The variation of concentration with time

To illustrate the kinds of considerations involved in dealing with a mechanism, lets suppose that a reaction takes place in two first-order steps, in one of which the intermediate I (the open-circle DNA, for instance) is formed from the 

For simplicity, we cue ignoring the reverse reactions, which is permissible if they are slow. The first of these rate laws implies that A decays with a first-order rate law and therefore that 

The net rate of formation of I is the difference between its rate of formation and its rate of consumption, so we C m write 

These solutions are illustrated in Fig. 7.6. We see that the intermediate grows in concentration initially, then decays as A is exhausted. Meanwhile, the concentration of P rises smoothly to its final value. As we see in the following Justification, the intermediate reaches its maximum concentration at 

Consider a manufacturing process of a pharmaceutical in which k =0.120 h and /Cb = 0.012 h . It follows that the intermediate is at a maximum at f = 21 h after the start of the process. This is the optimum time for a manufacturer trying to make the intermediate in a batch process to extract it. 

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FIRST-ORDER REACTIONS. In order for Equation 21 to be accurately valid beyond the very first stages of reaction, the variation of concentration with time must be taken into account. For a first-order reaction... 

Equations 13a and 13b are two forms of the integrated rate law for a first-order reaction. The variation of concentration with time predicted by Eq. 13b is shown in Fig. 13.9. This behavior is called an exponential decay. The change in concentration is initially rapid, but it changes more slowly as time goes on and the reactant is used up. 

The next question is what is the variation of concentration with time due to diffusion The variation is described by Fick s second law which, for a one-dimensional system, is... 

The equations that give the variation of concentration with time and the variation of potential with time are... 

Use as coordinates the points halfway between the interface of adjacent elements, calculating the variation of concentration with time at these points. In this case there is no problem with the positioning of the electrode surface, as it is at the edge of a box. The simulation formulae are the same. This method can be particularly advantageous for non-planar electrodes as it is not necessary to decide on the three-dimensional shape of the elements only the position of the points matters. 

Batch kinetic results are easiest to evaluate if the reactor is operated at constant volume. A variation of reaction volume with conversion contributes to the variation of concentrations with time (even the concentrations of inerts will vary ). 

Figure 2 shows results of calculations of spill behavior. The variation of concentration with time is shown at several stations downstream from the spill location. The units of concentration are parts per billion (ppb), which is equivalent to micrograms per liter. Several features can be seen in Fig. 2. The dispersion, or spreading out, of the material can be seen by the decreasing peak concentrations as the spill proceeds downstream. In addition, the time span of exposure increases moving downstream. Basically, this can result in a situation where peak concentrations are reduced, but the time above certain critical concentrations is actually greater at a downstream location. Assume a hypothetical line indicating the critical level. It can be seen that exposure to these levels is longer at, e.g., 300 miles, than at 100 miles. A final feature is the reduction of concentrations by decay. The effect of decay increases (in terms of percent reduction) as the travel time increases. If uncertainty exists, a conservative assumption can also be made by assuming that the decay coefficient is zero.

Diffusion constant n Symbol D. The constant of proportionality in Pick s laws of diffusion between the rate of diffusion and the concentration gradient (Pick s first law) and the variation of concentration with time and concentration gradient (Pick s second law). 

Sketch, without carrying out the calculation, the variation of concentration with time for the approach to equilibrium when both forward and reverse reactions are second order. How does your graph differ from that in Fig. 7.2 ... 

It is not possible to measure directly the rate of the reaction which must be deduced from the variation of concentrations with time. These variations can be obtained by the aid of any of the non-destructive physicochemical methods of analysis (continuous measurement of pressure, volume, index of refraction, optical density, conductivity, specific inductive power, pH, electrode potential, etc.) or even, in the case of reactions sufficiently... 

The velocity of this reaction for each time t can be obtained by tracing tangents to the cnrve of the variation of concentration with time for the particular time t of interest For the time f = 0, we obtain the initial rate. The plot of initial rate as a function of the initial concentration of haloalkane gives a straight line that passes through the origin. This means... 

Alternatively, descriptions of the time evolution of concentration can be obtained by mathematical integration of the kinetic laws rather than from the study of the variation of concentration with time. From this, it is possible to predict the concentrations of reactants or products from the initial rate at any instant during the reaction. These can be compared with the experimental results. We should note, however, that there are cases in which the kinetic law is very complex, such that its analytical integration is very difficult or even impossible. In these cases it is still possible to resort to numerical integration. 

In a reaction of rate constant k = W sec if we consider a timescale with units of 1 nsec and 10 events are accomplished in each cycle, the probability of a reaction occurring in each cycle will be p = 10 The description of the time evolution of the system can thus be given in terms of the dependence of the molar fraction as a function of the number of cycles or, using the previous relation and converting the initial molar fraction into the initial concentration, as the variation of concentration with time. 

This example models the dynamic behaviour of an non-ideal isothermal tubular reactor in order to predict the variation of concentration, with respect to both axial distance along the reactor and flow time. Non-ideal flow in the reactor is represented by the axial dispersion flow model. The analysis is based on a simple, isothermal first-order reaction. 

Potentiometric measurements give log(3/ values which are correct to within 10% but the relative accuracy is difficult to assess due to the successive, repetitive nature of the experiments. Probably the most significant source of error, as we have stated, is the variation of pH with time. At tracer concentrations the enthalpy AH0 can be determined only by the temperature differential method. However, the temperature differential method is not as precise as the calorimetric one. The enthalpy of formation for each complex is computed with the assumption that ACp is constant over the temperature range and that the range of error corresponds to 10%. 

The occurrence of partial differential equations in electrochemistry is due to the variation of concentration with distance and with time, which are two independent variables, and are expressed in Fick s second law or in the convective-diffusion equation, possibly with the addition of kinetic terms. As in the resolution of any differential equation, it is necessary to specify the conditions for its solution, otherwise there are many possible solutions. Examples of these boundary conditions and the utilization of the Laplace transform in resolving mass transport problems may be found in Chapter 5. 

Fig. III.l. Variation of concentration with time for two consecutive first-order reactions (A —> B —> C). Note The lower curve for B/Bo does not necessarily lie below that for A/Ao.

Hence, the net outflow of ions per unit volume per unit time is D d c/dx ). But this net outflow of ions per unit volume per unit time from the parallepiped is in fact the sought-for variation of concentration with time, i.e., dcldt. One obtains partial differentials because the concentration depends both on time and distance, but the subscripts X and r are generally omitted because it is, for example, obvious that the time variation is at a fixed region of space, i.e., constant x Hence,... 

A distinction is made between rates of change and process rates. The former are observable, measurable phenomena of nature the latter, abstractions expressing the contributions of physical processes to such rates of change. For example, variations of concentrations with time or distance are rates of change, whereas rates of chemical reactions or mass transfer are process rates. 

The function q, in rotary diffusion, is the analogue of the concentration, c, in translational diffusion. The variation of q with time, in a system of the type under consideration, is exactly analogous to... 

This is an important result connecting the variation in concentration with time at a given point in the system to the divergence of the flux at the same point. It is called the equation of continuity with respect to mass. 

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