Atkinson equation

Atkinson, K. E. A Sui oey of Numeiical Methods foi the Solution ofFied-holm Integial Equations of the Second Kind, SIAM, Philadelphia (1976). 

Microbes tend to form flocks as they grow, into which nutrients and dissolved oxygen must diffuse. The rate of growth thus depends on the diffusional effectiveness. This topic is developed by Atkinson (1974). Similarly enzymes immobilized in gel beads, for instance, have a reduced catalytic effectiveness analogous to that of porous granular catalysts that are studied in Chapter 7. For the M-M equation this topic is touched on in problems P8.04.15 and P8.04.16. 

Atkinson, K.E., 1997, The numerical solution of boundary integral equations, Cambridge University Press, Cambridge. 

The chemistry of the troposphere (the layer of the atmosphere closest to earth s surface) overlaps with low-temperature combustion, as one would expect for an oxidative environment. Consequently, the concerns of atmospheric chemistry overlap with those of combustion chemistry. Monks recently published a tutorial review of radical chemistry in the troposphere. Atkinson and Arey have compiled a thorough database of atmospheric degradation reactions of volatile organic compounds (VOCs), while Atkinson et al. have generated a database of reactions for several reactive species with atmospheric implications. Also, Sandler et al. have contributed to the Jet Propulsion Laboratory s extensive database of chemical kinetic and photochemical data. These reviews address reactions with atmospheric implications in far greater detail than is possible for the scope of this review. For our purposes, we can extend the low-temperature combustion reactions [Equations (4) and (5)], whereby peroxy radicals would have the capacity to react with prevalent atmospheric radicals, such as HO2, NO, NO2, and NO3 (the latter three of which are collectively known as NOy) ... 

In equation (C), A() (or A 111 as used earlier) is the low-pressure limiting rate constant and Ay is the high-pressure limiting rate constant. Fc is known as the broadening factor of the falloff curve its actual value depends on the particular reaction and can be calculated theoretically. Troe (1979) suggests that for reactions under atmospheric conditions, the value of Aft will be 0.7-0.9, independent of temperature. However, values as low as 0.4 are often observed. The NASA evaluations of stratospheric reactions (DeMore et al., 1997) take Aft = 0.6 for all reactions. The IUPAC evaluation (Atkinson et al., 1997a,b) does not restrict Fc to 0.6. However, it is important to note that the values of A0 and Ay will depend on the value of Fc used to match the experimental data. For example, for reaction (11)... 

The 1997 recommendations for the OH + N02 rate constants (DeMore et al., 1997 Atkinson et al., 1997a, 1997b) may be systematically high (e.g., Donahue et al., 1997) at temperatures below 240 K. Thus, recent measurements at temperatures characteristic of the upper troposphere give rate constants that are smaller than the recommendations by 10-30% (Brown et al., 1999a Dransfield et al., 1999). In addition, 02 appears to be only about 70% as efficient a third body as N2 in the termolecular reaction. Using a modified form of the semiempirical equation for the rate constant in the falloff region (Chapter 5, Eq. (C)), which takes into account the variable collision efficiency /3,... 

As Chapter 10 discusses in detail, chemical compounds in the atmosphere are partitioned between the gas and particle phases (Pankow, 1987 Bidleman, 1988), and the phase in which a chemical exists in the atmosphere can significantly influence its dominant tropospheric removal process(es) and lifetime (Bidleman, 1988 Atkinson, 1996). Gas/particle partitioning has been conventionally described by the Junge-Pankow adsorption model that depends on the liquid-phase (or sub-cooled liquid-phase) vapor pressure, Pu at the ambient atmospheric temperature, the surface area of the particles per unit volume of air, 9, and the nature of the particles and of the chemical being adsorbed (Pankow, 1987 Bidleman, 1988). The fraction of the chemical present in the particle phase, ( ), depends on these parameters through an equation of the form (Pankow, 1987 Bidleman, 1988) ... 

Mitigating this situation to some extent are the correlation equations for estimating kHO (water) from kHO (air) or from the Hammett relation (see below), making it possible to tap into the abundant database of measured values for kHO (air) (Atkinson, 1989), the excellent SARs for estimating kHO (air) values (Atkinson, 1987), and the utility of the Hammett equation with its large database of substituent constants (Exner, 1978 Perrin et al., 1981). 

The Elovich equation has been used to describe the kinetics of P04 sorption and desorption on soils and soil minerals (Atkinson et al., 1970 Chien and Clayton, 1980 Chien et al., 1980 Sharpley, 1983), potassium reactions in soils (Sparks et al., 1980b Martin and Sparks, 1983 Sparks and Jardine, 1984 Havlin and Westfall, 1985), borate dissolution from... 

Some investigators have also suggested that breaks or multiple linear segments in a q versus In t Elovich plot [Eq. (2.48)] could indicate a changeover from one type of binding site to another (Atkinson et al., 1970 Chien and Clayton, 1980). However, one should be cautious about making such mechanistic conclusions from a plot of an empirical equation. 

Atkinson, R. J., Hingston, F. J., Posner, A. M., and Quirk, J. P. (1970). Elovich equation for the kinetics of isotope exchange reactions at solid-liquid interfaces. Nature (London) 226, 148-149. 

An example not previously discussed is the Pitzer-Debye-Hiickel slope for apparent molar volume (Av) that is required in Eqs. 2.76, 2.80, and 2.81. A numerical equation for Ay as a function of temperature and pressure was derived from the database of Ananthaswamy and Atkinson (1984) over a temperature range of 273 to 298 K and over a pressure range of 1 to 1000 bars ... 

There are also some empirical equations for describing the yields of products formed in the cracking reactions of polymers. One of them is the Atkinson and McCaffrey kinetic model, which derives the weight loss of polymer for their initial degree of polymerization, weight of sample and reaction rate. As a matter of fact the reaction rate constant is calculated by using a first-order kinetic equation [33, 34]. 

From the viewpoint of the combustion chemist, mechanistic and theoretical studies of abstraction reactions serve two purposes. First, they can determine the overall rate coefficient for an abstraction over a range of temperatures, especially when there are limited experimental data. Second, the combustion modeller wishes to know the rate of abstraction at any particular site on a hydrocarbon molecule. For reaction (10) this is trivial as there is only one type of site a primary C—H bond. However, for more complex fuels there will be a variety of different sites which to a first-order approximation can be considered as primary, secondary and tertiary C—H bonds. As mentioned in the introduction to this section, Atkinson et al. [10] and Walker [11] have attempted to describe radical/ alkane kinetics with the following simple model based on equation (2.4)... 

The residuals may also be obtained from cross-validation. With the methods described in this book, both the X part and the y part of Equation (7.7) are modeled and hence have associated residuals. The residual ey of y is usually summed to a number of regression statistics such as percentage variance explained, the coefficient of determination, root mean squared error of prediction etc. Diagnostics based on y-residuals are well covered in standard regression literature [Atkinson 1985, Beebe et al. 1998, Cook 1996, Cook Weisberg 1980, Cook Weisberg 1982, Martens Ntes 1989, Weisberg 1985],... 

Consider flow into a circular channel or pipe with the velocity a constant at the entrance. As the fluid enters the channel, the velocity soon develops a parabolic profile. The velocity at the centerline increases from (v) (at the entrance) to 2 (v) (far downstream) while the velocity at the wall decreases suddenly from (v) to zero. The entry length is defined as the length of channel it takes before the centerline velocity is 99% of its ultimate value, 2 (v). Sometimes the entry length is defined using a 98% or 95% criterion. Atkinson et al. [2,4] have solved this problem using the hnite element method and correlated the results with the equation ... 

To calculate the isomer effect Atkinson et al. [67] used the Cvetanovic equation [68] ... 

Figure 7.1. Relationship of leaf Mo concentrations and cured-leaf yields of hurley tobacco. The fitted equation is of the form y = Z>o + hi[min(jr, / )], where is the intercept and is the slope of the line up to the point X=A. [Reproduced from Sims and Atkinson (1976) Lime, molybdenum, and nitrogen source effects on yield and selected chemical components of hurley tobacco. Tobacco Science 20 181-4, courtesy of Lockwood Trade Journal Co., Inc., New York, NY, and Tobacco Science.]...

Arsindoles are prepared by cyclization of aryl vinyl arsinous chlorides and aryl vinyl chlorides. Atkinson s review details the generality of these approaches <84CHEC-i(i)539>. In a method of limited scope, the arsindole (33) is prepared by reaction of ASCI3 with the acetylene salt (Equation (33)) <80JHC1341>. 

The solubilities of the scale-forming salts barium and strontium sulphates in aqueous solutions of sodium chloride have been reviewed by Raju and Atkinson (1988, 1989). Equations were proposed for the prediction of specific heat capacity, enthalpy and entropy of dissolution, etc., for all the species in the solubility equilibrium, and the major thermodynamic quantities and equilibrium constraints expressed as a function of temperature. Activity coefficients were calculated for given temperatures and NaCl concentrations and a computer program was used to predict the solubility of BaS04 up to 300 °C and SrS04 up to 125 °C. 

The results presented in Fig.14-16 show that irrespective of the corona polarity and the charging temperature the Friondlih and Lengmuir equations do not describe very well the results obtained in our experiments as the determination factor is 0.36 2 0.85. That is why we assumed that the desorption process might run together with another process like charge movement along the surface, for example. Our assumption is based on the results reported in various papers (Protodyakonov et al., Baum et al., Karmazova et al., Atkinson et al., 1976, Atkinson et al., 1980,1, Atkinson et al., 1980, II) and some additional experiments we have carried out with electrets with an island charge distribution. It is reported in... 

Atkinson and Boore (2006) had proposed a method to evaluate the amplification factors based on empirical studies of ground motion data. The evaluation of PGA values for south India was done using the method proposed by Raghu Kanth and Iyengar (2007). The amplification factors can be evaluated based on the following equation. 

Atkinson, G. M., Boore, D. M. (2006). Earthquake Ground-Motion Prediction Equations for Eastern North America. Bulletin of the Seismological Society of America, 96(6), 2181-2205. doi 10.1785/0120050245... 

---