First-order products

Reaction 1 is a second-order reaction, and Reaction 2 is first order. Product B is... 

The nitration of pyrene by N(IV) (N02, N204) in methylene chloride at 25 °C under conditions in which the N(IV)is primarily the tetroxide demonstrates first-order product formation both in pyrene and N(IV). The order in N(IV) shifts to second order when the starting N(IV) concentration is increased. N(IV) nitrates pyrene directly nitric acid, that is, N(V), does not. Under first-order conditions, the addition of NO [NO] [N(7Vj] shifts the product formation to second order and substantially accelerates the reaction. The reaction scheme involves the initial formation of a pyrene-N204 complex, followed by (1) the unimolecular collapse of the complex to product and (2) the addition of a second N204 to the pyrene in the complex. The second path leads directly to polynitropyrenes when the concentration of N(IV) is sufficiently high. 

The effect of added NO is more straightforward. In a single run under argon, 22 torr of NO was added to the gas cell. The cell contained 5 mL of a methylene chloride solution of pyrene (2.8 X 10 2 M and N(IV) (6.7 X 10 5 M) on a molar basis, [NO] >> [N(IV)]. In this case, the development of product smoothly followed second-order behavior, in contrast with the first-order production of 1-nitropyrene under the same conditions but in the absence of NO. 

A computer model has been constructed simulating the kinetics of the reaction sequence of Fig. 7. The isotopic enrichment of glycine is represented by the equation for the polyexponential die-away curve of hippurate shown in Fig. 3. That of the amide-N of glutamine is represented by a first order product curve derived from the enrichment of phenylacetylglutamine, which starts at zero, reaches a maximum at 3 hours, and thereafter closely approximates the die-away curve of Fig. 4. The model also includes an arbitrary convention which accelerates a late reaction of the sequence (IMP -> hypoxanthine) as a higher power function of [PP-ribose-P], in order to achieve a ratio of >1 in the expression (increase in rate of synthesis of 3-phosphoribosylamine)/(increase... 

The development in the previous section produced the functions / (z, t) and /2(z, z, t), which were very important in the calculation of the first and second moments of the population densities and other properties derived from them. Their probability interpretations were also recognized, although they were clearly not probability densities. Following developments in the early physics literature, we will refer to them as product densities. Thus, f z, t) is called the first order product density. The term product refers to the product taken of the number density, and the order refers to the number of times the product is taken of the number density. Thus, /2(z, z, t) is called product density of order 2. More generally, it is possible to define the product density of order r as... 

Equation (7.1.28) is merely a restatement of (7.1.25). It is useful to define the first-order product density, which yields the expected population density and is defined by... 

The boundary condition at x = 0 is obtained by argument that the probability there is a particle (nucleus) between 0 and X(0) dt on the size coordinate during time t and t + dt can be obtained in two different ways. First, it is given by the left-hand side of the equation appearing below by definition of the first-order product density. Second, it is also given by the right-hand side by definition of the nucleation rate, which is the transition probability for the appearance of a nucleus in the time interval t, t + dt). Thus,... 

Equation (7.3.2), however, provides only the average behavior of the system. It is of interest to observe here that the average behavior of the system could be obtained by dealing only with the first-order product density, viz., the expected population density. A truly stochastic formulation must consider, however, the higher order densities in order to calculate the average fluctuations about the mean behavior. The calculation of fluctuations was the subject of Sections 7.1.1.1 and 7.1.1.2. Since the higher order densities were the basic implements of this calculation, it will be our objective to first formulate the second-order product density equation for the breakage process under discussion. 

We have thus derived the second-order product density equation for the pure breakage process based purely on probabilistic considerations. It could also have been derived by averaging the master density equation (7.2.5), using the definition (7.1.11). The boundary conditions for the second-order product density can be obtained in much the same manner as the first-order product density by accounting for the formation of nuclei. Thus, recognizing the symmetry of the product density, we have the boundary condition... 

We shall first address the first-order product density or the expected population density, / (x, t). As before, the strategy is one of considering various circumstances at the instant t — dt that will lead to the presence of a particle at time t with mass between x and x -f dx. First, there may be a particle a,t t — dt with mass between x and x -f dx that may grow to the... 

Interestingly, this brings the first-order product density into the equation, thus initiating a closure problem. Averaging the boundary condition (7.3.13), one obtains... 

In accord with the definition (7.1.29) for the first-order product density, we integrate Eq. (7.3.12a) over all but one particle coordinates to obtain... 

Through an error in the derivation of the first order product density, the author did not recognize the appearance of the second-order product density in Ramkrishna (1979). 

We address applications here in which closure problems are not encountered. Thus, the average behavior of the population can be obtained from solving the first-order product density equation, and average fluctuations (of any order) about the mean can be calculated progressively by solving higher-order product density equations. In order to elucidate the nature of what can be obtained from such a theory, we shall consider a simple enough example for which analytical answers can be found. It is followed by a second example which has potential application to the study of cell death kinetics and hence to sterilization processes. 

The average number of individuals in any age bracket is obtained by the first order product density. The coefficient of variation about the mean requires the calculation of the second-order product density /2(t, t, t). We assume that the initial average number of individuals to be with an age distribution given by (t) and that initially their ages are not correlated. Neither assumption is a necessary attribute of the analysis. 

The first-order product density must satisfy the following partial differential equation and initial condition... 

The particle state includes only spatial coordinates along the radial and axial directions. We let the first-order product density be / (r, z) and let the second-order product density be /2(r, z r, z ). The differential equations are readily identified for the foregoing functions. Thus,... 

What is left in the sequel is the identification of the product density equations for the process above. Following Sampson and Ramkrishna (1986), we directly write the equations in the product densities integrated over the volume so that the densities are only in particle size coordinate but not in spatial volume and recognize them with a hat on top. This implies that we must use the redefined aggregation frequency a x, x ) in describing the aggregation process. The expected population density or the first-order product density must satisfy... 

Poisson process is a special case of a point process, whose increments dN(t) defined on disjoint time intervals dt are independent. The nonhomogeneous Poisson process is completely characterized by its first-order product density ... 

Consider an interval [0, t of the time axis. An ordinary renewal density hjj) (Cox 1962 Cox and Isham 1980) represents the probability that a random point (not necessarily the first) occurs in [t, t + dt], given that a random point occurs at the origin. A modified renewal density hm(f) (Cox 1%2) represents the probability that a random point (not necessarily the first) occurs in [t, t + dt, with arbitrarily chosen time origin. A modified renewal density is the first-order product density of the renewal point process ... 

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