Isotherm temperature independent

In the case of [Fe(Cp )2]2[Cu(dcdmp)2], yj is nearly temperature independent, decreasing slightly at low temperatures (T < 30 K). This suggests that the magnetic behavior is due to weak AF D+A interactions. The isothermal obtained at 1.7 K, presents magnetization values slightly below those predicted by the Brillouin function, confirming the presence of very weak AF interactions [63]. 

Method. The validity of Equation 7 for experimental isotherms at various temperatures can be demonstrated without any assumption concerning the dependance of p on T. The test consists, at each temperature, in looking for a value of qm giving a temperature-independent straight line in a plot In (q/(qm — q)) vs. e, according to ... 

We have mentioned above the tendency of atoms to preserve their coordination in solid state processes. This suggests that the diffusionless transformation tries to preserve close-packed planes and close-packed directions in both the parent and the martensite structure. For the example of the Bain-transformation this then means that 111) -> 011). (J = martensite) and <111> -. Obviously, the main question in this context is how to conduct the transformation (= advancement of the p/P boundary) and ensure that on a macroscopic scale the growth (habit) plane is undistorted (invariant). In addition, once nucleation has occurred, the observed high transformation velocity (nearly sound velocity) has to be explained. Isothermal martensitic transformations may well need a long time before significant volume fractions of P are transformed into / . This does not contradict the high interface velocity, but merely stresses the sluggish nucleation kinetics. The interface velocity is essentially temperature-independent since no thermal activation is necessary. 

Repeat Exercise 5.3.4 when the two-dimensional equation of state - RTr/(l - br) + ar2 applies, where a and b are temperature independent. Also specify the adsorption isotherm that corresponds to this equation of state. What is the analogue of this equation of state in three dimensions ... 

The fugacity equations are solved using absolute variables. As discussed previously, the single-gas isotherms at sub-atmospheric pressure provide the absolute isotherm in the form fiirif). Given the temperature of the isotherms, the independent variables are P and yi in the bulk gas. For a binary mixture the fugacity equations are written ... 

Single-component adsorption equilibria on activated carbon of the n-alkanes Q-C4 and of the odorant tert-butyl mercaptan were measured at the operating conditions expected in a large-scale facility for adsorbed natural gas (ANG) storage. The experimental data were correlated successfully with the Adsorption Potential theory and collapsed into a single temperature-independent characteristic curve. The obtained isotherm model should prove to be very useful for predicting the adsorption capacity of an ANG storage tank and to size and optimize the operation of a carbon-based filter for ANG applications. 

Figure 1 displays the experimental characteristic curve obtained and the p values employed to generate it. The existence of very little scatter in the data demonstrates that the isotherms of the various adsorbates were successfully correlated as a single temperature-independent characteristic curve. This fact corroborates the applicability of the Potential theory to the carbon under study. 

It is often desirable to operate the reactor and the catalyst under isothermal conditions to achieve high reactor performance. Heat requirement of an endothermic reaction in a membrane reactor to maintain an isothermal condition can be challenging as in most of the dehydrogenation reactions such as conversions of ethylbenzene to styrene and prc pane to propylene. Maintaining an isothermal condition implies that some means must be provided to make the adequate heat input (e.g., from a burner) that is longitudinally dependent It is not trivial to make the temperature independent of the longitudinal position because the permeate flow also varies with the location in the axial direction. 

Intuitively, surface and interfacial tensions may be expected to be related to a number of physical characteristics of the liquid or the liquid-vapour transition. Two of these are the enthalpy and entropy of evaporation, discussed in sec. 2.9. Other parameters that come to mind are the molar volume V, the isothermal compressibility and the expansion coefficient. The combination of certain powers of such parameters and y sometimes leads to products with interesting properties, like temperature independence or additivity. Severed of such scaling rules have been proposed over the past century, mostly with limited quantitative success. A few of these wUl now be discussed. 

The pure component adsorption equilibrium of ethane and propane are measured on Norit AC at three temperatures (30, 60 and 90 °C). All experimental data of two species at three temperatures are employed simultaneously to fit the isotherm equation to extract the isothermal parameters. Since an extended Langmuir equation is used to describe the local multicomponent isotherm, the maximum adsorbed capacity is forced to be the same for ethane and propane in order to satisfy the thermodynamic consistency. The saturation capacity was assumed to be temperature dependent while the other parameters, bo and u], are temperature independent but species dependent. The derived isotherm parameters for ethane and propane are tabulated in Table 1. The experimental data (symbols) and the model fittings (solid lines)... 

The correlation of data for the methane and pentane binary system is shown in Figure 4.2.1. In this case the van Laar excess Gibbs free-energy model has been used in the HVO model the two model parameters were fitted to VLE data on the 277 K isotherm, and the vapor-liquid equilibria at higher temperatures were predicted with the same temperature-independent parameters. The results are very good in this case and similar to those obtained with the IPVDW and 2PVDW models. 

Several aspects of the adsorption of self-associating molecules in microporous structures have been developed by Talu and Meunier [32], Their approach is similar to that of the chemical interpretation of nonideality of vapor and liquid phases. The theory leads to type V isotherm behavior and can explain the transition between types I and V. The data at one temperature are represented by three parameters Henry s law constant, saturation capacity, and reaction constant for cluster formation in the micropores. To describe a set of isotherms obtained over a certain temperature interval, the theory can be used with five temperature-independent parameters to determine the entire phase behavior including the heat of adsorption. Besides obtaining a good agreement with experimental data, they found that the dimerization enthalpy of water in the micropores is lower than that in the vapor phase. 

Since these spreading isotherms are independent of temperature, the coefficient of the first degree term is independent of temperature. Besides confirming the presence on the surface of a form different from that obtained with solvent I, the interaction energies between polymeric segments are smaller than in the preceding case. From earlier observations and the value of the first degree coefficient of the two-dimensional state equation (Table I) which is equal to or less than that of the a form (37), it seems improbable that the random coil form exists on the surface. A helical form more extended than the a one—possibly a p form—appears to be more sound. 

THE/EDDy DIFFUSIVITY OF HEAT. When there is no temperature gradient across the isothermal surface, all eddies have the same temperature independent of the point of origin, dT/dy = 0, and no net heat flow occurs. If a temperature gradient exists, an analysis equivalent to that leading to Eq. (3.17) shows that the eddies carry a net heat flux from the higher temperature to the lower, in accordance with the equation... 

For an isothermal process the Elovichian equation is still valid for a situation in which there is an exponential variation of the number of sites on the homogeneous patches and also a linear variation of activation energy with patch number, i.e., (i) Us = rio ex-p( bs) and (ii) Es = Eo- - as, where s is the reference number of the patch and a, b are temperature-independent constants. 

For a given adsorbent-adsorbate system the relationship between s and the fractional occupation of the micropore volume is defined by the characteristic curve , which is assumed to be independent of temperature (see Fig. 5). Such an assumption should be valid for systems dominated by dispersion-repulsion forces (which are temperature independent), but cannot be expected to hold when electrostatic forces (which are temperature dependent) are important. The characteristic curve generally has a Gaussian form leading to an isotherm of the form ... 

If the sorption isotherm is temperature-independent the heat of sorption is zero therefore a number of sorption isotherm equations used in agricultural sciences are useless from the point of view of dryer calculations unless drying is isothermal. It is noteworthy that in the model equations derived in this section the heat of sorption is neglected, but it can easily be added by introducing Equation 3.59 for the solid enthalpy in energy balances of the solid phase. 

The isothermal temperature coefficients were measured by taking the difference of reactivity at approximately 250 and 370°C during reactor start-up. The measured isothermal temperature coefficients were constant through the MK-II operation because they were determined mainly by radial expansion of the core support plate, which is independent of bumup. However, when the core region was gradually extended from the 32 cycle, the isothermal temperature coefficients were decreased as predicted with the mechanism. Table 4 shows these values. 

In practice almost exclusively VLE data are used to fit the required parameters. Since a distillation column works nearly at constant pressure, most chemical engineers prefer thermodynamically consistent isobaric VLE-data in contrast to isothermal VLE-data to fit the model parameters. But that can cause problems, in particular if the boiling points of the two compounds considered are very different [24], as for example, for the binary system ethanol-n-decane. The result of the Wilson equation after fitting temperature-independent binary parameters only to reliable isobaric data at 1 atm is shown in Figure 5.31 for the system ethanol-n-decane, where the sum of the relative deviations of the activity coefficients was used as objective function. 

When mass transfer resistance is important the dynamic behavior of an adsorption column can generally be predicted only by numerical solution of the coupled differential heat and mass balance equations which describe the system. However, in the extreme case of an irreversible isotherm with independent of temperature, the differential mass balance and mass transfer... 

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