Second-order integrated rate equation

Integrated rate equations second-order reactions... 

Choice of initial conditions. To give a very obvious example, in Chapter 2 we saw that a second-order reaction A -I- B —> products could be run with the initial conditions Ca = cb, thus permitting a very simple plotting form to be used. For complex reactions, it may be possible to obtain a usable integrated rate equation if the initial concentrations are in their stoichiometric ratio. 

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

Deriving the integrated rate equation of a second-order reaction is a little more complicated. Let us assume the second-order reaction... 

The integrated rate equation for a reaction that is second order with respect to N02 as the only reactant ... 

Since rate = [N02], we must use the second order integrated rate equation - [no2]0 =... 

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... 

For a second-order reaction, the form of the integrated rate equation is different ... 

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

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.

Derive an integrated rate equation similar to Equation 3.22 for the irreversible second-order reaction, when reactants A and B are introduced in the stoichiometric ratio ... 

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 is pressure/time data the appropriate integrated rate equation should be used. Reaction is stated to be second order ... 

Expressions similar to equation (17) may easily be derived for various second-, third-, and higher-order reactions. These expressions are readily integrated for all second-order reactions and for many third- and higher-order reactions, yielding (in many cases) relations analogous to equation (18), which define useful concentration-time graphs. The dimensions of the rate constant k for an nth order reaction are (concentration) (time) ... 

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... 

For reactions involving A —> products that are second order with respect to A and second order overall, the integrated rate equation is... 

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. 

Integrated rate equations can be derived similarly from other simple rate laws. For a reaction ah. —> products that is second order in reactant A and second order overall, we can write the rate equation as... 

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... 

The general model assumes instantaneous equilibria in the boundary layer of all solution species except C02> It uses a different diffusivity for each species. It accounts for the finite-rate, reversible reaction of CO2 and H2O to give IT " and HCO3- by iterative, numerical integration of a second-order, nonlinear differential equation and a set of nonlinear algebraic equations. 

In the case of Equation 5.9 the result of the integration depends on the numerical value of the partial order a. However, we do not need to be concerned with the details of the integration process itself it is the results of the process that are our main concern. The equations that result from integration are referred to as integrated rate equations and they are particularly important in the case of first-order (a= 1) and second-order (a =2) reactions. 

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