Equilibrium time example

The presence of color in many industrial effluent streams is highly undesirable. LDHs have been found to be particiflarly effective at removing various synthetic dyes (Table 1) [158]. For example. Acid Blue 29 could be adsorbed on the surface or enter the interlayer region of the LDH by anion exchange an equilibrium time of 1 h with 99 % dye removal was obtained. Furthermore,... 

Some ketones such as /3-dicarbonyls contain substantial amounts of the enol at equilibrium. For example, acetylacetone in aqueous solutions contains 13% of 4-hydroxypent-3-en-2-one, which is stabilized both by an intramolecular hydrogen bond and the inductive effect of the remaining carbonyl group.17 When bromine is added to such a solution, a portion is initially consumed very rapidly by the enol that is already present at equilibrium. The ketone remaining after consumption of the enol reacts more slowly via rate-determining enolization. The slow consumption of bromine is readily measured by optical absorption. In acidic solutions containing a large excess of the ketone the slow reaction follows a zero-order rate law the rate is independent of bromine concentration, because any enol formed is rapidly trapped by bromine (Scheme 1). In this case, the amount of enol present at equilibrium may be determined as the difference between the amount of bromine added and that determined by extrapolation of the observed rate law to time zero, as is shown schematically in Fig. 2. 

Since thermodynamics deals with systems at equilibrium, time is not a thermodynamic coordinate. One can calculate, for example, that if benzene(equilibrium with hydrogen(g) and carbon(s) at 298.15 K, then there would be very little benzene present since the equilibrium constant for the formation of benzene is 1.67 x 10-22. The equilibrium constant for the formation of diamond(s) from carbon(s, graphite) at 298.15 K is 0.310 that is, graphite is more stable than diamond. As a final example, the equilibrium constant for the following reaction at 298.15 K is 2.24 x 10-37 ... 

Skopp (1986) has noted that Eq. (2.5) or (2.6) alone, are only applicable far from equilibrium. For example, if one is studying adsorption reactions near equilibrium, back or reverse reactions are occurring as well. The complete expression for the time dependence must combine Eqs. (2.5) and (2.6) such that,... 

The equilibration time for the adsorption in some microporous materials, like CMS and carbonized chars, may be extremely long that may be a source of error for the evaluation of microporosity. For example, this occurs for N2 at 77 K in samples with narrow microporosity (size below 0.7 nm), where the size of the adsorbate molecule is similar to the size of the pore entrance. In this case, contrary to the exothermic nature of the adsorption process, an increase in the temperature of adsorption leads to an increase in the amount adsorbed. In this so-called activated diffusion process, the molecules will have insufficient kinetic energy, and the number of molecules entering the pores during the adsorption equilibrium time will increase with temperature [9,23],... 

At the same time, this means that near the equihbrium of the stepwise process under consideration, actually = 2 and hj = 1. More difficulties arise at the analysis of the stationary rate of the stepwise process in the sys tern that is far away from the equilibrium. For example, for a left to right direction of the process, equation (1.44) is simplified at R P ... 

Microscopic reversibility" as used in chemical kinetics is a classic misnomer. The name stems from a complicated derivation based on Onsager s axiom of reversibility at the molecular level [13-17] At that level there is no preferential direction of time and, therefore, all events are in principle reversible. What is called microscopic reversibility in chemical kinetics is the statement that there can be no net circular reaction in a loop at equilibrium. For example, at equilibrium there can be no net circulation A — B — C — D — A in the loop 2.9 ... 

Fig 1 Example of a typical N2-isotherm of Fig 2 Theoretical isotherms for in-an activated carbon with corres- creasing equilibrium times,... 

In classical physics we are familiar with another kind of stationary states, so-called steady states, for which observables are still constant in time however fluxes do exist. A system can asymptotically reach such a state when the boundary conditions are not compatible with equilibrium, for example, when it is put in contact with two heat reservoirs at different temperatures or matter reservoirs with different chemical potentials. Classical kinetic theory and nonequilibrium statistical mechanics deal with the relationships between given boundary conditions and the resulting steady-state fluxes. The time-independent formulation of scattering theory is in fact a quantum theory of a similar nature (see Section 2.10). 

A system that undergoes no net chemical changes over time is said to be in chemical equilibrium. Generally, this condition occurs when the forward and reverse rates of the chemical reactions taking place in a system are equal. Common phenomena, including many phenomena associated with living organisms, can be analyzed in terms of this concept of chemical equilibrium. Several examples are presented below. 

Internal (carrier) driving force coefficients, and K, <, or distribution coefficients, Ep and E, are determined by membrane-based extraction experiments. Membrane-based forward and backward extraction is carried out in two-compartment modules using the F and R compartments, separated by the same membranes as in the BAHLM tests. The experiments lasted up to equilibrium conditions, when the concentration of solutes in every compartment does not change with time. Examples of membrane-based extraction of copper, cadmium, and zinc from the concentrated phosphoric acid solution by PVSH and backward extraction by 2 M HCl are presented in Table 6.2 [7]. 

The pH of the solutions decrease with aging time until equilibrium pH values are ultimately achieved. The equilibrium pH values are consistent with known thermodynamic values for monomeric species and solubility products. The equilibrium pH achieved in a particular solution depends on its initial rn value. The lower the r value, the lower is the equilibrium pH. The rate at which equilibrium is approached depends on the rn value and also on the rate at which base is added during solution make-up. The lower the rn value, the slower is the progress toward equilibrium. For example, if rw value is around 2.8, equilibrium may be achieved in 6 or 8 months (depending on how much AP was initially formed during base addition). If rn value is 1.0 or less, equilibrium probably will not be achieved after several years aging. 

Molecules cannot move fast enough to keep the reaction as reported at equilibrium. For example, a recent paper on ammonia photosynthesis over oxide promoters by Augugliaro et al. [67] reports that ammonia was formed at 0.04 imol h-) in their system, and so this sets the minimal reverse rate at equilibrium. Nitrogen flowed into their system at 0.4 cm3 s-1 and so we can ask ourselves how fast ammonia molecules would have to move toward the surface of the catalyst in order to maintain the reverse rate. It transpires that the ammonia molecules would have to travel fast enough to cross the known universe (i.e., 2 x 1010 light years) about 1021 times in one second. This absurd result is intended only as another way of demonstrating that reaction 2, if it occurred, could not come to equilibrium. 

To compare equilibrium times evaluated by approximate Eq. (12.204) and the lower bound Eq. (12.209), the example of an ideal cascade to perform the separation of Table 12.8 will be considered. It is assumed, in addition, that the stage holdup time h is 1 s and the stage separation factor is 1.0043, the nominal value for separating from UF6. For this... 

This example shows that the equilibrium time in an ideal cascade with a — 1 < 1 may be relatively long, even when the stage holdup time h is very short. In a cascade that is not tapered at the product end, the equilibrium time will be even greater, because of the increased inventory of desired component in this part of the plant. Equation (12.197) may be used to estimate the equilibrium time of such a nonideal cascade Eq. (12.209) is restricted to ideal cascades. 

The relaxation of certain properties of the system can often be described by simple phenomenological equations called relaxation equations. In chemical kinetics, for example, the constrained state may be a mixture of gases in metastable equilibrium—for example, hydrogen and oxygen. A spark is then introduced and the gas mixture reacts. The concentration of the reactants and products change with time until a new equilibrium state is achieved. The relaxation equations are the familiar phenomenological equations of chemical kinetics and the relaxation times are related to the chemical rate constants. 

The attainment of equilibrium is important, especially for polymer blends that have low conductivity. The equilibrium times, for example, for cellular materials, are in the order of hours or tens of hours. For this reason, stable over long time-periods power supplies are necessary. 

Another laboratory observed a loss of Cd in the extract after an equilibrium time of 48 h which was, however, not confirmed by the other participants. This laboratory applied a second filtration step with a membrane filter (not included in the protocol), which could explain the losses observed. Losses due to precipitation were considered to be unlikely since the stability of the extracts over a period of 72 h was verified, providing that they were stored at 4°C. These examples demonstrated how essential it was that the extraction protocols be strictly followed, since a slight variation may lead to an incomparability of data. The protocol should state that the analysis be carried out with the samples as received and that the moisture content in the soil intake be corrected for by determining the loss of mass on drying a separate soil sample. 

Our analysis describes virus adsorption from the standpoint of chemical equilibrium. Since adsorption equilibrium appears to be approached closely in our systems in less than or equal to 2 hr, and since the residence time of viruses in natural water systems is greater than 2 hr for many cases (for example, lakes, groundwaters, rivers, etc.), equilibrium considerations are entirely appropriate. In other situations, where residence times of the virus in the system are small compared to expected times required for adsorption to approach equilibrium (for example, sand filters in water treatment, water distribution systems, etc.), the DLVO-Lifshitz theory may still be applied directly. The work of Fitzpatrick and Spielman (57) concerning filtration and that of Zeichner and Schowalter (58) concerning colloid stability in fiow fields demonstrate this clearly. Their developments of hydrodynamic trajectory analysis coupled to DLVO-Lifshitz considerations can be extended... 

Of the five main effects, the one for factor 3 (equilibrium time) is clearly secondary. As for the other four, we cannot discard the possibility that some of their values are mainly due to interactions. For example, the effect of stirring (4) is confounded with interaction 12 (sample volume and temperature) while the effect of pressurization (5) is confounded with the interaction of temperature and time (23). Actually,... 

In the experiment one often deals with large initial deviations from equilibrium for example, such is the case when a new oil-water interface is formed by the breaking of larger emulsion drops during emulsification. In the case of large perturbation there is no general analytical expression for the dynamic surface tension a(t) since the adsorption isotherms (except that of Henry, see Table 1) are nonlinear. In this case one can use eiflier a computer solution (44, 45) or apply the von Karman approximate approach (46,47). Analytical asymptotic expressions for the long time (t tj) relaxation of surface tension of a nonionic surfactant solution was obtained by Hansen (48) ... 

Solubility in an early discovery assay containing 1% DMSO can however exceed thermodynamic solubility by a lot more than 65%. However, this is very likely due to the time scale. Studies by the Avdeef (plon Inc) group show a close approximation of early discovery solubility (quantitated by UV) to literature thermodynamic solubility if the early discovery assay is allowed to approach equilibrium, for example by sitting overnight. The... 

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