Reciprocal initial concentration

The best correlation resulted from regression of three variables that combined four of the measured carrier properties. The variables, moisture divided by pOH, water absorption, and reciprocal initial concentration, resulted in the following equation ... 

Apparently the properties of moisture divided by pOH, reciprocal initial concentration, and water absorption as given in Equation 1 are only part of the story. Measurement of other properties, better able to indicate interactions between the surfaces of the toxicant and the carrier, might give significant correlations. These properties should probably include some function of available alkalinity and some function of the absorption capacity of the carrier and its degree of saturation. It is hoped that eventually someone will develop test methods for more useful properties of carriers that will correlate significantly with stability results. 

During the oxidation of glucose the concentration of H2O2 rises linearly with time at fixed oj. The initial velocity Vq can be calculated from the initial slope of the Id vs. time dependence. Later, the curve starts to show deviations from the linear behavior as a result of product inhibition or enzyme instabilization. Evaluation of rate constants k, and k2 proceeds obviously by plotting v vs. reciprocal initial concentrations of oxygen and glucose, respectively ... 

Fig. 8 Dependence of the forward rate constant of a reversible reaction (1) and an irreversible reaction (2) on the reciprocal initial concentration c (both reactant cone, were equal)...

It is well established that the release of electrolytes from multiple emulsions in the presence of monomeric emulsifiers takes place even if the droplets are very stable to coalescence and even if the osmotic pressure of the two phases has been equilibrated (Garti and Benichou, 2001 Garti and Benichou, 2003). The model adopted for the release of matter from polymeric matrices is the Stehle and Higuchi model (Stehle and Higuchi, 1972). The model was worked out and tested by a modified release equation. The release factor B was plotted against the time t, and reciprocal initial concentrations of the solute (1/Co). 

Fig. 14.—Reciprocal initial degree of polymerization plotted against the initial rate for undiluted styrene at 60°C using benzoyl peroxide. The dashed straight line corresponds to the line drawn through the points for low concentrations of initiator shown in the inset using an enlarged scale. (From the results of Mayo, Gregg, and Matheson. )...

Figure 6 shows plots of Mjj ol poly(MVE) versus the cumulative weight of added MVE, Wmve M s increase linearly with increasing Wmve and the lines pass through the origin. Mn is higher at a lower initiator concentration, [p-DCC]Q the slope of the plots is nearly proportional to the reciprocal of [p-DCC]0. 

Using a value of K (derived from Eq. 5.5) suggested that in the presence of ultrasound, and the absence of initiator, ultrasound was able to provide an equivalent initiator concentration of 6 x 10 mol dm. In general the work was important in that it showed a reciprocal volume dependence rather than a (l/Vh j dependence as shown by Kruus. It also showed that in the absence of initiator, it was possible to... 

The time (symbolized by t) needed for a concentration of a molecular entity to decrease, in a first-order decay process to e of its initial value. In this case, the lifetime (sometimes called mean lifetime) is equal to the reciprocal of the sum of rate constants for all concurrent first-order decompositions. If the process is not first-order, the term apparent lifetime should be used, and the initial concentration of the molecular entity should be provided. The terms lifetime and half-life should not be confused. See Half-Life Fluorescence... 

The rate of polymerization (in % conversion per hour) is plotted against the square root of the initiator concentration (in mol%), according to Eq. 3.6.The limiting viscosity numbers [ri] and hence the degrees of polymerization are also determined and plotted against the reciprocal square root of the initiator concentration. 

The potentially greater toxicity of peroxynitrite can be readily visualized by comparing the mean diffusion distances that various nitrogen and oxygen-centered species may traverse in one lifetime. The definition of lifetime (t) is the time required for 67% of the initial concentration to decompose, and is readily calculated as the reciprocal of the pseudo-first-order rate constant for the disappearance of the species in question. Distances were calculated from the following equation, which is readily derived from the Fick s laws of diffusion (Nobel, 1983 Pryor, 1992). 

Heat evolution starts after an induction period, the reciprocal of which increases linearly with the light intensity and is independent of the borate concentration. The initial rate of photopolymerization is proportional to the square root of the light intensity, and at constant intensity the rate of photopolymerization is independent of borate concentration. Based on these results we write Eq. (37) for the initial rate of photopolymerization, where [M]0 is the initial concentration of double bonds (10.8 M, Rj the rate of... 

Fig. 5. Lineweaver-Burk plot of the Orpinomyces BglA on hydrolysis of cellobiose (0.04-16 mM). The release of glucose was measured. Reciprocal initial velocities (mg/U) are plotted against the reciprocal concentrations of substrates (1 mM). The inset is a plot of initial velocities (U/mg) against the log cellobiose concentrations (mM).

The relations between escaping concentration and time, column length and breakdown time, and flow rate and breakdown time, as predicted by the theory, were found to be valid for the solid-gas reaction between activated silver permanganate and carbon monoxide. However, the predicted linear relation between critical column length and the logarithm of the initial concentration was not confirmed. Similarly, the reciprocal of the critical flow rate did not vary linearly with log Co. The critical column length increased with increasing flow rate but was unaffected by the initial concentration. 

Lifetime (t) The lifetime of a molecular entity which decays in a first-order process is the time needed for a concentration of the entity to decrease to 1/e of its original value. Statistically, it represents the life expectation of the entity. It is equal to the reciprocal of the sum of the (pseudo)unimolecular rate constants of all processes which cause the decay. Lifetime is used sometimes for processes which are not first order. However, in such cases, the lifetime depends on the initial concentration of the entity, or of a quencher and therefore only an initial or a mean lifetime can be defined. In this case it should be called apparent lifetime, instead. Occasionally, the term half-life (T1/2) is used, representing the time needed for the concentration of an entity to decrease to one half of its original val-... 

This last equation tells us that, in the absence of transfer agents other than M, the reciprocal mean degree of polymerization is a linear function of / 1V(M). Rt can, of course, be varied at constant (M) by using photoinitiation, whence Rtoz lay the absorbed light intensity. Alternatively, where chemical initiation is used, R cc (In), the initiator concentration. In this last case, transfer to initiator must be investigated separately in order to distinguish this effect from monomer transfer. 

In this analysis of first-order kinetics, any value can be plotted against time of reaction, unless the value is proportional to the molar amount of a substrate or product for example, absorbance of a compound at a given wavelength can be used even if the molar absorption (extinction) coefficient is unknown and thereby absolute concentration can not be determined. Moreover, taking the logarithm means a reciprocal value can also be used, e.g., [Ao]/[A], where [Aq] is initial concentration of a substrate A (See Fig. 6). 

Furthermore, the initial rate of the oxidation showed a saturation at high substrate ccMicentration (Fig. 4), characteristic of Michaelis — Menten type kinetics. Using the plot of the reciprocal initial rate versus the reciprocal substrate concentration (Fig. 7), the Michaelis constant (corre nding to the dissociation constant)... 

Fig. 7. Oxidation of DOPA catalyzed by poly(S-lysine) - copper(II) complex, reciprocal initial rate (Vo) versus reciprocal DOPA concentration. pH 10.5, 20 C, (Cul/[N1 = 0.13, ICu) = 2.0 X 10 M (a) RS-DOPA, (b) S-DOPA (29)...

This is an approximation because the OH concentration does change during the reaction, but since the change is not very great the equation is adequate to illustrate the importance of the two reaction mechanisms. Equation (4.31) is the solution for a reversible reaction that begins with an initial concentration of [C02]° and progresses toward an equilibrium value of [C02]° + BjA. The value represented in A is the reciprocal of the residence time of CO2 with respect to chemical reaction and incorporates both mechanisms of reaction. 

Let the integral transformation of Laplace-Carson be applied to system (50). The functions of time x t) can then be substituted by the x (q) transformation functions, and the time derivatives by the qx (g) — f/xlO) values, where q represents reciprocal time and the x(0) are the initial concentrations of the intermediates. The Laplace-Carson procedure transforms the system of differential equations (50) into a system of algebric equations with respect to X (q) thus ... 

The reactivity of the amine as a polymerization accelerator depends upon the a+ value of the meta- or para-substituent of the amine, where a+ is the electrophilic substituent parameter previously described and tabulated (23). When the kinetic rate constant (or the reciprocal of the polymerization time with amine and peroxide initial concentrations held constant) is plotted against the a+ value on a semi-logarithmic plot (Figures 1 and 2), an inverted "V" shaped curve results. The kinetic data for Figure 1 were taken from a published article describing the polymerization of methyl methacrylate in the presence of BP and various tertiary aromatic amines as shown in Figure 2 ( 5). ... 

Fig. 11. Dependence of the polymerization induction period r on reciprocal initial monomer concentration 1/(M]0...

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