The Average Rate of Change

Define the average rate of change of the dependent variable... 

Figure 4.1 Defining the average rate of change of /(x) as x is incremented from xb to Xb+Ax...

Although the box matrix has nine elements, only six of them are independent. Thus, in setting up a box matrix for equilibrium simulations from crystal cell parameters, three of the off-diagonal elements can be arbitrarily chosen to be zero. This choice cannot be arbitrary for planar Couette flow. This is because, if the initial values of 21 aud 23 are chosen to be nonzero, and will evolve boundlessly in time. On the other hand, if 21 and f 23 were to be zero at t = 0, then during NPT dynamics they would oscillate around zero, and thus the average rate of change of and hi would be zero. 

Short-rate models are composed of two components. The first attempts to capture the average rate of change, also called the drift, of the short-term rate at each instant, while the second component measures this drift as a function of the volatility of the short-term rate. This is given by ... 

Here k(t) is the average rate of change of HB population for those trajectories where the bond is broken at time t later. Figure 3.11 shows k(t) determined from their simulation. At short times, k(t) manifests an assortment of motions leading to HB breaking. The most important of these motions are librations, at timescale of less... 

The average rate of change of enthalpy due to reaction over the batch reaction will bc. 

Because the derivative component of output varies as the difference between two successive values of the controlled variable, it is fundamentally a measure of the average rate of change during a sampling Interval. Put In other terms. It may be said to represent the rate of change of the controlled variable midway between sampling Intervals. The effect Is that derivative action Is delayed by At. It can be seen that derivative... 

The quantity above is the average rate of change of change of [ Fe ] in this interval. It is a negative quantity because [Fe " "] decreases with time. The average rate of disappearance of Fe is defined as follows. 

Similarly, with cycling temperatures, large fast cycles can cause discomfort. To avoid this, if the time to complete one cycle is less than 15 minutes and the peak-to-peak temperature variation is greater than 1.1 °C, the average rate of temperature change should be less than 2.2 °C/h (Fig. 5.9). Very slow rates of temperature change dT/dt < 0.5 °C/h) are much less difficult to adjust to and the change can go unnoticed until the temperature is beyond the comfort zone temperature. 

The average rate of a reaction is the change in concentration of a species divided by the time over which the change takes place the unique average rate is the average rate divided by the stoichiometric coefficient of the species monitored. Spectroscopic techniques are widely used to study reaction rates, particularly for fast reactions. 

The average rate of a reaction can be expressed as the change in concentration (A c) over some time interval... 

Because rates change continuously with time, accurate rate determinations must use small time intervals. At the outset of the NO2 decomposition experiment shown in Figure 15-6. the time interval between measurements is 5 seconds. For the first 5 seconds, the average rate of O2 production is as follows ... 

The average rate of a reaction is the average change in the concentration of a reactant or product per unit time over a given time interval. For example, using the data in Table 6.1, you can determine the average rate of the reaction from t = 0.0 s to t = 5.0 s. 

It s also important to specify the time when quoting a rate because the rate changes as the reaction proceeds. For example, the average rate of formation of N02 is 3.7 X 10-5M/s during the time period 300 to 400 s, but it is only 2.2 X 10-5 M/s during the period 600 to 700 s. Ordinarily, reaction rates decrease as the reaction mixture runs out of reactants, as indicated by the decreasing slopes of the curves in Figure 12.1 as time passes. 

As a final example, consider line D of Table 9.1. We represent this problem as a body of density p, and heat capacity cp and whose surface is in contact with another medium of temperature Ts. Assume the initial body temperature is the same as the temperature of the other medium at I m = Ts. From the fundamental equation we can write, pcpLdTb/dt = X(TS — Tb)/L, where L is the characteristic conductive length, and X is the thermal conductivity. We now scale this problem over the entire time of the thermal transient. Once the entire time of the transient passes fe - t ), the body will have reached the new temperature of 7, 2. For the overall transient, the temperature rate of change is (Tb2 - Tb )/Gi -t ). and the average driving potential for the thermal conduction will be TS2 - T, )/2 = ( 7),2 - Tj, )/2. We now define the first-order relationship between the parameter as... 

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