The Instantaneous Rate of Change

If the limit in equation (4.2) exists, it is called the derivative of the function y =/(x) at the point x0. The value of the derivative varies with the choice of xq, and we define it in general terms as  

The requirement that, for the limit in equation (4.2) to exist, the function does not undergo any abrupt changes is sometimes overlooked, yet it is an important one. An example of a function falling into this category is the modulus function, y = x, defined by  

Q Differentiate y — fix) = x2 using the definition of the derivative given in equation (4.4). 

Since Ax tends to zero, but never takes the value zero, cancellation of Ax from all terms in the numerator and denominator yields  

Differentiate the function y=Jlx), where fx) = 3, using the definition of the derivative given in equation (4.4). Hint the function y =/(x) = 3 requires that y — 3 for all values of x thus if fx) = 3, then f(x + Ax) must also equal 3. 

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

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. 

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

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 ... 

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

Notice that Eq. 2.1-1 is concerned with an instantaneous rate of change, and it requires data on the rates at which flows occur. Equation 2.1-2, on the other hand, is for computing the total change that has occurred and requires data only on the total flows over the time interval. These two equations, one for the instantaneous rate of change of a system property (here the amount of water) and the other for the change over an interval of time, illustrate the two types of change-of-state problems that are of interest in this book and the forms of the balance equations that are used in their solution. 

The balance equation (Eq. 2.1-3) is useful for computing the change in the extensiv e property 9 over the time interval Ar. We can also obtain an equation for computing the instantaneous rate of change of 9 by letting the time interval Ar go to zero. This is done as follows. First we use the symbol 9 t) to represent the amount ef 9 in the system at time r, and we recognize that for a very small time interval A/ (over which the flows into and out of the system are constant) we can write... 

I. The calculation of the momentary states from the complete law. Before the instantaneous rate of change, dyjdx, can be determined it is necessary to know the law, or form of the function connecting the varying quantities one with another. For instance, Galileo found by actual measurement that a stone falling vertically downwards from a position of rest travels a distance of s = gt2 feet in t seconds. Differentiation of this, as we shall see very shortly, furnishes the actual velocity of the stone at any instant of time, V = gt. In the same manner, Newton s law of inverse squares follows from Kepler s third law and Ampere s law, from the observed effect of one part of an electric circuit upon another. 

This value, which is an estimate that depends on how well the tangent has been drawn, is the instantaneous rate of change of concentration of BrO with time, d[BrO-]/dr, at 1 500 s. Notice that the units for this quantity have been derived by including appropriate units at all stages in the calculation. 

The instantaneous rate of change of the concentration of a reactant, or a product, at a particular instant in a chemical reaction is equal to the slope of the tangent drawn to the kinetic reaction profile at that time. 

In most situations, it is difficult to determine numerical values for the instantaneous rate of change of reactant concentration at specific times. To circumvent this problem, Swinboume (66) suggested the following equation ... 

The final term in the rate vector is reserved to indicate the system residence time. All other terms present in the vector are simply individual species rate functions. Observe that species rate expressions can be viewed as the instantaneous rate of change with respect to residence time. Thus for rate... 

The quantity (dn P/ >t) in Eq. ( 25j represents the instantaneous rate of change of surface excess of component i at Z and t. This replaces the local rate of adsorption term in the conventional energy balance equation derived by using the actual amount adsorbed as the base variable. A simple model to describe the rate of change of GSE is the analog of the conventional linear driving force model [12]. 

The instantaneous rates of change of these variables are explicitly... 

The first derivative of a function /(x) at point x gives the instantaneous rate of change of the function with respect to variations in x and represents a generalization of the slope for... 

Derivative Function derived from a given function by means of the limit process, whose output equals the instantaneous rates of change of the given function. 

An extensional flow is steady if the instantaneous rate of change of length per unit length is constant. 

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