Temperature dependence of parameters

It should be also noted that eqs 18b and 18c provide a temperature dependence for parameter b. In contrast, the conventional expressions (eq 18b), used in numerous mixing rules, provide no temperature dependence of parameter b, even though the direct calculation of a and b from experimental data indicated that b is slightly temperature-dependent.23... 

Table 5.1 presents the most representative cubic EOS. It should also be remembered that an important feature is the temperature dependency of parameters. Thus, the parameter a can be expressed as a product of its value at the critical point Oc, and a dimensionless correction function a(T), such as expressed by the equation (5.8). Table 5.2 gives more details about the temperature dependency. 

Kozlov, G. V Shogenov, V. N. Kharaev, A. M. Mikitaev, A. K. The temperature dependence of parameters, characterizing inelastic deformation of polymers, in the conditions of impact loading. High-Molecular Compounds. B, 1987, 29(4), 311-314. 

Phi-phi. If a reliable PvTx equation of state is available, then we may use the phi-phi method to compute gas solubilities. Thermodynamically, this is merely phi-phi applied to VLB and the general approach has been discussed in 10.1.1 and 12.1.1. But in practice, this is a relatively recent development because reliable equations of states have only recently been devised for supercritical solutes in subcritical solvents. When the phi-phi method is used, computed solubilities are found to be sensitive to the temperature dependence of parameters in the equation of state they are also sensitive to the mixing rules used for those parameters. In particular, when cubic equations are used, the temperature dependence and mixing rule for the parameter a must be chosen with care. However, we judge this to be a modeling problem, not a thermodynamic problem. 

The analysis has been done for a continuous wave, 5 kW incident beam that is assumed to have a Gaussian profile with a 14 mm diameter at 14% (1/e ) of maximum intensity. For this level of power the temperature increases are relatively low and the temperature dependence of parameters such as dn/dT [54] have been ignored, but the temperature dependence of the thermal conductivity for optical grade CVD diamond (Fig. 11), although not very pronounced, has been taken into account. 

Another problem usually encountered with force field parameters in molecular dynamics simulations involves the implicit temperature dependence of parameters derived from experimental data. Since experimental data are usually collected at room temperature, parameters fitted to reproduce these experimental data will work best at room temperature and therefore will contain temperature effects implicitly. In molecular dynamics simulations, however, the temperature is an explicit variable, and it is assumed that the force field reproduces the structure at 0 K. The temperature effects obtained in molecular dynamics are added to properties for absolute zero. Parameters derived from data measured at higher temperatures therefore will necessarily introduce a small error. Parameters derived from ab initio calculations are free of these errors, since they are fitted to data for 0 K and do not implicitly contain temperature effects. If only experimental data are available, molecular dynamics calculations should be used to derive force field parameters.i >20 ... 

Numbering of substituent positions, see benzene. ) Theoretical discussion of a, spin density calculations. ) Further ESR-data of alkylated benzene anions, see [Pi 7]. t) Temperature dependence of /-parameters, tt) Observation of ion pairs. ... 

For transitions which involve reanangemenl of the structure, sudi as the glass transitions, the temperature dependence of parameter T in the Kohlrausch equation is... 

The temperature dependence of parameters a and T eff shown in Fig. 13. It can be seen that a will be greater than unity for all Debye temperatures of the model oxide surface, i.e., the phonon-vibration interaction will accelerate the dissociation of the bond according to Eq. (137). Also, from Fig. 13a, it can be seen that higher values of the Debye temperature decrease the rate of the reaction. 

To find models and physical descriptions of the charge transport in conjugated polymers, it is useful to consider the temperature dependence of parameters, such as conductivity and thermopower, of those polymers. Some typical features of metals and semiconductors have already been pointed out. 

The temperature dependence of parameter cp (Fig. 35) was determined from Eqs. 129-132, assuming that the continuous phase is a PU-enriched phase. The value of cp was determined as the average of two

values obtained from the experimentally found values of E and Applying Eqs. 129-132, a reasonable value (p < could be observed only if one assumes that the PU... 

The effect of the flUer concentration was studied (up to 84% — in copolymer of butadience with acrylic acid) in the presence of 0-84% solid filler on the temperature dependence of parameters of mechanical dynamic properties. For the first time a principle was stated in that wodc to the effect that the filler influence m dso be described by the reduction method similar to temperature reduction. For this, use was made of an equation equivalent in its form to the WLF equation but with different numerical values of the constants Ci and C2. Its most generd result is the stress-strain representation in a twice-normalized state-by temperature and by concentration of the flUer. However, not all the parameters can be represented in the concentration-invariable form by the use of tl same method of filler concentration reduction. Specifically, the effect of the filter at the initial section of the stress-strain curve is greater than was predicted by the gemtal reduction method. The relative strain at rapture can also be represented in the concentration invariant from, if tiie vertical shift of the experimental curves is used in addition to tiie commm horizontal one. The author has proposed empirical formulas descrfliing the concentrational dependence of the reduction coefficient. 

FIG U RE 11.7 Temperature dependence of parameters of the continuous kinetic model. 

When there are sufficient data at different temperatures, the temperature dependence of the parameters is reflected in the confidence ellipses (Bryson and Ho, 1969 Draper and Smith,... 

Temperature Dependence of UNIQUAC Parameters for Ethanol(1)/Cyclohexane(2) Isothermal Data (5-65°C) of Scatchard (1964)... 

Vector (length 20) of stream composition (I = 1,N). Contribution from temperature dependence of UNIQUAC binary interaction parameters, here taken as 0. 

In many applications the phase stmcture as a function of the temperature is of interest. The discussion of this issue requires the knowledge of the temperature dependence of the Flory-Huggins parameter x (T). If the interactions... 

In order to Introduce thermal effects into the theory, the material balance equations developed in this chapter must be supplemented by a further equation representing the condition of enthalpy balance. This matches the extra dependent variable, namely temperature. Care must also be taken to account properly for the temperature dependence of certain parameters In... 

Although the Arrhenius equation does not predict rate constants without parameters obtained from another source, it does predict the temperature dependence of reaction rates. The Arrhenius parameters are often obtained from experimental kinetics results since these are an easy way to compare reaction kinetics. The Arrhenius equation is also often used to describe chemical kinetics in computational fluid dynamics programs for the purposes of designing chemical manufacturing equipment, such as flow reactors. Many computational predictions are based on computing the Arrhenius parameters. 

For the same polymer this parameter has values of 4.47 X 10" and 5.01 X 10 " kg sec" at 298 and 398 K, respectively. Since density is far less sensitive to temperature, these results show that the primary temperature dependence of viscosity is described by the temperature dependence of f. 

Cullinan presented an extension of Cussler s cluster diffusion the-oiy. His method accurately accounts for composition and temperature dependence of diffusivity. It is novel in that it contains no adjustable constants, and it relates transport properties and solution thermodynamics. This equation has been tested for six very different mixtures by Rollins and Knaebel, and it was found to agree remarkably well with data for most conditions, considering the absence of adjustable parameters. In the dilute region (of either A or B), there are systematic errors probably caused by the breakdown of certain implicit assumptions (that nevertheless appear to be generally vahd at higher concentrations). 

Figure 9.6. (a) The temperature dependence of the flow stress for a Ni-Cr-AI superalloy containing different volume fractions of y (after Beardmore et al. 1969). (b) Influence of lattice parameter mismatch, in kX (eflectively equivalent to A) on creep rupture life (after Mirkin and Kancheev... 

The temperature dependence of A predicted by Eq. (5-11) makes a very weak contribution to the temperature dependence of the rate constant, which is dominated by the exponential term. It is, therefore, not feasible to establish, on the basis of temperature studies of the rate constant, whether the predicted dependence of A is observed experimentally. Uncertainties in estimates of A tend to be quite large because this parameter is, in effect, determined by a long extrapolation of the Arrhenius plot to 1/T = 0. 

Figure 3. The temperature dependence of the crystal distortion for VjSi, (0), derived from the data in Figure 2 [curves (a) and (c)], as discussed in reference 5. The dot-dashed curve shows the same distortion parameter for In-26.5 at%Tl derived from the data in Figure 1. The inset shows in detail the data below the super-c-nducting critical temperature, Tc. (From reference 5)...

Fig. 102 shows the temperature dependence of K5Nb30Fi8 and Rb5Nb30Flg unit cell parameters [438]. Parameter a increases with the increase in temperature, while parameter c displays a maximum at a temperature of about 490-500K. The changes in parameter values with respect to the temperature are reversible. 

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