Carlson equation

Carlson and Helquist410 were the first to perform the alkylation of 2-lithio 1,3-dithian-S-oxide 323 (equation 180). The yields of this reaction appeared, however, to be low. In spite of the fact that dithian-S-oxides have been intensively investigated268-411, their synthetic applications are rather limited. 

Equation 6.36 for the adiabatic potential is exact within the framework of the mean field description. However, the structure of the electric part P1 is too complex to disclose its analytic properties. Here we examine the adiabatic potential numerically following the Carlson theory of elliptic integrals [15-21], To proceed with numerical computation, it is necessary to enter a set of parameters designed to describe an experimental situation. It will not surprise the reader who has made it this far that we use values of the chemically fixed parameters specified by the n-butylammonium vermiculite gels [22], namely m+ = 74 mp and m = 36 mp. The average density n0 of the small ions is given by... 

The formula for the adiabatic Gibbs potential (Equation 6.51) is exact within the framework of the mean held description. As with the adiabatic Helmholtz potential, we examine it numerically following the Carlson theory of elliptic integrals [15-21], In order to make clear comparisons between the two types of free energy, the width D of the container is set to be 200 nm throughout (except for Figure 6.21). 

The multi-component van Laar equation takes the following form (Carlson and Colburn, 1942 Wohl, 1946 Prausnitz et al., 1967) ... 

The expression for the mean local energy has recently been utilized by Harriss and Carlson to derive (1) and prove the upper bound relationship existing between the eigenvalue solutions to these equations and the corresponding equations in which Hij is replaced by the symmetric H jj —... 

Because of its flexibility, simplicity, and ability to fit many systems well, the van Laar equation is widely used in practice. It can be derived from the general energy expansion of Wohl, which considers effective volume fractions and molecular interactions. The so-called Carlson and Colburn natural logarithm version of the van Laar equation is given in Table 5.3. However, a common logarithm form is more common. The Margules and Scatchard-Hamer equations in Table 5.3 can also be derived from the Wohl expansion by a set of different assumptions. 

The data were consistent with the transition temperatures of 1653 and 2033 K reported earlier by Hash and Carlson [1960EAS/CAR], and these were used to calculate the enthalpies of transition and fusion, which were estimated to be (3590 + 120) and (13800 + 1250) J-mor respectively. The enthalpy data were fitted to the equations ... 

Indole (64) reacted with aqueous chlorine under mildly acidic conditions (Baben-zian et al., 1963 Lin and Carlson, 1984). The products are shown in Equation 5.28. The initial product formed is presumably N-chloroindole (DeRosa and Alonso, 1978). Carbazole (65) also reacted with free chlorine to produce a complex mixture of CI1-CI4 substitution products (Lin and Carlson, 1984). 

Inhaled ozone is known to initiate free-radical autooxidation of unsaturated fatty acids in animal pulmonary lipids (Pryor et al., 1981). These reactions lead to the formation of such typical autooxidation products as conjugated dienes and short-chain alkanes like ethane and pentane. Whether these reactions also occur in water treatment is uncertain. Glaze et al. (1988) showed that 9-hexadecenoic acid (83) reacted readily in aqueous solution to form the expected C, and C, aldehydes and acids. Linoleic acid (84) was converted to a mixture of aldehydes and acids (Carlson and Caple, 1977) notably, 3-nonenal (85) was among the products. Isolation of an unsaturated aldehyde is significant because of the high reported toxicity of these compounds. Carlson and Caple (1977) also implied that the epoxide of stearic acid was formed when an aqueous solution of oleic acid was ozonized the product probably derives from an indirect attack on the double bond by peracids or peroxy radicals (Equation 5.39). Nevertheless, it is conceivable that similar reactions could occur in natural waters. 

Indole was attacked by CIO2 at the 2,3-double bond to form a diketone and an anhydride (Equation 5.58 Lin and Carlson, 1984). Carbazole (65) underwent an interesting reaction with low concentrations of CIO2 to produce the quinonoid compound 111 at higher concentrations, a colored solution was formed from which no identifiable compounds could be isolated (Lin and Carlson, 1984). 

The activity coefficient depends on tenperature, pressure, and concentratiom Excellent correlation procedures for activity coefficients such as the Margules, Van Laar, Wilson, NRTL, and UNIQUAC methods have been developed fPoling et al.. 2001 Prausnitz et al.. 1999 Sandipr 2008 Tpstpr and Modell. 1997 Van Ness and Abbott. 1982 Walas. 1985). The coefficients for these equations for a wide variety of mixtures have been tabulated along with the experimental data (see Table 2-2). When the binary data are not available, one can use infinite dilution coefficients fTable 2-2 Carlson. 1996 T pt... 

For unsymmetrical systems beyond the capabilities of the van Laar equations, the Scatchard-Hamer equations, although less convenient, are better. If Va = Vb, they reduce to the Margules equations, while if Aab/Aba = Va/Vb, they reduce to those of van Laar. Carlson and Col-bum (5) consequently suggest that the ratio of Va/Vb may be taken as a guide as to which of the equations are applicable. For systems that cannot be handled by any of these, the more complex equations suggested by Wohl (35) may be tried. 

Activity coefficients calculated by these methods agree fairly well for systems where the original equations, i.e., Eqs. (3.50) to (3.66), apply. Carlson and Colburn (5) and Colburn, Schoenborn, and Shilling (8) have shown that the van Laar constants cannot be calculated from solubility data for n-butanol-water and isobutanol-water, but the van Laar equations do not satisfactorily describe the activity coefficients obtained from vapor-liquid data in these systems either. 

B. G. Carlson, Solution of the transport equation by approximations, Los Alamos Scientific Laboratory Report, LA-1891, 1955. 

AkahamaY, Kawamura H, Carlson S et al (2000) Structural stabUity and equation of state of simple-hexagonal phosphorus to 280 GPa phase transition at 262 GPa. Phys Rev B61 3139-3142... 

The first relations have already been proposed by Margules (95MAR1) more than one eentury ago, and later on they have appeared many others, e.g. the equations by Van Laar, Wohl, Seatehard-Hammer, Carlson-Colbum etc. some of them were modified in mareh of time. At present, several newer ones beeame popular because of their easy application to the deseription of multieomponent systems and for the phase behavior prediction methods based on knowledge of binary data. 

Carlson B. g and G. L bell, solution of Transport Equation by the Sn Method, Prop. Intern. Cont Peaceful Uses Atomic Energy, 2nd, Geneva, 1958, 16,... 

Miller, P.J., and Carlson, K.E. Determination JWL Equation of State Parameters Using the Gurney Equation Approximation, 9th Symposium (Intmiational) on Detonation, Portland, MD, 1989, pp. 930-936. 

Some of the applicable muscle models include the Maxwell, Voigt, Hill and Carlson models (Figure 1). In particular, the Carlson (1957) equation is used in much of this work to describe the stress-velocity relationship of cardiac muscle over the entire cardiac cycle. Min et al. (1978) found very little difference in analyzing ventricular dynamics when he alternately used Carlson s equation only during isotonic contraction and Hill s equation during isovolumic contraction. 

Using the Carlson (1957) equation to describe the stress-velocity relationship of cardiac muscle... 

Substituting this equation into Carlson s equation and introducing Eq. (2) for volume, a pressure-volume equation for this model of the left ventricle results. 

In terms of circumferential tension, Carlson s equation can be expressed as... 

In the event that there is a localized infarct, the effected cylinders can be represented by a segment of normal contractile tissue (obeying Carlson s equation) and a segment of abnormal (non-contractile) tissue with only elastic properties. Using this description, Watts (1976) has shown that the individual cylinder equivalent cricuit is modified as illustrated in Figure 12 where ... 

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