Average and instantaneous rates

When we talk about rates (or speeds or velocities) we often distinguish between two different forms of rates (or speeds or velocities). Most people are probably familiar with the concept of an average rate. For example the distance between Canterbury and London is approximately 60 miles. If it takes us to drive this distance in 1 V2 hours we can easily calculate our average speed to 

In mathematical terms the average rate is given by the equation  

Xinitiai = amount of something at the beginning of the process tfinai = endpoint of the process tinitiai = starting point of the process. 

The A symbolises the difference between Xfmai and Xmitiai or tfmai and initial. 

So what s the difference between the average rate (or speed or velocity) and the instantaneous equivalent  

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Describe experimental methods for measuring average and instantaneous rates (Section 18.1, Problems 1-2). 

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 hr. 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. 

The rate of a chemical reaction is expressed in terms of the changes in concentration per unit time, and a careful definition must account for the stoichiometry of the reaction. Reaction rates change as a reaction proceeds, so we also need to distinguish between average and instantaneous rates. 

In this section, you learned how to express reaction rates and how to analyze reaction rate graphs. You also learned how to determine the average rate and instantaneous rate of a reaction, given appropriate data. Then you examined different techniques for monitoring the rate of a reaction. Finally, you carried out an investigation to review some of the factors that affect reaction rate. In the next section, you will learn how to use a rate law equation to show the quantitative relationships between reaction rate and concentration. 

Distinguish between average rate and instantaneous rate. Which of the two rates gives us an unambiguous measurement of reaction rate Why ... 

Define reaction rate. Distinguish between the initial rate, average rate, and instantaneous rate of a chemical reaction. Which of these rates is usually fastest The initial rate is the rate used by convention. Give a possible explanation as to why. 

Define reaction rate. Distinguish between the initial rate, average rate, and instantaneous rate of a chemical reac-... 

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

Stoichiometry and Rate Average Rate and Instantaneous Rate... 

The following travel analogy helps to distinguish between average rate and instantaneous rate. The distance by car from San Francisco to Los Angeles is 512 mi along a certain route. If it takes a person 11.4 h to go from one city to the other, the average... 

The rate changes as the reaction proceeds fastest at the beginning, when reactant concentration is highest, and slowest at the end. Average rate is the concentration change over a period of time, and instantaneous rate is the change at any instant. Kinetic studies typically measure the initial rate, the rate at the moment the reactants are mixed, so product is absent. (Section 16.2)... 

To understand the difference between average rate and instantaneous rate, it may help to think of the speed of an automobile. Speed can be defined as the rate of change of position, x that is, speed equals Ax At, where Ax is the distance traveled. If an... 

To better understand the difference between average and instantaneous reaction rates, think of taking a 190 km highway trip in 2.00 h. The average speed is 95 km/h. The instantaneous speed is the speedometer reading at any instant. 

Similar behavior was observed for LNG clouds during both continuous and instantaneous tests, but average flame speeds were lower the maximum speed observed in any of the tests was 10 m/s. Following premixed combustion, the flame burned through the fuel-rich portion of the cloud. This stage of combustion was more evident for continuous spills, where the rate of flame propagation, particularly for LNG spills, was very low. In one of the continuous LNG tests, a wind speed of only 4.5 m/s was sufficient to hold the flame stationary at a point some 65 m from the spill point for almost 1 minute the spill rate was then reduced. 

Given that, from the penetration theory for mass transfer across an interface, the instantaneous rale ol mass transfer is inversely proportional to the square root of the time of exposure, obtain a relationship between exposure lime in the Higbie mode and surface renewal rate in the Danckwerts model which will give the same average mass transfer rate. The age distribution function and average mass transfer rate from the Danckwerts theory must be deri ved from first principles. 

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... 

Monomer concentrations Ma a=, ...,m) in a reaction system have no time to alter during the period of formation of every macromolecule so that the propagation of any copolymer chain occurs under fixed external conditions. This permits one to calculate the statistical characteristics of the products of copolymerization under specified values Ma and then to average all these instantaneous characteristics with allowance for the drift of monomer concentrations during the synthesis. Such a two-stage procedure of calculation, where first statistical problems are solved before dealing with dynamic ones, is exclusively predetermined by the very specificity of free-radical copolymerization and does not depend on the kinetic model chosen. The latter gives the explicit dependencies of the instantaneous statistical characteristics on monomers concentrations and the rate constants of the elementary reactions. 

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,... 

However, the average rates calculated by concentration versus time plots are not accurate. Even the values obtained as instantaneous rates by drawing tangents are subject to much error. Therefore, this method is not suitable for the determination of order of a reaction as well as the value of the rate constant. It is best to find a method where concentration and time can be substituted directly to determine the reaction orders. This could be achieved by integrating the differential rate equation. 

As an excellent, simple example of how fluctuating parameters can affect a reacting system, one can examine how the mean rate of a reaction would differ from the rate evaluated at the mean properties when there are no correlations among these properties. In flow reactors, time-averaged concentrations and temperatures are usually measured, and then rates are determined from these quantities. Only by optical techniques or very fast response thermocouples could the proper instantaneous rate values be measured, and these would fluctuate with time. 

In your previous courses in science or physics, you probably learned the difference between instantaneous velocity and average velocity. How did you use a displacement-time graph to determine instantaneous velocity and average velocity Write a memo that explains instantaneous rate and average rate to a physicist, by comparing reaction rate with velocity. 

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. 

Why are the units for the average rate and the instantaneous rate the same ... 

Distinguish between an average rate and an instantaneous rate of a reaction. 

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