Concentration, equilibrium

3 It is worthy of note that the introduction of a separate Boltzmann estimate for D and yields an equation that — owing to mass conservation — simulates Fermi-Dirac statistics which is valid over the entire concentration range for the building elements, i.e. the electrons in the acceptor term. The same would be true for the band-band transition, if we formulated Eq. (5.31) in terms of structure elements (see Eqs. (5.94c, 5.94d)). However here the correction would only be qualitatively correct (cf. Eq. (5.209) and Fig. 5.69). 

3 This is obtained, for instance, in the donor case from Ep = Ec + RTln (Boltzmaim approximation suffices, since No- = Ne — 0), Ep = Ep -I- RThr rj and Nd- = Ne. The result is Ep = -I- In 1 corresponding to Eq. (5.54). See also the semiconductor physics literature 

The standard value is the molar free enthalpy of grain boimdary formation ArG, i.e., the G-change during the reaction  

58) we have incorporated the factor 3 into ArG as RTln3. It does not play any role in the following order of magnitude consideration. 

If we compare this with the concentration of point defects N and make the statement that ArG is a times the free energy of formation of a single defect, then with Eq. (5.19) we get  

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However, when carboxylic acids are present in a mixture, fugacity coefficients must be calculated using the chemical theory. Chemical theory leads to a fugacity coefficient dependent on true equilibrium concentrations, as shown by Equation (3-13). ... 

Irreversible adsorption discussed in Section XI-3 poses a paradox. Consider, for example, curve 1 of Fig. XI-8, and for a particular system let the equilibrium concentration be 0.025 g/lOO cm, corresponding to a coverage, 6 of about 0.5. If the adsorption is irreversible, no desorption would occur on a small dilution on the other hand, more adsorption would occur if the concentration were increased. If adsorption is possible but not desorption, why does the adsorption stop at 6 = 0.5 instead of continuing up to 0 = 1 Comment on this paradox and on possible explanations. 

With equilibrium concentrations the (small) deviation from equilibrium is given by... 

Figure B2.5.7 shows the absorption traces of the methyl radical absorption as a fiinction of tune. At the time resolution considered, the appearance of CFt is practically instantaneous. Subsequently, CFl disappears by recombination (equation B2.5.28). At temperatures below 1500 K, the equilibrium concentration of CFt is negligible compared witli (left-hand trace) the recombination is complete. At temperatures above 1500 K (right-hand trace) the equilibrium concentration of CFt is appreciable, and thus the teclmique allows the detennination of botli the equilibrium constant and the recombination rate [54, M]. This experiment resolved a famous controversy on the temperature dependence of the recombination rate of methyl radicals. Wliile standard RRKM theories [, ] predicted an increase of the high-pressure recombination rate coefficient /r (7) by a factor of 10-30 between 300 K and 1400 K, the statistical-adiabatic-chaunel model predicts a...

The ortho- and meto-isomers are bulkier than the para-iaomer and diffuse less readily in the zeolite pores. The transport restriction favours their conversion into the /lara-isomer, which is fonned in excess of the equilibrium concentration. Because the selectivity is transport influenced, it is dependent on the path length for transport, which is the length of the zeolite crystallites. 

The vacancy is very mobile in many semiconductors. In Si, its activation energy for diffusion ranges from 0.18 to 0.45 eV depending on its charge state, that is, on the position of the Fenni level. Wlrile the equilibrium concentration of vacancies is rather low, many processing steps inject vacancies into the bulk ion implantation, electron irradiation, etching, the deposition of some thin films on the surface, such as Al contacts or nitride layers etc. Such non-equilibrium situations can greatly affect the mobility of impurities as vacancies flood the sample and trap interstitials. 

The distributions of excess, or injected, carriers are indicated in band diagrams by so-called quasi-Fenni levels for electrons or holes (Afp). These functions describe steady state concentrations of excess carriers in the same fonn as the equilibrium concentration. In equilibrium we have... 

If the equilibrium concentrations for A, B and C are a, b and c, respectively, the concentration changes resulting from the application of the perturbation will be... 

These equations, relating to oi,s, and E t,g to Egy, show that 3od can be calculated for a reaction proceeding through the equilibrium concentration of a free base if the thermodynamic quantities relating to the ionisation of the base, and the appropriate acidity function and its temperature coefficient are known (or alternatively, if the ionisation ratio and its temperature coefficient are known under the appropriate conditions for the base. )... 

Because of these difficulties, special mechanisms were proposed for the 4-nitrations of 2,6-lutidine i-oxide and quinoline i-oxide, and for the nitration of the weakly basic anilines.However, recent remeasurements of the temperature coefficient of Hq, and use of the new values in the above calculations reconciles experimental and calculated activation parameters and so removes difficulties in the way of accepting the mechanisms of nitration as involving the very small equilibrium concentrations of the free bases. Despite this resolution of the difficulty some problems about these reactions do remain, especially when the very short life times of the molecules of unprotonated amines in nitration solutions are considered... 

For the nitration of the very weak base, acetophenone, there is reasonable agreement between observed and calculated activation parameters, and there is no doubt that nitration of the free base occurs at acidities below that of maximum rate. In this case the equilibrium concentration of free base is much greater than in the examples just discussed and there is no question of reaction upon encounter. ... 

Since the first-order rate constant for nitration is proportional to y, the equilibrium concentration of nitronium ion, the above equations show the way in which the rate constant will vary with x, the stoichiometric concentration of dinitrogen tetroxide, in the two media. An adequate fit between theory and experiment was thus obtained. A significant feature of this analysis is that the weak anticatalysis in pure nitric acid, and the substantially stronger anticatalysis in partly aqueous nitric acid, do not require separate interpretations, as have been given for the similar observations concerning nitration in organic solvents. 

Substituting the equilibrium concentrations into the solubility product expression (equation 6.33)... 

How might we solve equation 6.34 if we do not have access to a computer One possibility is that we can apply our understanding of chemistry to simpKfy the algebra. From Le Chatelier s principle, we expect that the large initial concentration of Pb will significantly decrease the solubility of Pb(I03)2. In this case we can reasonably expect the equiKbrium concentration of Pb to be very close to its initial concentration thus, the following approximation for the equilibrium concentration of Pb seems reasonable... 

Before accepting this answer, we check to see if our approximation was reasonable. In this case the approximation 0.10 -l-x 0.10 seems reasonable since the difference between the two values is negligible. The equilibrium concentrations of Pb3+ and IO3-, therefore, are... 

Letting x equal the change in the concentration of Pb +, the equilibrium concentrations are... 

Chaston, S. Calculating Complex Equilibrium Concentrations by a Next Guess Factor Method, /. Chem. Educ. 1993, 70, 622-624. 

The solubility of AgCl, therefore, is the sum of the equilibrium concentrations for all soluble forms of Ag+. 

Substituting the equilibrium constant expressions for reactions 8.3-8.S into equation 8.6 defines the solubility of AgCl in terms of the equilibrium concentration of Ch. 

Clearly the equilibrium concentration of chloride is an important parameter if the concentration of silver is to be determined gravimetrically by precipitating AgCl. In particular, a large excess of chloride must be avoided. 

Equation 8.7 shows how the solubility of AgCl varies as a function of the equilibrium concentration of Ch. Derive a similar equation to describe the solubility of AgCl as a function of the equilibrium concentration of Ag+. Graph the resulting solubility function and compare it with that shown in Figure 8.1. 

To determine the equilibrium constant s value, we prepare a solution in which the reaction exists in a state of equilibrium and determine the equilibrium concentration of H3O+, HIn, and Im. The concentration of H3O+ is easily determined by measuring... 

Protonic acids dissociate to some extent in the nonaqueous reaction mixtures to produce an equilibrium concentration of protons ... 

Depending on the final polymerization conditions, an equilibrium concentration of monomers (ca 8%) and short-chain oligomers (ca 2%) remains (72). Prior to fiber spinning, most of the residual monomer is removed. In the conventional process, the molten polymer is extmded as a strand, solidified, cut into chip, washed to remove residual monomer, and dried. In some newer continuous processes, the excess monomer is removed from the molten polymer by vacuum stripping. 

Essential for synthesis considerations is the abiUty to determine the amount of ammonia present ia an equiUbrium mixture at various temperatures and pressures. ReHable data on equiUbrium mixtures for pressures ranging from 1,000 to 101,000 kPa (10 —1000 atm) were developed early on (6—8) and resulted ia the determination of the reaction equiUbrium constant (9). Experimental data iadicates that is dependent not only on temperature and pressure, but also upon the ratio of hydrogen and nitrogen present. Table 3 fists values for the ammonia equilibrium concentration calculated for a feed usiag a 3 1 hydrogen to nitrogen ratio and either 0 or 10% iaerts (10). 

Isomerization of sorbitol, D-mannitol, L-iditol, and dulcitol occurs in aqueous solution in the presence of hydrogen under pressure and a nickel—kieselguhr catalyst at 130—190°C (160). In the case of the first three, a quasiequiUbrium composition is obtained regardless of starting material. Equilibrium concentrations are 41.4% sorbitol, 31.5% D-mannitol, 26.5% L-iditol, and 0.6% dulcitol. In the presence of the same catalyst, the isohexides estabUsh an equihbrium at 220—240°C and 15.2 MPa (150 atm) of hydrogen pressure, having the composition 57% isoidide, 36% isosorbide, and 7% isomannide (161). 

The partition coefficient P, defined as the equilibrium concentration of the compound in n-octanol divided by that in the aqueous phase, has been measured for pyrazole and indazole (B-79MI40416). It was found that log F = 0.13-0.26 for pyrazole and 1.82 for indazole, clearly showing the greater hydrophobicity (lipophilicity) of the indazole ring, due to the benzenoid moiety. 

A theoretical or equihbrium stage is a device or combination of devices that accomplishes the effect of intimately mixing two immiscible liquids until equilibrium concentrations are reached, then physically separating the two phases into clear layers. Crosscurrent extraction (Fig. 15-4) is a cascade, or series of stages, in which the raffinate R from one extraction stage is contacted with additional fresh solvent S in a subsequent stage. 

Dry scrubber pollutant gas equilibrium concentration over sorbent ... 

Hundreds of metabohc reac tions take place simultaneously in cells. There are branched and parallel pathways, and a single biochemical may participate in sever distinct reactions. Through mass action, concentration changes caused by one reac tion may effect the kinetics and equilibrium concentrations of another. In order to prevent accumulation of too much of a biochemical, the product or an intermediate in the pathway may slow the production of an enzyme or may inhibit the ac tivation of enzymes regulating the pathway. This is termed feedback control and is shown in Fig. 24-1. More complicated examples are known where two biochemicals ac t in concert to inhibit an enzyme. As accumulation of excessive amounts of a certain biochemical may be the key to economic success, creating mutant cultures with defective metabolic controls has great value to the produc tion of a given produc t. 

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