Distribution isotherm, definition

The yield of a gas-solid heterogeneous reaction depends not on the total time that molecules spend in the reactor but on the time that they spend on the catalyst surface. The contact time distribution provides a standardized measure of times spent in the absorbed state. A functional definition is provided by the following equation applicable to a first-order, heterogeneous reaction in an isothermal reactor ... 

N2 adsorption-desorption isotherms and pore size distribution of sample II-IV are shown in Fig. 4. Its isotherm in Fig. 4a corresponds to a reversible type IV isotherm which is typical for mesoporous solids. Two definite steps occur at p/po = 0.18, and 0.3, which indicates the filling of the bimodal mesopores. Using the BJH procedure with the desorption isotherm, the pore diameter in Fig. 4a is approximately 1.74, and 2.5 nm. Furthermore, with the increasing of synthesis time, the isotherm in Fig. 4c presents the silicalite-1 material related to a reversible type I isotherm and mesoporous solids related to type IV isotherm, simultaneously. These isotherms reveals the gradual transition from type IV to type I. In addition, with the increase of microwave irradiation time, Fig. 4c shows a hysteresis loop indicating a partial disintegration of the mesopore structure. These results seem to show a gradual transformation... 

When two immiscible solvents are placed in contact, any substance soluble in both of them will distribute or partition between the two phases in a definite proportion. According to the Nernst partition isotherm, the following relationship holds for a solute partitioning between two phases a and b ... 

There seem to be no literature data potentially capable of showing definite consistency with Frenkel-Halsey type isotherms, with the exception of the results for nitrogen on titanium dioxide (2). Here we find that no site energy distribution can be constructed using such a local isotherm type which will lead to a fit over the entire range of data. The same appears to be true of local isotherm equations of the polarization theory type ... 

On the other hand, it is assumed in the stationary theory of the thermal explosion, on which the F-K equation is based, that the term, dT dt, in Eq. (20) takes the value of zero, but the term, A T, takes a definite value. In other words, as stated in Section 1.3, it is assumed in the latter theory that the temperature of a solid (powdery, in reality) chemical of the TD type, including every gas-permeable oxidatively-heating substance, having an arbitrary shape and an arbitrary size, placed in the atmosphere maintained at a temperature situated below the critical state for the thermal explosion, does not vary with time as a whole, but there is a spatially gradient distribution of temperature in the chemical (see the F-K model in Fig. 3 in Section 1.5). It must, thus, be admitted that the spatial distribution of temperature in a powdery chemical of the TD type, placed in the atmosphere under isothermal conditions, is a gradient one even in the early stages of the self-heating process. 

The Freundlich isotherm has no definite maximum adsorption or saturation value therefore, when 0 approaches a monolayer, the Freundlich equation necessarily breaks down. For high coverages the exponential distribution function is conveniently replaced by a linear distribution... 

The preceding discussion and Eqs. (2-l)-(2-5) assume that the sample distribution coefficient K is constant throughout separation. Quite frequently K varies with sample concentration, however. Separations in which K is independent of sample concentration are referred to as linear isotherm separations, from the definition of an isotherm as a plot of X versus (20 in an equilibrium system at a given temperature. Linearity, of the plot is equivalent to constant K. A nonlinear sample isotherm... 

The Langmuir isotherm is by definition free from interactions between adsorbate molecules (2) and surface heterogeneity (3). It is therefore an ideal model for illustrating the contribution of surface saturation (1) to isotherm nonlinearity. Beginning with the Langmuir isotherm for liquid-solid systems [Eq. (3-5a)], we can define a distribution coefficient K equal to S/iVa , so that ... 

Theoretically the use of the desorption branch of the isotherm is applicable for materials with pores predominandy comprised of independent capillaries, because this was the system used for the deduction of the Kelvin equation. If bottle-hke pores are anticipated, where the Hquid in the narrow opening prevents evaporation of the condensate from larger compartments, it is better to calculate pore size distribution using the adsorption branch of the hysteresis loop. Since the shape of pores is usually a priori unknown, the selection of the adsorption or desorption branch of the hysteresis loop largely remains arbitrary. Concerning the choice between dWo/dD or dfFo/dlog(D) plots, which by definition are different functions, one may note that the difference may be small for materials with narrow pore size distribution, but become more significant for a broad and unsymmetrical pore size distribution. 

We arrived at this definition after discovering that other techniques, such as calculating pore size distributions from the BJH method via nitrogen isotherms, do not account for a large portion of the total pore volumes within these types of materials. The amorphous structures seen in SEM images indicate a porosity that is difficult to describe mathematically, whereas the field of microscopy has given us tools that are readily available, which help us to quantify these structures. 

The type IV isotherms displayed by most clays are, in gena-al, amenable to surface area measurements using BET analysis, as long as attention is paid to catain constants in the BET equation (49). Furthermore, only the adsorption branch of the isotherm is reliably used to measure pore size distributions, which in most cases is quite broad. Finally, for the interested reader, guidelines for the charactCTization of porous solids containing both definitions and appropriate method selection are provided by Rouquerol et al. (54). 

For an isothermal, homogeneous system of constant density with well-defined boundaries and a single inlet and outlet stream, a residence time distribution, F(t), is defined as the probability that a fluid element has a residence time less than t. At steady state F(t) is the fraction of the fluid in the outflow that has resided in the vessel less than time t. Here we assume that each fluid element has a definite point of entry into the vessel and that age (residence time) is acquired by the fluid element only while it resides within the system s boundaries. 

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