First-order removal rate constant

Since the oil drop removal rate (drops removed per unit time per unit of vessel volume) was ditectly proportional to oil drop concentration and independent ol inlet oil concentration and vessel residence time, the flotation process was studied by considering the first-order removal-rate constants for each oil drop si/e. 

The ability to separate the removal rates due to air bubbles from drop aggregation/coalcscencc for each oil drop size permitted a detailed study of the system variables. These variables and their ranges of variation are shown in Table I. Note that the first-order removal rate constants were independent of residence time and oil droplet population in the feed and effluent. The variables which may influence the rate constants are air flowrate, temperature, NaCI concentration, bubble diameter, cationic polymer concentration, and oil drop diameter. 

Injection of a cationic polyclcctrolylc into the flotation feed did not affect the first-order removal rate constants for bubble/drop interaction. 

K and tu to be depth independent. Analogous to that in box-models, it has also been common to assume that the removal rate, J(z), is a first order process i.e., J(z) = i >C where t is a first order scavenging rate constant. The removal is speculated to occur through an irreversible j n situ scavenging or absorption process on to settling particulate matter. In this case, the scavenging residence time of the nuclide would be ... 

FIGURE 4.9 Disposition model representing the elimination of a unit impulse drug dose (Hq = 1) from a single body compartment. Drug in this compartment (H) is removed as specified by the first-order elimination rate constant k. 

Some dryers also provide heat energy to the powder mass by a jacketed vessel, thereby increasing overall heat transfer. Moisture can be removed via vacuum or hot air fluidization depending on the design of the dryer allowing for improved evaporative drying and vapor mass transfer. Fig. 13 shows the relationship between power input (W) and first-order drying rate constant in a microwave fluid-bed processor. ... 

The rate of heat removal from the washcoat is proportional to the heat transfer coefficient and to the temperature difference between the gas and the washcoat, while the rate of heat generation is the product of the rate of mass transfer times the heat of reaction (-AH), as indicated by Eqs. (3) and (4) below, where kr is a first-order reaction rate constant. At the steady state the rates of heat generation and removal are equal. 

The bulk mantle has a constant volume (a free parameter depends on D" and RDM volumes). The starting U concentration is that of the bulk silicate Earth and that of He is calculated for a closed system with a Loihi He/ He ratio. The main He outflow is to the atmosphere by degassing a constant volume with time (i.e., first order degassing). U is largely transferred to the continental crust at a set first-order removal rate. Smaller fluxes of He and U to D" and RDM (via the altered ocean crust and depleted oceanic lithosphere, respectively) also occur. Small inputs occur from subduction of continental crust and from D" and RDM. Present isotopic compositions are those seen in MORE. 

A one-compartment model (Figure 6.15) is based on a single compartment— here, the plasma—also referred to as the central compartment. A first-order rate constant kg determines how fast the drug is absorbed into the bloodstream, while the first-order elimination rate constant kei describes the speed at which the drug is removed. A logarithmic plot of plasma concentration versus time is linear with a slope of - kei/2.303. The plasma concentration as a function of time is described by the relationship ... 

Thus, for an element whose removal from seawater follows first-order reaction kinetics, its MORT is the inverse of its removal rate constant. This relationship predicts that reactive elements should have short residence times. As shown in Figure 21.3, the actual data do demonstrate a linear relationship (r = 0.79, p = 0.00), although a log-log plot is required to cover the several orders of magnitude diversity of MORT and concentrations exhibited by the solutes in seawater. A similar relationship exists between the MORT and the seawater-crustal rock partition coefficient (Ay). The latter is defined as the ratio of the mean seawater concentration of an element to its mean concentration in crustal rocks. Elements with high partitioning coefficients would be expected to have low seawater concentrations. As shown in Figure 21.4, this is seen in the data and... 

If each type of sulfur compound is removed by a reaction that was first-order with respect to sulfur concentration, the first-order reaction rate would gradually, and continually, decrease as the more reactive sulfur compounds in the mix became depleted. The more stable sulfur species would remain and the residuum would contain the more difficult-to-remove sulfur compounds. This sequence of events will, presumably lead to an apparent second-order rate equation which is, in fact, a compilation of many consecutive first-order reactions of continually decreasing rate constant. Indeed, the desulfurization of model sulfur-containing compounds exhibits first-order kinetics, and the concept that the residuum consists of a series of first-order reactions of decreasing rate constant leading to an overall second-order effect has been found to be acceptable. 

Thus the lifetime of a constituent with a first order removal process is equal to the inverse of the first order rate constant for its removal. Taking an example from atmospheric chemistry, the major removal mechanism for many trace gases is reaction with hydroxyl radical, OH. Considering two substances with very different rate constants for this reaction, methane and nitrogen dioxide... 

Thus to estimate the lifetime of 85Kr does not even require knowledge of its atmospheric abundance Q, but only of the radioactive decay constant. Consequently, it does not matter whether 85Kr is uniformly mixed throughout the entire atmosphere or not to estimate its lifetime. In a case where the removal process is first-order, then even for a poorly mixed species a simple and accurate estimate for its lifetime can be obtained provided that its removal rate constant can be accurately estimated. 

As with the case of energy input, detergency generally reaches a plateau after a certain wash time as would be expected from a kinetic analysis. In a practical system, each of its numerous components has a different rate constant, hence its rate behavior generally does not exhibit any simple pattern. Many attempts have been made to fit soil removal (50) rates in practical systems to the usual rate equations of physical chemistry. The rate of soil removal in the Launder-Ometer could be reasonably well described by the equation of a first-order chemical reaction, ie, the rate was proportional to the amount of removable soil remaining on the fabric (51,52). In a study of soil removal rates from artificially soiled fabrics in the Terg-O-Tometer, the percent soil removal increased linearly with the log of cumulative wash time. 

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