First-order integrated rate equation

In a curve-fitting method the concentration of a reactant or product is monitored continuously as a function of time, and a regression analysis is used to fit an appropriate differential or integral rate equation to the data. Eor example, the initial concentration of analyte for a pseudo-first-order reaction, in which the concentration of a product is followed as a function of time, can be determined by fitting a rearranged form of equation 13.12... 

Using calculus, it is possible to develop integrated rate equations relating reactant concentration to time. We now examine several such equations, starting with first-order reactions. 

Students who have taken calculus will recognize that Equation results from integration of the first-order rate law. 

This is the integrated rate equation for a first-order reaction. When dealing with first-order reactions it is customary to use not only the rate constant, k for the reaction but also the related quantity half-life of the reaction. The half-life of a reaction refers to the time required for the concentration of the reactant to decrease to half of its initial value. For the first-order reaction under consideration, the relation between the rate constant k and the half life t0 5 can be obtained as follows ... 

Since rate = [N205], we must use the first order integrated rate equation ln( ) = akt with tm = where a is the stoichiometric coefficient of N2Os... 

Rate data for the condensation of formaldehyde (F) with sodium paraphenolsulfonate (M) were taken by Stults et al (CEP Symp Series 4 38, 1952) at 100°C and pH = 8.35. Equal quantities of the reactants were present initially. Check first and second order mechanisms with the tabulated data. Integrated rate equations are... 

Integrated rate equations for a first-order reaction... 

Figure 8.7 shows the way the concentration of hydrogen peroxide decreases with time. The trace is clearly curved, and Figure 8.8 shows a graph constructed with the linear form of the first-order integrated rate equation, Equation (8.26). This latter graph is clearly linear. 

Figure 8.22 Kinetic graph for a reversible first-order reaction with the axes for an integrated rate equation ln([A], — A fcq ) (as 3/ ) against time (as V). The gradient is —5.26 x 10 3 min 1...

SAQ 8.23 Consider a reversible first-order reaction. Its integrated rate equation is given by Equation (8.50). People with poor mathematical skills often say (erroneously ) that taking away the infinity reading from both top and bottom is a waste of time because the two infinity concentration terms will cancel. Show that the infinity terms cannot be cancelled in this way take [A](eq) = 0.4 moldrrT3, [A]o = 1 moldrrT3 and [A]t = 0.7 mol dm 3. 

The k values are found using all three differential rate equations. First of all, Eq. 34, which is of simple first order, is integrated to give... 

Figure 1 Time-resolved evolution of ethylene during homometathesis of methyl-3-butenoate, catalyzed by 10 mg MeRe03/Si02-Al203 (8.8 wt% Re) in pentane at 15°C. The solid line is the curve-fit to the first-order integrated rate equation.

Figure 2 Kinetics of gas-phase propylene homometathesis at 0°C, catalyzed by (a) perrhenate/silica-alumina activated by SnMe4 (10 mg, 0.83 wt % Re) and (b) MeReOs on HMDS-capped silica-alumina (10 mg, 1.4 wt % Re). Solid lines are curve-fits to the first-order integrated rate equation. Solid squares first addition solid circles second addition open circles third addition of propylene (30 Torr) to the catalyst.

The simplest reactions have the one-step unimolecular or bimolecular mechanisms illustrated in Table 4.1 along with their differential rate equations, i.e. the relationships between instantaneous reaction rates and concentrations of reactants. That simple unimolecular reactions are first order, and bimolecular ones second order, we take as self-evident. The integrated rate equations, which describe the concentration-time profiles for reactants, are also given in Table 4.1. In such simple reactions, the order of the reaction coincides with the molecularity and the stoichiometric coefficient. 

Data given is concentration/time data, use the integrated rate equation to confirm first order kinetics. 

Sufficient DO data were not obtained from basalt-synthetic Grande Ronde groundwater experiments to allow determination of a definitive rate law. A first order kinetic model with respect to DO concentration was assumed. Rate control by diffusion kinetics and by surface-reaction mechanisms result in solution composition cnanges with different surface area and time dependencies (32,39). Therefore, by varying reactant surface area, determination of the proper functional form of the integrated rate equation for basalt-water redox reactions is possible. 

The emission intensity is proportional to I so that, from the integrated rate equation for these first-order decay processes, the fluorescence intensity will decrease according to the relation... 

The following equation needs to be solved in order to give the integrated rate equation for a first-order reaction ... 

In efifect, the determination of heat capacities of activation involves the second derivative of the reaction rate with respect to temperature, and the reactions have usually been followed by standard methods such as the analysis of aliquot samples or conductance measurements (see Kohnstam, 1962 Robertson, 1966). Most of the recent experiments have been concerned with first-order processes so that the rate coefficient could be obtained either by the method of Guggenheim (1926) or directly fi om the integrated rate equation. When the development of the product has been measured, this equation takes the form... 

On a practical point, the fact that the units of concentration cancel out for a first-order reaction means that any physical quantity that is proportional to the concentration may be used in the equation in place of concentration, e.g. light absorbance or titration volume. This is very useful, since it means data measured in the laboratory can be inserted directly into the integrated rate equation. 

We first determine the concentration of NOBr that remains after 1.50 X 10 M is used up. Then we use the second-order integrated rate equation to determine the time required to reach that concentration. 

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. 

The derivation of the integrated rate equation is an example of the use of calculus in chemistry. The following derivation is for a reaction that is assumed to be first order in a reactant A and first order overall. If you do not know calculus, you can still use the results of this derivation, as we have already shown in this section. For the reaction... 

This is the integrated rate equation for a reaction that is first order in reactant A and first order overall. 

The integrated rate equation can help us to analyze concentration-versus-time data to determine reaction order. A graphical approach is often used. We can rearrange the integrated first-order rate equation... 

Radionuclides have different stabilities and decay at different rates. Some decay nearly completely in a fraction of a second and others only after millions of years. The rates of all radioactive decays are independent of temperature and obey first-order kinetics. In Section 16-3 we saw that the rate of a first-order process is proportional only to the concentration of one substance. The rate law and the integrated rate equation for a first-order process (Section 16-4) are... 

Because Aq/A is a ratio, Aq and A can represent either molar concentrations of a reactant or masses of a reactant. The rate of radioactive disintegrations follows first-order kinetics, so it is proportional to the amount of A present we can write the integrated rate equation in terms of N, the number of disintegrations per unit time ... 

We determine the value of the specific rate constant, k, from the given half-life. This value is then used in the first-order integrated rate equation to calculate the amount of cobalt-60 remaining after the specified time. 

The adsorption of CO occurs sequentially with Au CO as an intermediate product. As the carbon monoxide concentration in the trap is constant, all reaction steps are taken to be pseudo-first-order in the simulations. Purely consecutive reaction steps do not fit the experimental data, and it is therefore essential to introduce a final equilibrium (1.2). The fits of the integrated rate equations to the data are represented by the solid lines in Fig. 1.36b and are an excellent match to the experimental results. 

Solution The volume of the reaction mixture is constant for any one run. If plug-flow behavior is assumed, Eq. (12-2) is applicable. The data in Table 12-1 represent integral reactor results. To use them Eq. (12-2) must be integrated, and this requires expressing the rate in terms of Q. When external composition differences are significant and the rate equation is not first order, this is difficult. The difference Q — will vary in the axial direction, so that a troublesome trial-and-error stepwise procedure is necessary to accomplish the integration. In this case the intrinsic rate is first order, so that the integration of Eq. (12-2) is simple and analytical. 

The integrated rate equations describing enzyme reaction pathways under first-order (or pseudo-first-order) conditions will always be a sum of exponential terms ... 

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