Integrated rate laws zero-order

Tlie integrated zero-order rate law is [A] = -kt + [A]q. Tlierefore, a plot of [A] versus time should be a... 

If this differential equation is integrated it gives an equation which is often called the integrated zero-order rate law. 

As we did with the zero-order rate law, this differential equation can be integrated to obtain an equation that directly expresses how the concentration of the reactant varies as a function of time during the reaction process. For a general first-order reaction involving the consumption of a reacting species A, this integration yields. 

Types of Rate Laws Determining the Form of the Rate Law Method of Initial Rates Half-Life of a First-Order Reaction Second-Order Rate Laws Zero-Order Rate Laws Integrated Rate Laws for Reactions with More Than One Reactant 12.7 Catalysis Heterogeneous Catalysis Homogeneous Catalysis... 

For the inner zone, in which both transport and reaction occurs, the differential equations are those of the first stage, but the boundary conditions arc dC Jdr = 0 at r = 0 and Eq. 4.2-11 at the boundary with the outer zone. This mt el corresponds to that set up by Ausman and Watson, to describe the rate of burning of carbon deposited inside a catalyst particle [8]. Analytical integration of this fairly general two-stage model is only possible for a zero-order, first-order or pseudo-first-order rate law, whereby Eq. 42-8 reduces to... 

A second-order reaction is one for which the overall reaction order is 2. If a second-order rate law depends on the concentration of only one reactant, then rate = k[A], and the time dependence of [A] is given by the integrated form of the rate law 1/[A], = 1/[A]q + kt. In this case a graph of 1/[A] t versus time yields a straight line. A zero-order reaction is one for which the overall reaction order is 0. Rate = fc if the reaction is zero order. 

Because equation 20.37 has only one variable (the concentration of A), it can be integrated just like any other first-order rate law. In doing so (and in making the same assumptions that the initial time is set to zero so that the variable t stands for elapsed time), we get... 

We can integrate this differential rate law to obtain the zero-order integrated rate law ... 

The Landolt reaction (iodate + reductant) is prototypical of an autocatalytic clock reaction. During the induction period, the absence of the feedback species (Irere iodide ion, assumed to have virtually zero initial concentration and fomred from the reactant iodate only via very slow initiation steps) causes the reaction mixture to become kinetically frozen . There is reaction, but the intemiediate species evolve on concentration scales many orders of magnitude less than those of the reactant. The induction period depends on the initial concentrations of the major reactants in a maimer predicted by integrating the overall rate cubic autocatalytic rate law, given in section A3.14.1.1. 

The integrated rate law for a zero-order reaction is easy to find. Because the rate is constant (at k), the difference in concentration of a reactant from its initial value, [A]0, is proportional to the time for which the reaction is in progress, and we can write... 

The values of a A, and EA must be determined from experimental data to establish the form of the rate law for a particular reaction. As far as possible, it is conventional to assign small, integral values to a2, etc., giving rise to expressions like first-order, second-order, etc. reactions. However, it may be necessary to assign zero, fractional and even negative values. For a zero-order reaction with respect to a particular substance, the rate is independent of the concentration of that substance. A negative order for a particular substance signifies that the rate decreases (is inhibited) as the concentration of that substance increases. 

We will develop the integrated rate laws individually for the cases n = 1 (first order), n = 2 (second order), and n = 0 (zero order). 

The expression for the half-life of a zero-order reaction can be obtained from the integrated rate law. By definition, [A] = [A]0/2 when t = tyj, so... 

In the integral method of analysis of rate data we are looking for the appropriate function of concentration correspondmg to a particular rate law that is linear with time. You should be thoroughly familiar with the methods of obtaining these linear plots for reactions of zero, first, and second order. 

You must choose the form of the rate-law expression or the integrated rate equation —zero, first, or second order—that is appropriate to the order of the reaction. These are summarized in Table 16-2. One of the following usually helps you decide. 

Integrating over time gives the integrated rate law for a zero-order reaction ... 

Consider the zero-, first-, and second-order integrated rate laws. If you have concentration versus time data for some species in a reaction, what plots would you make to "prove" a reaction Is either zero, first, or second order How would the rate constant, k, be determined from such a plot What does thej-intercept equal In each plot When a rate law contains the concentration of two or more species, how can plots be used to determine k and the orders of the species in the rate law ... 

Derive expressions for the half-life of zero-, first-, and second-order reactions using the integrated rate law for each order. How does each half-life depend on concentration If the half-life for a reaction is 20. seconds, what would be the second half-life assuming the reaction is either zero, first, or second order ... 

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