Hansen approximation

C. F. Hansen, Approximations for the Thermodynamic and Transport Properties of High-Temperature Air, NASA Tech. Rep. R-50, 1959. 

For large molecules it is no longer feasible to carry out the complete anharmonic vibrational analysis implied by Equation 12.16. One is forced to the approximate relation, Equation 12.17, which seems to work pretty well in spite of the criticisms discussed above. Numerous examples abound in the literature. The interested student is referred to the review of Hansen. 

Figure 7.6. Comparison of (a) experimental phase diagram for the Cu-Au system (Hansen 1958) with (b) predictions for the Cu-Au system calculated using the tetrahedron approximation but including asymmetric four-body interactions (de Fontaine and Kikuchi 1978).

Wiedemeier, T.H., M.A. Swanson, J.T. Wilson, D.H. Kampbell, R.N. Miller, and J.E. Hansen. 1996. Approximation of biodegradation rate constants for monoaromatic hydrocarbons (BTEX) in ground water. Ground Water Monit. Remed. 16 186-194. 

Aqueous extracts of soil samples Arsenic Arsenite, arsenate, monomethyl arsonate dimethylarsenite, arsenbetaine, arsenocholine, tetramethyl arsonium ion Ion exchange (both cationic and anionic) Approximately 1 jug cnT1 Hansen et al. (1992)... 

The F-method is not applicable to the calculation of <5h- It has already been stated by Hansen that the hydrogen bonding energy Ehi per structural group is approximately constant. This leads to the form of Eq. (7.24). For molecules with several planes of symmetry, <5 = 0. The group contributions F i, Fpi and Ehi for a number of structural groups are given in Table 7.10. 

Chemically bound water is most reasonably defined as including that present in interlayer spaces, or more firmly bound, but not that present in pores larger than interlayer spaces. As will be seen in Chapter 8, the distinction between interlayer space and micropores is not sharp water adsorbed on surfaces of pores further blurs the definition. From the experimental standpoint, the determination is complicated by the fact that the amount of water retained at a given RH depends on the previous drying history of the sample and on the rate at which water is removed. An approximate estimate is obtained by equilibrating a sample, not previously dried below saturation, with an atmosphere of 11% RH (F12,F13,F14). Saturated aqueous LiCl HjO gives the required RH (partial pressure of water vapour = 2,7 torr at 25°C). To achieve apparent equilibrium in a reasonable time (several days), the sample must be crushed and the system evacuated the salt solution should be stirred, at least intermittently. Young and Hansen (Y5) found the composition of the C-S-H in C3S paste thus... 

The C-S-H present in a saturated paste is assumed to include that part of the porosity, and its content of water, without which it cannot be formed. The H20/Si02 ratio is approximately 4.0. Assuming the water in excess of that present at 11% RH to have a density of 1000 kg m", Hansen (H28) calculated the density of the C-S-H to be 1850-1900 kg m . This value is near those of 1900-2100kgm" for cement pastes under saturated conditions (P20), if allowance is made for the CH and other hydrated phases which these contain. 

Bessey OA, Adam DJD, and Hansen AE (1957) Intake ofvitamin Be and infantile convulsions a first approximation to the requirements of pyridoxine in infants. Pediatrics 20, 33-44. 

In the first paper on this sutgect (Hansen and Ugelstad, 1978) desorption of monomer radicals was neglected. Furthermore, in order to simplify Eq. (74), P was set equal to 1 (i.e., terinination and absorption of initiator radicals was neglected). AD values were set equal to some average absorption constant K (when micelles are absent). As an approximation. could then be expressed by... 

Fig. 10.10. Determination of thermogenin amount in brown adipose tissue mitochondria by the enzyme-linked immunosorbent assay (ELISA) system. The amount of thermogenin was determined as elsewhere described (Cannon et al. [13] Sundin et al. [40] Hansen et al. [56]) in an assay system based on the competition between absorbed and added thermogenin for rabbit on/r-rat-thermogenin antibodies. The interaction was followed with a sheep onri-rabbit-IgG antibody conjugated to alkaline phosphatase. The reaction was linearized as indicated (abs 0 is the absorbance developed in the absence of competing thermogenin). It is seen that this assay can detect less than 0.25 fig thermogenin, i.e., the content in less than 10 fig of mitochondria. It is also seen that the thermogenin content of rat brown fat mitochondria is approximately doubled after a 24 h cold stress. (Our unpublished observations.)...

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