Using Concentration to Describe Rate

Reactions are usually described in terms of changes in concentration over time. In the hypothetical reaction 

Remember that the Greek letter delta, A, means change in a variable. For this case, the equation can be rewritten as  

Sample Consider the reaction aA + bB — cC + dD. The concentration of C was recorded for a period of time, and some of the data is shown in the table below  

Calculate the average rate of reaction of C during this time interval. 

C]Mame [ClMame l 0-00275 - 0.00230 = 2 3 x 1Q-4 M -i final time-initial time J 4-2 

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The rate constants for micelle-catalyzed reactions, when plotted against surfactant concentration, yield approximately sigmoid-shaped curves. The kinetic model commonly used quantitatively to describe the relationship of rate constant to surfactant, D, concentration assumes that micelles, D , form a noncovalent complex (4a) with substrate, S, before catalysis may take place (Menger and Portnoy, 1967 Cordes and Dunlap, 1969). An alternative model... 

Let us now deduce the factors that control the rate of conversion of B and C to D and E by imagining the transformation process is portrayed well by what is known as a collision rate model. (Strictly speaking, the collision rate model applies to gas phase reactions here we use it to describe interactions in solution where we are not specifying the roles played by the solvent molecules.) First, in order to be able to react, the molecules B and C have to encounter each other and collide. Hence, the rate of reaction depends on the frequency of encounters of B and C, which is proportional to the product of their concentrations. The rate is also related to how fast B and C move in the aqueous solution. Next, the rate is proportional to the probability that B and C meet with the right orientation to be able to react, which we may refer to as the orientation probability . Third, only a fraction of collisions have a sufficient amount of energy (greater then or equal to Ea) to break the relevant bonds in B and C... 

With deeper understanding of the rate laws applicable to these hydrolases, now we need to deduce the parameters that combine to give corresponding khl0 values for Michaelis-Menten cases (Eq. 17-80). We may now see that the mathematical form we used earlier to describe the biodegradation of benzo[f]quinoline (Eq. 17-82) could apply in certain cases. Further we can rationalize the expressions used by others to model the hydrolysis of other pollutants when rates are normalized to cell numbers (e.g., Paris et al., 1981, for the butoxyethylester of 2,4-dichlorophenoxy acetic acid) or they are found to fall between zero and first order in substrate concentration (Wanner et al., 1989, for disulfoton and thiometon). 

Regression analysis in time series analysis is a very useful technique if an explanatory variable is available. Explanatory variables may be any variables with a deterministic relationship to the time series. VAN STRATEN and KOUWENHOVEN [1991] describe the dependence of dissolved oxygen on solar radiation, photosynthesis, and the respiration rate of a lake and make predictions about the oxygen concentration. STOCK [1981] uses the temperature, biological oxygen demand, and the ammonia concentration to describe the oxygen content in the river Rhine. A trend analysis of ozone data was demonstrated by TIAO et al. [1986]. 

Equation (67), which describes the change in the experimental rate constant k [228] in this model, is formally very similar to the equation which was used earlier to describe the inhibition [294, 295] of electrode reactions. Such an equation was used hy Kisova [296]. Later it was found that Eq. (68) better describes the rate constant-solvent composition dependence. However, it fails to describe this dependence at high concentrations of an organic solvent which has a donor number lower than that of water. In general, it fails when the rate constant-solvent composition dependence exhibits a minimum. 

This rate equation is called Michaelis-Menten double substrate kinetics . It is a formal multiplication of two Michaelis-Menten models for both substrates A and B. This model can be used to describe rate kinetics of two substrate reactions in the absence of the product(s). The kinetic measurements have to be performed by varying the concentration of one substrate keeping the concentration of the second substrate at a constant value well above the Km value. The model cannot be used if back reactions occur and an equilibrium has to be described by an appropriate Haldane equation. 

Like in most other physical chemical topics this is not about math but about using math to describe what is known and what can be experimentally accomplished. In an enzyme kinetics experiment, measuring the concentration of the enzyme-substrate complex, [TiS], is usually difficult if not impossible so you try to re-write the whole expression in terms of the quantities which are known, [S] and [ ]tot. for example, or which can be measured, the rate of reaction at maximum [S], V ax. for example. Keeping this in mind you can reshuffle (10-9) to obtain an expression for [ S] ... 

At the simplest level, a quick comparison of rates with two different concentrations of a reactant can indicate the order of the reaction. Thus, if the rate doubles when the concentration is doubled, we can infer that the reaction is first order in that reactant, and if it quadruples, then it is second order. However, to assess the data more fully, we need to be more systematic. There are two principal approaches. In one, we use rate measurements directly in the other, we use concentration measurements, not rates. In this section we prepare the ground for the first approach and describe its implementation. The second approach needs more preparation and is described in Section 6.5. 

Equation (5.11) is the differential form of the rate law which describes the rate at which A groups are used up. To test a proposed rate law and to evaluate the rate constant it is preferable to work with the integrated form of the rate law. The integration of Eq. (5.11) yields different results, depending on whether the concentrations of A and B are the same or different ... 

Fouling is the term used to describe the loss of throughput of a membrane device as it becomes chemically or physically changed by the process fluid (often by a minor component or a contaminant). A manifestation of fouling in cross-flow UF is that the membrane becomes unresponsive to the hydrodynamic mass transfer which is rate-controlling for most UF. Fouling is different from concentration polarization. Both reduce output, and their resistances are additive. Raising the flow rate in a cross-flow UF will increase flux, as in Eq. 

Kinetic studies at several temperatures followed by application of the Arrhenius equation as described constitutes the usual procedure for the measurement of activation parameters, but other methods have been described. Bunce et al. eliminate the rate constant between the Arrhenius equation and the integrated rate equation, obtaining an equation relating concentration to time and temperature. This is analyzed by nonlinear regression to extract the activation energy. Another approach is to program temperature as a function of time and to analyze the concentration-time data for the activation energy. This nonisothermal method is attractive because it is efficient, but its use is not widespread. ... 

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