Isotherms exponent

Isotherm exponent flow ratio in TMB or SMB systems [Eq. (16-207)] Molecular mass, kg/kmol Mass of adsorbent, kg... 

The borderline dimensionality, d. is the hi iest at whidi the d-dependent exponent relations, such as (9.58), (9.59), and (9.61), hold, and the lowest at whidi the exponents have thdr dassical values. Thus, d can always be found as the solution d of the algebraic equation that results from setting the exponents in any of ffie d-dependent exponent relations equal to their dassical values. For example, setting v= and a = 0 in (9.58), or t =0 and 5 = 3 in (9.59), or v= and i. = in (9.61), yidds d = 4. Since t = 0 classically, it follows from (9.59) that the borderline dimensionality d is related to tiie dassical value 5 of the critical-isotherm exponent 5 by... 

The intercept on the adsorption axis, and also the value of c, diminishes as the amount of retained nonane increases (Table 4.7). The very high value of c (>10 ) for the starting material could in principle be explained by adsorption either in micropores or on active sites such as exposed Ti cations produced by dehydration but, as shown in earlier work, the latter kind of adsorption would result in isotherms of quite different shape, and can be ruled out. The negative intercept obtained with the 25°C-outgassed sample (Fig. 4.14 curve (D)) is a mathematical consequence of the reduced adsorption at low relative pressure which in expressed in the low c-value (c = 13). It is most probably accounted for by the presence of adsorbed nonane on the external surface which was not removed at 25°C but only at I50°C. (The Frenkel-Halsey-Hill exponent (p. 90) for the multilayer region of the 25°C-outgassed sample was only 1 -9 as compared with 2-61 for the standard rutile, and 2-38 for the 150°C-outgassed sample). 

When n = 1, the compression is isothermal when n=k, it is adiabatic. The slope of the compression curve is a function of the exponent n. Figure 12-60 illustrates the effect of the n and k values on the gas compression and the work associated with this compression. 

P the total pressure, aHj the mole fraction of hydrogen in the gas phase, and vHj the stoichiometric coefficient of hydrogen. It is assumed that the hydrogen concentration at the catalyst surface is in equilibrium with the hydrogen concentration in the liquid and is related to this through a Freundlich isotherm with the exponent a. The quantity Hj is related to co by stoichiometry, and Eg and Ag are related to - co because the reaction is accompanied by reduction of the gas-phase volume. The corresponding relationships are introduced into Eqs. (7)-(9), and these equations are solved by analog computation. 

The form of equations (8.11) and (8.12) turns out to be general for properties near a critical point. In the vicinity of this point, the value of many thermodynamic properties at T becomes proportional to some power of (Tc - T). The exponents which appear in equations such as (8.11) and (8.12) are referred to as critical exponents. The exponent 6 = 0.32 0.01 describes the temperature behavior of molar volume and density as well as other properties, while other properties such as heat capacity and isothermal compressibility are described by other critical exponents. A significant scientific achievement of the 20th century was the observation of the nonanalytic behavior of thermodynamic properties near the critical point and the recognition that the various critical exponents are related to one another ... 

Results of adsorption experiments for butylate, alachlor, and metolachlor in Keeton soil at 10, 19, and 30°C were plotted using the Freundlich equation. A summary of the coefficients obtained from the Freundlich equation for these experiments is presented in TABLE IV. Excellent correlation using the Freundlich equation over the concentration ranges studied (four orders of magnitude) is indicated by the r values of 0.99. The n exponent from the Freundlich equation indicates the extent of linearity of the adsorption isotherm in the concentration range studied. If n = 1 then adsorption is constant at all concentrations studied (the adsorption isotherm is linear) and K is equivalent to the distribution coefficient between the soil and water (Kd), which is the ratio of the soil concentration (mole/kg) to the solution concentration (mole/L). A value of n > 1 indicates that as the solution concentration increases the sorption sites become saturated, resulting in a disproportionate amount of chemical being dissolved. Since n is nearly equal to 1 in these studies, the adsorption isotherms are nearly linear and the values for Kd (shown in TABLE IV) correspond closely to K. These Kd values were used to calculate heats of adsorption (AH). 

In the case of adiabatic flow we use Eqs. (9-1) and (9-3) to eliminate density and temperature from Eq. (9-15). This can be called the locally isentropic approach, because the friction loss is still included in the energy balance. Actual flow conditions are often somewhere between isothermal and adiabatic, in which case the flow behavior can be described by the isentropic equations, with the isentropic constant k replaced by a polytropic constant (or isentropic exponent ) y, where 1 < y < k, as is done for compressors. (The isothermal condition corresponds to y= 1, whereas truly isentropic flow corresponds to y = k.) This same approach can be used for some non-ideal gases by using a variable isentropic exponent for k (e.g., for steam, see Fig. C-l). 

Just as for isothermal flow, this is an implicit expression for the choke pressure (P ) as a function of the upstream pressure (Pi), the loss coefficients (J] Kf), and the isentropic exponent (7c), which is most easily solved by iteration. It is very important to realize that once the pressure at the end of the pipe falls to P and choked flow occurs, all of the conditions within the pipe (G = G, P2 = P, etc.) will remain the same regardless of how low the pressure outside the end of the pipe falls. The pressure drop within the pipe (which determines the flow rate) is always Pt — P when the flow is choked. 

The Freundlich isotherm (or Freundlich model) is an empirical description of species sorption similar to the K, approach, but differing in how the ratio of sorbed to dissolved mass is computed. In the model, dissolved mass, the denominator in the ratio, is raised to an exponent less than one. The ratio, represented by the Freundlich coefficient Kf, is taken to be constant, as is the exponent, denoted f, where 0< f <1. As before, the masses of dissolved and sorbed species are entered, respectively, in units such as moles per gram of dry sediment and moles per cm3 fluid. Since the denominator is raised to an arbitrary exponent rif, the units for Kf are not commonly reported, and care must be taken to note the units in which the ratio was determined. 

The effect of the exponent rtf is to predict progressively less effective sorption as concentration increases. Where f approaches one, the isotherm reverts to the K model, in which sorption is equally effective at any concentration. For n less than one, a smaller fraction of a component sorbs at high than at low concentration. This effect is taken as reflecting a heterogeneous distribution of sorbing sites in... 

To confirm the shape of the spherulites described by the Avrami exponent, polarized optical micrographs of the isothermal crystallized melt blends were taken, and are shown in Figure 20.26 [44],... 

We review Monte Carlo calculations of phase transitions and ordering behavior in lattice gas models of adsorbed layers on surfaces. The technical aspects of Monte Carlo methods are briefly summarized and results for a wide variety of models are described. Included are calculations of internal energies and order parameters for these models as a function of temperature and coverage along with adsorption isotherms and dynamic quantities such as self-diffusion constants. We also show results which are applicable to the interpretation of experimental data on physical systems such as H on Pd(lOO) and H on Fe(110). Other studies which are presented address fundamental theoretical questions about the nature of phase transitions in a two-dimensional geometry such as the existence of Kosterlitz-Thouless transitions or the nature of dynamic critical exponents. Lastly, we briefly mention multilayer adsorption and wetting phenomena and touch on the kinetics of domain growth at surfaces. 

Cv Pc 8 a an P Y c f specific heat reduced density critical exponent for the critical isotherm critical exponent for the specific heat critical exponent for the specific heat along isot r critical exponent for the order parameter critical exponent for the susceptibility reduced temperature friction coefficient... 

Huang et al. (1997) measured sorption isotherms for phenanthrene on 21 soils and sediments. All isotherms were nonlinear with Freundlich exponents n,- (Eq. 9-1) between 0.65 and 0.9. For example, for a topsoil (Chelsea 1) and for a lake sediment (EPA-23), interpolating the isotherm data yields the following observed sorbed concentrations, Cis, in equilibrium with dissolved concentrations, Ciw, of 1 /zg L-1 and 100 /zg-L-1, respectively ... 

The exponent n usually is less than unity. Both gas and liquid adsorption data are fitted by the Freundlich isotherm. Many liquid data are fitted thus in a compilation of Landolt-Bornstein (II/3, Numerical Data and Functional Relationships in Science and Technology, Springer, New York, 1956, pp. 525-528), but their gas... 

When applied to the Freundlich isotherm, q = (7p1//n, Ya.B. s conception leads to an unusually simple conclusion the exponent is directly proportional to the absolute temperature,... 

The exponent 1/n of isotherm (1) is linearly dependent on the temperature. At sufficiently high temperatures when, by (18), 1/n > 1, the theory gives simply a linear course of the isotherm at the beginning an isotherm of the form q = Cp1/", 1/n > 1, cannot be obtained by superposition of the Langmuir isotherm. Indeed, in this case a(b) still has the form of equation (9) with 1/n > 1, but we may no longer seek the asymptotic form of the beginning of the isotherm by formulas (11) and (12) since... 

In the literature there have been repeated reports on an apparent mean-field-like critical behavior of such ternary systems. To our knowledge, this has first been noted by Bulavin and Oleinikova in work performed in the former Soviet Union [162], which only more recently became accessible to a greater community [163], The authors measured and analyzed refractive index data along a near-critical isotherm of the system 3-methylpyridine (3-MP) + water -I- NaCl. The shape of the refractive index isotherm is determined by the exponent <5. Bulavin and Oleinikova found the mean-field value <5 = 3 (cf. Table I). Viscosity data for the same system indicate an Ising-like exponent, but a shrinking of the asymptotic range by added NaCl [164],... 

The Freundlich equation is similar to a linear equation, expect for the presence of the exponent n. For linear distributions, n = 1. With Freundlich isotherms usually have n < 1, which causes the adsorption isotherm to curve downward at higher concentrations as the readily available adsorption sites are filled and lower proportions of the arsenic from the aqueous solutions are adsorbed (Figure 2.8). The distribution coefficient for a Freundlich isotherm is often written as Kt to stress that the isotherm is not linear (Drever, 1997), 89. 

---