Rate expression, meaning

The rate expression just written is unsatisfactory in that it cannot be checked against experiment. The species NOCl2 is a reactive intermediate whose concentration is too small to be measured accurately, if at all. To eliminate the [NOCl2] term from the rate expression, recall that the rates of forward and reverse reactions in step 1 are equal, which means that... 

Calculational problems with the Runge-Kutta technique also surface if the reaction scheme consists of a large number of steps. The number of terms in the rate expression then grows enormously, and for such systems an exact solution appears to be mathematically impossible. One approach is to obtain a solution by an approximation such as the steady-state method. If the investigator can establish that such simplifications are valid, then the problem has been made tractable because the concentrations of certain intermediates can be expressed as the solution of algebraic equations, rather than differential equations. On the other hand, the fact that an approximate solution is simple does not mean that it is correct.28,29... 

As given above, the statements that adjust the exponents m and n have been commented out and the initial values for these exponents are zero. This means that the program will fit the data to. = k. This is the form for a zero-order reaction, but the real purpose of running this case is to calculate the standard deviation of the experimental rate data. The object of the fitting procedure is to add functionality to the rate expression to reduce the standard deviation in a manner that is consistent with physical insight. Results for the zero-order fit are shown as Case 1 in the following data ... 

It is important to recognize that by no means can all reactions be said to have an order. For example, the gas phase reaction of H2 and Br2 to form HBr has a rate expression of the following form ... 

The fractional life approach is most useful as a means of obtaining a preliminary estimate of the reaction order. It is not recommended for the accurate determination of rate constants. Moreover, it cannot be used for systems that do not obey nth order rate expressions. 

Thus mechanism B, which consists solely of bimolecular and unimolecular steps, is also consistent with the information that we have been given. This mechanism is somewhat simpler than the first in that it does not requite a ter-molecular step. This illustration points out that the fact that a mechanism gives rise to the experimentally observed rate expression is by no means an indication that the mechanism is a unique solution to the problem being studied. We may disqualify a mechanism from further consideration on the grounds that it is inconsistent with the observed kinetics, but consistency merely implies that we continue our search for other mechanisms that are consistent and attempt to use some of the techniques discussed in Section 4.1.5 to discriminate between the consistent mechanisms. It is also entirely possible that more than one mechanism may be applicable to a single overall reaction and that parallel paths for the reaction exist. Indeed, many catalysts are believed to function by opening up alternative routes for a reaction. In the case of parallel reaction paths each mechanism proceeds independently, but the vast majority of the reaction will occur via the fastest path. 

In order for the overall rate expression to be 3/2 order in reactant for a first-order initiation process, the chain terminating step must involve a second-order reaction between two of the radicals responsible for the second-order propagation reactions. In terms of our generalized Rice-Herzfeld mechanistic equations, this means that reaction (4a) is the dominant chain breaking process. One may proceed as above to show that the mechanism leads to a 3/2 order rate expression. 

Reactor inlet conditions are particularly useful as reference conditions for measuring the input volumetric flow rate in that they not only give physical meaning to CA0 and but also usually lead to cancellation of CA0 with a similar term appearing in the reaction rate expression. 

Table 8.1 summarizes the fundamental design relationships for the various types of ideal reactors in terms of equations for reactor space times and mean residence times. The equations are given in terms of both the general rate expression and nth-order kinetics. 

The reaction was stated to proceed by means of an associative mechanism, similar to the one proposed for the rhodium-triphenylphosphine system, with HPt(CO)(SnCl3)(PPh3) as the active intermediate. The high selectivity to linear aldehyde was attributed to steric requirements. The kinetic data led to the following general rate expression ... 

The motions of the individual fluid parcels may be overlooked in favor of a more global, or Eulerian, description. In the case of single-phase systems, convective transport equations for scalar quantities are widely used for calculating the spatial distributions in species concentrations and/or temperature. Chemical reactions may be taken into account in these scalar transport equations by means of source or sink terms comprising chemical rate expressions. The pertinent transport equations run as... 

The rate expression can be further simplified because in a real system the partial pressures of H20 and H2 are much higher than the partial pressures of CO and C02, which mean that the partial pressures of H20 and H2 are practically constant. 

Rates are most often expressed as unit rates. The keyword for rates is per. Speed is known as miles per hour and is actually a rate that means miles per one hour. In the same way, unit price, dollars per pound means dollars per one pound. Unit rates are easier to work with than other ratios, because the denominator is always 1, and so the denominator has no effect when performing cross-multiplication. 

Table 15.4. Mean over-all crystal growth rates expressed as a linear velocity ...

This result could be improved by assuming a more appropriate distribution function of T instead of a simple sinusoidal fluctuation however, this example—even with its assumptions—usefully illustrates the problem. Normally, probability distribution functions are chosen. If the concentrations and temperatures are correlated, the rate expression becomes very complicated. Bilger [47] has presented a form of a two-component mean-reaction rate when it is expanded about the mean states, as follows ... 

For non-rapid-equilibrium cases (i.e., steady-state cases) the enzyme rate expression is much more complex, containing terms with [A] and with [I]. Depending on the relative magnitude of those terms in the initial rate expression, there may be nonlinearity in the standard double-reciprocal plot. In such cases, computer-based numerical analysis may be the only means for obtaining estimates of the magnitude of the kinetic parameters involving the partial inhibition. See Competitive Inhibition... 

Reaction rate expressions are always empirical, which means that we use whatever expression gives an accurate enough description of the problem at hand. No reactions are as simple as these expressions predict if we need them to be correct to many decimal places. Further, all reaction systems in fact involve multiple reactiorrs, and there is no such thing as a tmly irreversible reaction if we coitld measure all species to sufficient accuracy. If we need a product with impurities at the parts per biUion (ppb) level, then aU reactions are in fact reversible and involve many reactions. 

Most real reactors are not homogeneous but use catalysts (1) to make reaction occur at temperatures lower than would be required for homogeneous reaction and (2) to attain a higher selectivity to a particular product than would be attained homogeneously. One may then ask whether any of the previous material on homogeneous reactions has any relevance to these situations. The answer fortunately is yes, because the same equations are used. However, catalytic reaction rate expressions have a quite different meaning than rate expressions for homogeneous reactions. 

In the very short time limit, q (t) will be in the reactants region if its velocity at time t = 0 is negative. Therefore the zero time limit of the reactive flux expression is just the one dimensional transition state theory estimate for the rate. This means that if one wants to study corrections to TST, all one needs to do munerically is compute the transmission coefficient k defined as the ratio of the numerator of Eq. 14 and its zero time limit. The reactive flux transmission coefficient is then just the plateau value of the average of a unidirectional thermal flux. Numerically it may be actually easier to compute the transmission coefficient than the magnitude of the one dimensional TST rate. Further refinements of the reactive flux method have been devised recently in Refs. 31,32 these allow for even more efficient determination of the reaction rate. 

Eqs. 42 and 43 provide a uniform expression for the partial rates, the decay rate and the diffusion coefficient in terms of the energy loss 5, the quantum parameter a and the rate expression in the spatial diffusion limit. The mean squared traversal distance may be obtained directly from the ratio of the diffusion coefficient to the escape rate. 

By repeating the experiment at different steady-state temperatures one can determine Zi as a function of T. From this a plot can be made of vs. T for a constant A, which means a constant vapor pressure, p. Changing the temperature in the side tube can establish a new value of p or A in the tube, and another d vs. T curve can be determined. From such a family of 0 vs. T curves one can plot a family of A vs. 9 curves for constant values of T. Since the 0 values were determined for a steady-state condition, the arrival rate. A, must be equal to the evaporation rate, expressed in atoms per square centimeter per second. Hence the A vs. 9 curves are also Fo vs. 6 curves at a constant T. 

In heterogeneous systems, the rate expressions have to be developed on the basis of (a) a relation between the rate and concentrations of the adsorbed species involved in the rate-determining step and (b) a relation between the latter and the directly observable concentrations or partial pressures in the gas phase. In consequence, to obtain adequate kinetic rate expressions it is necessary to have a knowledge of the reaction mechanism, and an accurate means of relating gas phase and surface concentrations through appropriate adsorption isotherms. The nature and types of adsorption isotherm appropriate to chemisorption processes have been discussed in detail elsewhere [16,17] and will not be discussed further except to note that, in spite of its severe theoretical limitations, the Langmuir isotherm is almost invariably used for kinetic interpretations of surface hydrogenation reactions. The appropriate equations are... 

The next step in the reaction sequence must involve attack of acetate or chloride to give the Pd (II) a-bonded carbon intermediate. First consider the case of acetate using vinyl ester exchange as an example. If acetate were attacking from the coordination sphere, it must enter the coordination sphere by means of the equilibrium shown in Equation 12. What would the rate expression for this route be First the equilibria shown in Equation 10 would require a [Li2Pd2Cl6] [C2H3OOCR]/[LiCl] term in the rate expression while Equation 12 would require a [LiOAc] / [LiCl] term so the complete rate expression would be that given by Equation 13. Actually the rate expression contained a [LiCl] term to the first power,... 

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