Concentration scale

The Landolt reaction (iodate + reductant) is prototypical of an autocatalytic clock reaction. During the induction period, the absence of the feedback species (Irere iodide ion, assumed to have virtually zero initial concentration and fomred from the reactant iodate only via very slow initiation steps) causes the reaction mixture to become kinetically frozen . There is reaction, but the intemiediate species evolve on concentration scales many orders of magnitude less than those of the reactant. The induction period depends on the initial concentrations of the major reactants in a maimer predicted by integrating the overall rate cubic autocatalytic rate law, given in section A3.14.1.1. 

Ramsden J J 1993 Concentration scaling of protein deposition kinetics Phys. Rev. Lett. 71 295-8... 

Fig. 4.1. Variation of NO2+ ion concentration with the concentration of mixed acid (nitric sulphuric, i mole i mole) inorganic solvents (a) in sulpholan (6)in aceticacid (c) in nitromethane. Curves (a) and (6) were determined by Raman measurements using the 1400 cm band while curve (c) was derived from infra-red measurements on the 237s cm band. Unity on the NO2+ concentration scale was determined to be 5-6 molar ( 2S 8 weight %). (From Olah et...

The numerical values of AG and A5 depend upon the choice of standard states in solution kinetics the molar concentration scale is usually used. Notice (Eq. 5-43) that in transition state theory the temperature dependence of the rate constant is accounted for principally by the temperature dependence of an equilibrium constant. 

The standard state chosen for the calculation of controls its magnitude and even its sign. The standard state is established when the concentration scale is selected. For most solution kinetic work the molar concentration scale is used, so A values reported by different workers are usually comparable. Nevertheless, an important chemical question is implied Because the sign of AS may depend upon the concentration scale used for the evaluation of the rate constant, which concentration scale should be used when A is to serve as a mechanistic criterion The same question appears in studies of equilibria. The answer (if there is a single answer) is not known, though some analyses of the problem have been made. Further discussion of this issue is given in Section 6.1. 

A first-order rate constant has the dimension time, but all other rate constants include a concentration unit. It follows that a change of concentration scale results in a change in the magnitude of such a rate constant. From the equilibrium assumption of transition state theory we developed these equations in Chapter 5 ... 

These apply to a bimolecular reaction in which two reactant molecules become a single particle in the transition state. It is evident from Eqs. (6-20) and (6-21) that a change in concentration scale will result in a change in the magnitude of AG. An Arrhenius plot is, in effect, a plot of AG against 1/T. Because a change in concentration scale alters the intercept but not the slope of an Arrhenius plot, we conclude that the values of AG and A, but not of A//, depend upon the concentration scale employed for the expression of reactant concentrations. We, therefore, wish to know which concentration scale is the preferred one in the context of mechanistic interpretation, particularly of AS values. 

In very dilute solutions of solute i, the several concentration scales become proportional to each other as expressed in Eqs. (6-25) and (6-26). 

For a substance in a given system the chemical potential gi has a definite value however, the standard potentials and activity coefficients have different values in these three equations. Therefore, the selection of a concentration scale in effect determines the standard state. 

We suppose that the molar concentration scale is used brackets signify concentration, and parentheses signify conventional activity as determined by pH measurements. First, consider an acid-catalyzed reaction having the rate equation... 

It is obvious, and verified by experiment [73], that above a critical trap concentration the mobility increases with concentration. This is due to the onset of intertrap transfer that alleviates thermal detrapping of a carrier as a necessary step for charge transport. The simulation results presented in Figure 12-22 are in accord with this notion. The data for p(c) at ,=0.195 eV, i.e. EJa—T), pass through a minimum at a trap concentration c—10. Location of the minimum on a concentration scale depends, of course, on , since the competition between thermal detrapping and inter-trap transport scales exponentially with ,. The field dependence of the mobility in a trap containing system characterized by an effective width aeff is similar to that of a trap-free system with the same width of the DOS. 

Figure 4.21. Residuals for linear (left) and quadratic (right) regressions the ordinates are scaled +20 mAU. Note the increase in variance toward higher concentrations (heteroscedacity). The gray line was plotted as the difference between the quadratic and the linear regression curves. Concentration scale 0-25 /ag/ml, final dilution.

To establish the operational pH scale, the pH electrode can be cahbrated with a single aqueous pH 7.00 phosphate buffer, with the ideal Nernst slope assumed. As Eqs. (2a)-(2d) require the free hydrogen ion concentration, an addihonal electrode standardization step is necessary. That is where the operational scale is converted to the concentration scale pcH (= -log [H ]) as described by Avdeef and Bucher [24] ... 

Figure 5.2 Concentration-response plot for an enzyme inhibitor displayed on linear (A) and logarithmic (B) concentration scales. The IC50 is identified from the midpoint (i.e., fractional activity = 0.5) of the semilog plot.

Because of the electroneutrality condition, the individual ion activities and activity coefficients cannot be measured without additional extrather-modynamic assumptions (Section 1.3). Thus, mean quantities are defined for dissolved electrolytes, for all concentration scales. E.g., for a solution of a single strong binary electrolyte as... 

It is assumed that the activities are based on the same concentration scale and that limCr 0fllVc/ = 1. For a tHCl(s) = a HCl(w), the difference between E(s) and E(w) is given by the relationship... 

Figure 12.19 shows the implication of concentration taken as a primary constraints and time taken as a secondary constraints. The concentration scale represents the maximum inlet and outlet concentrations in the given processes. This concentration is increasing in the direction of the arrow on the scale. Processes A and D precede processes B and C, respectively. This implies that the secondary constraints (time) is satisfied in both case (i) and case (ii). Moreover, water from process A has concentration less than the maximum concentration allowed in process B. Therefore, in case (i) both the concentration and time constraints are met and water from process A can safely be reused in process B. However, water from process D has concentration higher than the maximum concentration allowed in process C, which implies that the primary constraints (concentration) is not met. Therefore, water from process D cannot be reused in process C, although the secondary constraints (time) is met. 

The number of calibration points, p, their distance and measure at the concentration scale, the number of replicate measurements,... 

The term p[H] represents the pH value of the solution expressed on a concentration scale [5], where [H] represents the concentration of hydrogen ions H+. P° represents the partition coefficient of the unprotonated species, P1 is the mono-protonated species, and so on. P may also be expressed as... 

A major advantage of the simple model described in this paper lies in its potential applicability to the direct evaluation of experimental data. Unfortunately, it is clear from the form of the typical isotherms, especially those for high polymers (large n) that, even with a simple model, this presents considerable difficulty. The problems can be seen clearly by consideration of some typical polymer adsorption data. Experimental isotherms for the adsorption of commercial polymer flocculants on a kaolin clay are shown in Figure 4. These data were obtained, in the usual way, by determination of residual polymer concentrations after equilibration with the solid. In general, such methods are limited at both extremes of the concentration scale. Serious errors arise at low concentration due to loss in precision of the analytical technique and at high concentration because the amount adsorbed is determined by the difference between two large numbers. 

Fig. 10. DLTS spectra for MBE-grown GaAs. The DLTS time constant is r = 8 ms and the filling pulse width 50 ps. The top spectrum is for as-grown material. Lower spectra were taken following hydrogen plasma exposure and subsequent annealing as indicated. A concentration scale applicable to all four spectra is shown in the upper right. (Dautremont-Smith et al.,...

Both approaches lead to identical standard thermodynamic values of exchange (9-10). Such a difference in the choice of the surface concentration scale is of course only important for heterovalent exchange equilibria. For the heterovalent case the numerical value for both selectivity coefficients, Kg (Gaines Thomas) and (Vanselov) differ and, consequently, their variation with surface composition also differs. 

A graph is plotted between the concentration values and the ratios obtained from the physical value (i.e., peak area of absorption) of the internal standard and the series of known concentrations, thereby producing a straight line. Any unknown concentration may be determined effectively by adding the same amount of internal standard and locating exactly where the ratio obtained falls on the concentration scale. 

Equation A1.3 shows that isotope effects calculated from standard state free energy differences, and this includes theoretical calculations of isotope effects from the partition functions, are not directly proportional to the measured (or predicted) isotope effects on the logarithm of the isotopic pressure ratios. Rather they must be corrected by the isotopic ratio of activity coefficients. At elevated pressures the correction term can be significant, and in the critical region it may even predominate. Similar considerations apply in the condensed phase except the fugacity ratios which define Kf are replaced by activity ratios, a = Y X and a = y C , for the mole fraction or molar concentration scales respectively. In either case corrections for nonideality, II (Yi)Vi, arising from isotope effects on the activity coefficients can be considerable. Further details are found in standard thermodynamic texts and in Chapter 5. 

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