Dissolution rate, equation

Equation (1) predicts that the rate of release can be constant only if the following parameters are constant (a) surface area, (b) diffusion coefficient, (c) diffusion layer thickness, and (d) concentration difference. These parameters, however, are not easily maintained constant, especially surface area. For spherical particles, the change in surface area can be related to the weight of the particle that is, under the assumption of sink conditions, Eq. (1) can be rewritten as the cube-root dissolution equation ... 

Consider first the case where the chemical compound ApBq is not formed between initial substances A and B during dissolution. The rate of dissolution of any solid in the liquid phase is known to be described by the equation... 

The first of these reactions is a hydrolysis process, the second is a carbonic acid-promoted dissolution, and the third is a proton-promoted dissolution. Equations 3.59b and 3.59c are the forward reactions in Eqs. 3.17 and 3.15, respectively. They provide a mechanistic underpinning for the dependence of kd in Eq. 3.14 on pH or pc0, as discussed in Section 3.1. Indeed, if Eq. 3.7 is applied to the forward reaction in Eq. 3.14 and rate laws for Eq. 3.59 are developed consistently with the hypothesis leading to Eq. 3.7, the result is7,33,34... 

The quantities kc and ks in the last two equations result from a calculation of an exponential, and thus have no physical dimensions. The effective dissolution probability rate constant k eff is calculated by multiplying the above three factors, so that /c e// = kdkcks. Thus, k eff has dimension of time-1 and denotes the fraction of the total number of drug particles that can be dissolved per MCS. The mass of the tablet that will break off at any moment is given by multiplying the value of k eff by the undissolved mass of the tablet. If qd (t) is this mass, then qd (t) = [go — g (i)] d,eff and qd (t) /ip particles of the tablet with mass ip will break off, and will get separated from the larger mass. The dissolved particles now flow on their own, with the same characteristics (forward probability) as the undissolved particles. The mass qo — q (t) of the undissolved drug is then reduced by qd (t). 

These equations are indistinguishable for rate measurements at a single temperature and pressure, but predict different results for the variation of the rate constant. A , with environmental variables (i.e., r, P, and [Ca ]). Because [Ca ] is not constant in laboratory experiments during the course of dissolution. Equation (10) is normally used to interpret these results (e.g., Morse and Arvidson, 2002 Keir, 1980). In the ocean, where [Ca ] is nearly a constant but K p varies... 

The Crystal Growth of a CaCOj (Calcite) The crystal growth of calcite has been studied by Plummer and et al. (1978), by Kunz and Stumm (1984), and by Chou et al. (1989) to correspond to the reverse of the rate of dissolution (equation 32) ... 

Tabb 1.6 How the parameters of the dissolution equation can be changed to increase (+) or decrease (-) the rate of solution ... 

Table 9.4. The Effect of Changing Parameters from the Dissolution Equation on the Rate of Solution ...

In a recent paper W.D. Hinsberg(4) demonstrated that the Nan-ion concentration plays a key role in the dissolution rate of both exposed and unexposed photoresist. While the exact machanism of this effect is not yet known, it was possible to obtain a function dissolution rate equation rate for unexposed resist(4) ... 

The thickness of passive films at steady state depends on the applied potential and their rate of dissolution (equation 6.37). If chloride ions accelerate the rate of dissolution, a higher chloride concentration in the electrolyte should lead to thinner passive films. This behavior has been confirmed by XPS measurements, as shown in Figure 6.44. This figure presents the effect of several halogen-anions on the thickness of the passive films formed on iron in a phthalate buffer solution (pH 5). In this case, Cr ions have a greater effect than either Br or 1 ions. 

In equation 2 Qt = actinide content on filter at time t (Bq) Qo = initial actinide content on filter at time t = 0 (Bq) fi = actinide fraction of rapid dissolution Sr = rate constant of rapid dissolution (d ) fs = actinide fraction of slow dissolution and Ss = rate constant of slow dissolution (d ). Plotting Ln (Qt/Qo) versus time yields Sr and Ss from the slopes of the rapid and slow dissolution and and fs from the crossover from rapid to slow dissolution. 

Hydrated amorphous silica dissolves more rapidly than does the anhydrous amorphous silica. The solubility in neutral dilute aqueous salt solutions is only slighdy less than in pure water. The presence of dissolved salts increases the rate of dissolution in neutral solution. Trace amounts of impurities, especially aluminum or iron (24,25), cause a decrease in solubility. Acid cleaning of impure silica to remove metal ions increases its solubility. The dissolution of amorphous silica is significantly accelerated by hydroxyl ion at high pH values and by hydrofluoric acid at low pH values (1). Dissolution follows first-order kinetic behavior and is dependent on the equilibria shown in equations 2 and 3. Below a pH value of 9, the solubility of amorphous silica is independent of pH. Above pH 9, the solubility of amorphous silica increases because of increased ionization of monosilicic acid. 

Data on Illinois No. 6 and Kentucky No. 9 coals were used by Wen and Han (Prepi Pap.—Am. Chem. Soc., Div. Fuel Chem. 20(1) 216-233, 1975) to obtain a rate equation for coal dissolution under hydrogen pressure. These data included a temperature range of 648 to 773 K (705 to 930°F) and pressures up to 13.8 MPa (2000 psia). An empirical rate expression was proposed as... 

It follows from equation 1.45 that the corrosion rate of a metal can be evaluated from the rate of the cathodic process, since the two are faradai-cally equivalent thus either the rate of hydrogen evolution or of oxygen reduction may be used to determine the corrosion rate, providing no other cathodic process occurs. If the anodic and cathodic sites are physically separable the rate of transfer of charge (the current) from one to the other can also be used, as, for example, in evaluating the effects produced by coupling two dissimilar metals. There are a number of examples quoted in the literature where this has been achieved, and reference should be made to the early work of Evans who determined the current and the rate of anodic dissolution in a number of systems in which the anodes and cathodes were physically separable. 

The dissolution of passive films is, in the main, controlled by a chemical activation step in contrast to film-free conditions at. Many protective anodic films are oxides and hydroxides whose dissolution depends upon the hydrogen ion concentration, and the rate follows a Freundlich adsorption equation ... 

The implication of the foregoing equations, that stress-corrosion cracking will occur if a mechanism exists for concentrating the electrochemical energy release rate at the crack tip or if the environment in some way serves to embrittle the metal, is a convenient introduction to a consideration of the mechanistic models of stress corrosion. In so far as the occurrence of stress corrosion in a susceptible material requires the conjoint action of a tensile stress and a dissolution process, it follows that the boundary conditions within which stress corrosion occurs will be those defined by failure... 

If crack propagation occurs by dissolution at an active crack tip, with the crack sides rendered inactive by filming, the maintenance of film-free conditions may be dependent not only upon the electrochemical conditions but also upon the rate at which metal is exposed at the crack tip by plastic strain. Thus, it may not be stress, per se, but the strain rate that it produces, that is important, as indicated in equation (8.8). Clearly, at sufficiently high strain rates a ductile fracture may be propagated faster than the electrochemical reactions can occur whereby a stress-corrosion crack is propagated, but as the strain rate is decreased so will stress-corrosion crack propagation be facilitated. However, further decreases in strain rate will eventually result in a situation where the rate at which new surface is created by straining does not exceed the rate at which the surface is rendered inactive and hence stress corrosion may effectively cease. 

It was found that the dissolution rate of the material depends both on its surface area and on its crystalline size, but the importance of the crystalline size seems to be greater. The empiric equation describing the above dependence of the leaching rate on these two parameters is as follows (130) ... 

C) for cast iron and up to 140 °F for marstenitic SS (60 °C). It is widely used where silicates are present with the iron oxides. Typically, 5 to 7.5% HC1 is employed. The ammonium bifluoride normally is present at 0.5%, but it may be increased to a maximum of 1.5% for a boiler that has not been cleaned for many years. The presence of hydrofluoric acid (HF), which is formed by the reaction of ammonium bifluoride with HC1 (see equation), tends to increase the rate of iron oxide dissolution and reduce the corrosion rate of exposed steel, when compared to using HC1 alone. This is due to the stability of the hexafluoroferric ion (FeFg3 ), which prevents the ferric ion from corroding exposed steel. 

A different melting point, and hence supercooling, is predicted for the strained sector. This is the basis for a different interpretation of the (200) growth rates a regime //// transition occurs on (110) but not on (200). This is despite the fact that the raw data [113] show a similar change in slope when plotted with respect to the equilibrium dissolution temperature (Fig. 3.15). It is questionable whether it is correct to extrapolate the melting point depression equation for finite crystals which is due to lattice strain caused by folds, to infinite crystal size while keeping the strain factor constant. 

Sundararajan et al. [131] in 1999 calculated the slurry film thickness and hydrodynamic pressure in CMP by solving the Re5molds equation. The abrasive particles undergo rotational and linear motion in the shear flow. This motion of the abrasive particles enhances the dissolution rate of the surface by facilitating the liquid phase convective mass transfer of the dissolved copper species away from the wafer surface. It is proposed that the enhancement in the polish rate is directly proportional to the product of abrasive concentration and the shear stress on the wafer surface. Hence, the ratio of the polish rate with abrasive to the polish rate without abrasive can be written as... 

The microstmcture appeared well mixed although co-continuity of the phases was not obvious. The blends appeared to have a continuous PP phase containing extended, yet isolated, SBR components as shown in Figure 11.17. It appeared to be similar to the microstmcture of the TPV-based on nylon and EPDM. The presence of entrapped air or mumal dissolution was not observed. As the fraction of PP increased, the microstmctures became clustered into larger PP and SBR single phases, with lower SBR-PP interface area. Both the materials were shear thinning. There is a large decrease in the viscosity of the composites at small shear rate. The viscosity values of the phases followed the equation... 

Organic hydroperoxides have also been used for the oxidation of sulphoxides to sulphones. The reaction in neutral solution occurs at a reasonable rate in the presence of transition metal ion catalysts such as vanadium, molybdenum and titanium - , but does not occur in aqueous media . The usual reaction conditions involve dissolution of the sulphoxide in alcohols, ethers or benzene followed by dropwise addition of the hydroperoxide at temperatures of 50-80 °C. By this method dimethyl sulphoxide and methyl phenyl sulphoxide have been oxidized to the corresponding sulphone in greater than 90% yields . A similar method for the oxidation of sulphoxides has been patented . Unsaturated sulphoxides are oxidized to the sulphone without affecting the carbon-carbon double bonds. A further patent has also been obtained for the reaction of dimethyl sulphoxide with an organic hydroperoxide as shown in equation (19). 

During the lifetime of a root, considerable depletion of the available mineral nutrients (MN) in the rhizosphere is to be expected. This, in turn, will affect the equilibrium between available and unavailable forms of MN. For example, dissolution of insoluble calcium or iron phosphates may occur, clay-fixed ammonium or potassium may be released, and nonlabile forms of P associated with clay and sesquioxide surfaces may enter soil solution (10). Any or all of these conversions to available forms will act to buffer the soil solution concentrations and reduce the intensity of the depletion curves around the root. However, because they occur relatively slowly (e.g., over hours, days, or weeks), they cannot be accounted for in the buffer capacity term and have to be included as separate source (dCldl) terms in Eq. (8). Such source terms are likely to be highly soil specific and difficult to measure (11). Many rhizosphere modelers have chosen to ignore them altogether, either by dealing with soils in which they are of limited importance or by growing plants for relatively short periods of time, where their contribution is small. Where such terms have been included, it is common to find first-order kinetic equations being used to describe the rate of interconversion (12). 

To provide a kinetics side of the dissolution process with water, it may be added that the rate equation for the process goes by the following equation ... 

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