Instantaneous rate change

Velocity. A measure of the instantaneous rate of change of position in space with respect to time. Velocity is a vector quantity. 

To determine the reaction rate at a given instant in the course of the reaction, we should make our two concentration measurements as close together in time as possible. In other words, to determine the rate at a single instant we determine the slope of the tangent to the plot of concentration against time at the time of interest (Fig. 13.4). This slope is called the instantaneous rate of the reaction. The instantaneous reaction rate changes in the course of the reaction (Fig. 13.5). 

To set up expressions for the instantaneous rate of a reaction, we consider At to be very small so that t and t + At are close together we determine the concentration of a reactant or product at those times and find the average rate from Eq. 1. Then we decrease the interval and repeat the calculation. We can imagine continuing the process until the interval At has become infinitely small (denoted d/) and the change in molar concentration of a reactant R has become infinitesimal (denoted d R]). Then we define the instantaneous rate as... 

FIGURE 13.4 The rate ol reaction is the change in concentration of a reactant (or product) divided by the time interval over which the change occurs (the slope of the line AB, for instance). The instantaneous rate is the slope of the tangent to the curve at the time of interest. 

Patterns in reaction rate data can often be identified by examining the initial rate of reaction, the instantaneous rate of change in concentration of a species at the instant the reaction begins (Fig. 13.6). The advantage of examining the initial rate is that the products present later in the reaction may affect the rate the interpretation of the rate is then quite complicated. There are no products present at the start of the reaction, and so any pattern due to the reactants is easier to find. 

Because an instantaneous rate is a derivative of concentration with respect to time, we can use the techniques of integral calculus to find the change in [A] as a function of time. First, we divide both sides by A and multiply through by — dt ... 

Since the reaction rate almost invariably changes with time, it is necessary to use the time derivative to express tfye instantaneous rate of reaction. 

From Equations 16.1 and 16.2, we can write the instantaneous rate of change in the system s bulk composition as,... 

In stepping forward from t to a new point in time t, the instantaneous rate will change as the fluid s chemistry evolves. Rather than carrying the rate at t over the step, it is more accurate (e.g., Richtmyer, 1957 Peaceman, 1977) to take the average of the rates at t and t. In this case, the new bulk composition (at t) is given from its previous value (at t ) and Equations 16.7-16.9 by,... 

Further work on the absorption of sulphur dioxide by Uchida et aln5> has shown that the absorption rate changes with the surface area of the limestone particles which in turn varies with the size and the number of particles, and that the rate of dissolution plays a very important role on the absorption. It was further found the absorption rate does not vary significantly with temperature and that the reactions involved may be considered as being instantaneous. 

One way to ensure that back reactions are not important is to measure initial rates. The initial rate is the limit of the reaction rate as time reaches zero. With an initial rate method, one plots the concentration of a reactant or product over a short reaction time period during which the concentrations of the reactants change so little that the instantaneous rate is hardly affected. Thus,by measuring initial rates, one can assume that only the forward reaction in Eq. (35) predominates. This would simplify the rate law to that given in Eq. (36) which as written would be a second-order reaction, first-order in reactant A and first-order in reactant B. Equation (35), under these conditions, would represent a second-order irreversible elementary reaction. 

In the following ThoughtLab, you will use experimental data to draw a graph that shows the change in concentration of the product of a reaction. Then you will use the graph to help you determine the instantaneous rate and average rate of the reaction. 

Careful reading of papers is required to determine which definition has been used. Measurements of the continuous phase resistance around bubbles frequently use photographic, volumetric, or pressure change techniques to yield instantaneous rates of mass transfer, and thus kA. Here too, both definitions of the Sherwood number, Eqs. (7-43) and (7-45), have been used. 

Equation 1.3 can be generalized further by considering a time-dependent field c(r, t) the instantaneous rate of change of c with velocity v(r) is then... 

Use the concepts of limits to define the instantaneous rate of change... 

If we now reconsider the general situation shown in Figure 4.1, we can determine the instantaneous rate of change by examining the limiting behaviour of the ratio, QR/PR, the change in y divided by the change in x, as Ax tends to zero ... 

We note from Equation 3.4 that the changing concentrations of the components in the reaction with time are, strictly speaking, only proportional to the rate of reaction, as rigorously defined in Equations 3.1 and 3.2. Experimentally, the instantaneous rates of changes of the concentrations of the components in a chemical equation are usually proportional to the instantaneous concentrations of the components themselves, each raised to some power. Such a relationship is called the rate law, or rate equation, of the reaction. For example, the rate law for the reaction of Equation 3.3 might be Equation 3.5 ... 

When the stoichiometric coefficients, va, vy, etc., are included in the rate law, as in Equation 3.5, the reaction has a unique rate constant (k) under specified conditions regardless of whether the rate is measured by monitoring the changing concentration of A, B or C. It also follows from Equation 3.5 that (except for zero-order reactions) the instantaneous rate of a reaction changes as the reaction proceeds, as will be illustrated later in Fig. 3.1. Thus, k is the parameter which measures whether the reaction (imprecisely expressed) is fast or slow . In any case, it follows that any property of a reacting system which relates (preferably directly) to the concentration of any component in the chemical reaction maybe monitored to measure the rate and, hence, to investigate the rate law and quantify the rate constant. 

The rate of movement of a toxicant across a membrane may be expressed as the change in amount of toxicant, A, (dA) or toxicant concentration, C, (dC) per unit of time (dt), which equals dA/dt. Calculus can be used to express instantaneous rates over very small time intervals (dt). Thus rate processes may then be generally expressed as... 

With this method, the concentration of a reactant or product is plotted versus time for a very short initial period of the reaction during which the concentrations of the reactants change so little that the instantaneous rate is hardly affected (Bunnett, 1986). 

For this simple uniaxial extensional flow to be steady, the instantaneous rate of change of the 1 direction length (/) must be constant... 

The rate is defined as an intensive variable, and the definition is independent of any particular reactant or product species. Because the reaction rate changes with time, we can use the time derivative to express the instantaneous rate of reaction since it is influenced by the composition and temperature (i.e., the energy of the material). Thus,... 

The instantaneous rate of reaction is defined as the change in concentration of reactant during some specified time period, or instantaneous rate of reaction = [N205]/t. What is the instantaneous rate of reaction for the decomposition of N205 for the time period between the first and second hours of the reaction Between the second and third hours Between the sixth and seventh hours ... 

Reaction rate — The (instantaneous) rate of a reaction can be expressed by the derivative of any quantity X, which changes during a chemical reaction, with respect to time [i]. 

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