment) and this effect is incorporated in the Valensi [470]—Carter [474] approach it is found that eqn. (12) is modified by replacement of t by In t. This equation is obeyed [77] by many spinel formation reactions. Zuravlev et al. [476] introduced the postulate that the rate of interface advance under diffusion control was also proportional to the amount of unreacted substance present and, assuming a contracting sphere (radius r) model... 

The work of Melander and Carter (1964) on 2,2 -dibromo-4,4 -di-carboxybiphenyl-6,6 -d2 (1) has been referred to above in the introductory and theoretical sections, where it was pointed out that the availability of two detailed theoretical computations of the inversion barrier (Westheimer and Mayer, 1946, Westheimer, 1947 Hewlett, 1960) made this system especially attractive for the study of steric isotope efifects. Furthermore, in the preferred initial-state conformation the two bromines are probably in van der Waals contact (cf. Hampsoii and Weissberger, 1936 Bastiansen, 1950), and thus initial-state steric effects are unaffected by deuterium substitution in the 6 and 6 positions. The barrier calculations provided two different theoretical values for the non-bonded H Br distance in the transition state which, together with the corresponding H Br potential function, could be inserted in equation (10) to yield values for A AH. For... 

Handy-Carter (HC) equation, Renner-Teller effect, triatomic molecules, 611-615, 618-619... 

As pointed out by Peeters et al. (1996), based on their own experiments and on the reinterpretation of published field data, the adequate model to describe horizontal diffusion in lakes and oceans is the shear diffusion model by Carter and Okubo (1965). The model is described in Box 22.4. The most important consequence of this model is that the 4/3 law and the equivalent t3-power law for c2(t) expressed by Eq. 22-42 are replaced by an equation which corresponds to a continuous increase of the exponent m from 1 to 2 (Box 22.4, Eq.l) ... 

In the literature, one can find other empirical or semi-empirical equations representing the kinetics of powder reactions. One can certainly take into account grain size distribution, contact probability, deviations from the spherical shape, etc. in a better way than Carter has done. Even more important are parameters such as evaporation rate, gas transport, surface diffusion, and interface transport in this context. As long as these parameters are neglected in quantitative work, the kinetic equations are inadequate. Nevertheless, considering its technological relevance, a particular type of powder reaction will be discussed in the next section. 

An interesting Fe-catalyzed SN2 -like carbene insertion reaction using diazo compounds and allyl sulfides (the Doyle-Kirmse reaction) was reported by Carter and Van Vranken in 2000 [20], Various allyl thioethers were reacted with TMS-diazomethane in the presence of catalytic amounts of Fe(dppe)Cl2 to furnish the desired insertion products with moderate levels of stereocontrol [Equation (7.6), Scheme 7.14]. The products obtained serve as versatile synthons in organic chemistry, e.g. reductive desulfurization furnishes lithiated compounds that can be used in Peterson-type oleftnations to yield alkenes [Equation (7.7), Scheme 7.14] [21]. 

The problem of approximate dependence of upon N, M and K was addressed as early as in 1949 [21]. The equation proposed by Carter reads... 

Equations (7) and (8) both apply to a sphere. j3 is the ratio of product volume to reactant volume in the layer of product through which diffusion takes place. When 3 = 1, (8) reduces to eqn. (7). This should be the case if reduced material retains its gross geometry, with pores produced in it to account for the entire decrease in volume upon reaction. Equation (7) is known as the Ginstling—Brounshtein equation eqn. (8) is Carter s modification of it [88] and must be used if there is a volume increase on reaction. This does not often happen in reductions, but can occur in other types of reaction [89]. 

Scott [72] utilized a variant of Eq. (13). Carter et al. [73] have derived equations for the association constants of solvated complexes. Caution must be applied to cases where a large amount of donor may affect the behavior of the solvent, which can seriously affect the application of Eq. (13) [74,75]. 

The calculation of temperatures and equilibrium compositions of gas mixtures involves simultaneous solution of linear (material balance) and nonlinear (equilibrium) algebraic equations. Therefore, it is necessary to resort to various approximate procedures classified by Carter and Altman (Cl) as (1) trial and error methods (2) iterative methods (3) graphical methods and use of published tables and (4) punched-card or machine methods. Numerical solutions involve a four-step sequence described by Penner (P4). 

When corrections are made for these two simplifications. Carter [29, 30] has shown the following equation to be applicable ... 

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