Rates instantaneous

the rate is directly proportional to the concentration of bromine  

We can use the concentration and rate data from Table 14.1 for any value of t to calculate the value of k for this reaction. For example, using the data for t = 50.0 s gives 

Slight variations in the calculated values of k are due to experimental deviations in rate measurements. To two significant figures, we get k = 3.5 X 10 for this reaction, regardless of which line of data we chose from Table 14.1. It is important to note that the value of k does not depend on the concentration of bromine. The rate constant is constant at constant temperature. (In Section 14.4 we discuss how k depends on temperature.) 

We now consider another specific reaction, the decomposition of hydrogen peroxide  

Because one of the products is a gas, we can monitor the progress of this reaction by measuring the pressure with a manometer. Pressure is converted to concentration using the ideal gas equation  

How does the instantaneous rate of reaction change as the reaction proceeds  

In discussions that follow, the term rate means instantaneous rate unless indicated otherwise. The instantaneous rate at t = 0 is called the initial rate of the reaction. To understand the difference between average and instantaneous rates, imagine you have just driven 98 mi in 2.0 h. Your average speed for the trip is 49 mi/hr, but your instantaneous speed at any moment during the trip is the speedometer reading at that moment. 

Analyze We are asked to determine an instantaneous rate from a graph of reactant concentration versus time. 

Solve The tangent line falls from [C4H9CI] = 0.100 M to 0.060 M in the time change from 0 s to 210 s. Thus, the initial rate is 

What is the difference between average rate and instantaneous rate In a given reaction, can these two rates ever have the same numeric value  

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In counting experiments, the instantaneous rate at which particles arrive at the detector can be significantly... 

As already mentioned, the motion of a chaotic flow is sensitive to initial conditions [H] points which initially he close together on the attractor follow paths that separate exponentially fast. This behaviour is shown in figure C3.6.3 for the WR chaotic attractor at /c 2=0.072. The instantaneous rate of separation depends on the position on the attractor. However, a chaotic orbit visits any region of the attractor in a recurrent way so that an infinite time average of this exponential separation taken along any trajectory in the attractor is an invariant quantity that characterizes the attractor. If y(t) is a trajectory for the rate law fc3.6.2] then we can linearize the motion in the neighbourhood of y to get... 

For some materials the linear constitutive relation of Newtonian fluids is not accurate. Either stress depends on strain in a more complex way, or variables other than the instantaneous rate of strain must be taken into account. Such fluids are known collectively as non-Newtonian. Many different types of behavior have been observed, ranging from fluids for which the viscosity in the Navier-Stokes equation is a simple function of the shear rate to the so-called viscoelastic fluids, for which the constitutive equation is so different that the normal stresses can cause the fluid to flow in a manner opposite to that predicted for a Newtonian fluid. 

Instantaneous Rate This is a short-term rate when the eqiup-ment operates at the design rate or faster. Typical is the average weight handled over a short period of time, not exceeding 5 min. 

It is evident from their definitions that /, and hence d and w depend on the instantaneous rate of deformation of the current configuration. On the other hand, F and hence U and R relate the current configuration to the reference configuration. In order to find relations for d and w in terms of material derivatives of U and R, the material derivative of (A. 13) may be inserted into (A. 10)... 

The rate is defined as an intensive variable, and the definition is independent of any partieular reaetant or produet speeies. Beeause the reaetion rate ehanges with time, we ean use the time derivative to express the instantaneous rate of reaetion sinee it is influeneed by the eomposition and temperature (i.e., the energy of the material). Thus,... 

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

Determination of the instantaneous rate at a particular concentration. To determine the rate of reaction, plot concentration versus time and take the tangent to the curve at the desired point. For the reaction N20s(g) — 2N02(g) + 02(g), it appears that the reaction rate at [N205] = 0.080 M is 0.028 mol/L - min. 

Compare the instantaneous rate at 30 s with the average rate over the thirty-second intervaL... 

The instantaneous rate of monomer consumption in binary copolymerization is then given by eq. 62 ... 

According ro the penetration theory, the instantaneous rate of mass transfer per unit area LV, > at some time i after the commencement of transfer is given by ... 

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). 

From now on, whenever we speak of a reaction rate, we shall always mean an instantaneous rate. The definitions in Eqs.l and 2 can easily be adapted to refer to the instantaneous rate of a reaction. 

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. 

For the unique instantaneous rate. Three significant figures. 

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 ... 

When chain transfer is considered and following the derivation previously given (1, 2) the instantaneous rate of conversion is then given by... 

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,... 

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