Order with respect to time

A reaction order determined by plotting the integrated rate equation is sometimes called the order with respect to time f this order has an unambiguous meaning only if the order is independent of time, which means that the plotted function is linear... 

The rate of a second-order reaction may be proportional to two concentrations, v = [A][B] with [B]0 s> [A]o it follows first-order kinetics. Some authors refer to these as the order with respect to concentration and the order with respect to time. 

Comparison of formulae (2.51) and (2.64) allows one to understand the limits and advantages of the impact approximation in the theory of orientational relaxation. The results agree solely in second order with respect to time. Everything else is different. In the impact theory the expansion involves odd powers of time, though, strictly speaking, the latter should not appear. Furthermore the coefficient /4/Tj defined in (2.61) differs from the fourth spectral moment I4 both in value and in sign. Moreover, in the impact approximation all spectral moments higher than the second one are infinite. This is due to the non-analytical nature of Kj and Kf in the impact approximation. In reality, of course, all of them exist and the lowest two are usually utilized to find from Eq. (2.66) either the dispersion of the torque (M2) or related Rq defined in Eq. (1.82) ... 

Since the diffusion equation is also first order with respect to time, a condition (extra to the boundary conditions.) must he specified. It almost invariably describes the initial density. 

Lyster (1970) found denaturation of /3-lactoglobulin to be second order with respect to time. The kinetic constant K2 in log-1 sec-1 is described by the equation... 

Although not very commonly used (with the exception of the initial rate procedure for slow reactions), the differential method has the advantage that it makes no assumption about what the reaction order might be (note the contrast with the method of integration, Section 3.3.2), and it allows a clear distinction between the order with respect to concentration and order with respect to time. However, the rate constant is obtained from an intercept by this method and will, therefore, have a relatively high associated error. The initial rates method also has the drawback that it may miss the effect of products on the global kinetics of the process. 

Pigs. 18 and 21). As in the case of Ni0(200°), the initial total order is close to zero when NiO(250°) is used as a catalyst and the reaction rate on the fresh sample decreases with time according to the kinetics of order one (74). Kinetics of order one are not followed, however, on regenerated catalysts. Reaction orders were determined in this case by the differential method and were found to vary from 1 (fresh catalyst) to 0.77 (constant activity). Since the initial total order is, in all cases, zero, it was concluded that, as in the case of the same reaction on NiO(200°), the reaction order with respect to time is apparent and results from the inhibition of the catalyst by carbon dioxide, the reaction product. Modification of the apparent order with the runs indicates that regenerated samples of Ni0(250°) are less inhibited than the fresh catalyst. 

The reaction order with respect to time was determined by the differential method. A fractional order (1.3) is obtained for the catalytic reaction on both doped samples. However, as in the case of the same reaction on pure oxides, the initial reaction rate does not depend upon the pressure of either reagent (order zero). Since these results are similar to those obtained on pure samples, NiO(200°) and NiO(250°), we believe that the order with respect to time is, as in the former case, apparent and that it results from the inhibition of surface sites by carbon dioxide, the reaction product. The slowest step of the reaction mechanism on doped oxides should occur, therefore, between adsorbed species. 

Jumarie, G. 1992. A Fokker-Plank equation of fractional order with respect to time. J. Math. Phys. 33 3536-3542. 

In reality, of course, atoms obey quantum mechanics rather than classical mechanics. As you will discover in other chapters in this book, great advances have been made recently in the quantum mechanical treatment of molecular systems. However, one should realize just how much care has to go into the selection of correct coordinates and the necessity to choose appropriate systems for quantum mechanical study. For arbitrarily large systems, or for systems containing several heavy atoms, quantum methods are not yet readily applicable. It is in such cases that classical mechanical approaches can be utilized with profit. Furthermore, even in systems for which quantum mechanical treatments are now feasible, comparisons with classical data often help researchers to isolate those phenomena which arise solely in the quantum mechanics, yielding fundamental insight into the two different dynamics. In the classical approach, the motion of each atom is calculated by numerically solving the classical differential equations of motion (1), either second order with respect to time in the positions, x (Newton s law), or,... 

A reaction order determined by using either equations 9 or 10 is termed order with respect to time. 

When conversion reaches about 70%, all the remaining monomer is absorbed in the polymer particles and there are no more droplets left. At this point the reaction rate becomes first order with respect to time. 

Note that the left-hand side is identical with (22), which is the forward difference formula. The difference is in the right-hand side, which is a function of values at the next time level, to which the same left-hand-side approximation now pertains. Intuitively, one might expect this form to be about as inefficient as the forward difference and this is in fact true this formula, too, is first order with respect to time, as is the forward difference formula. The big difference is that this formula, applied to the diffusion equation, yields a method that is stable for aU k. This is... 

This extra monomer supplied is sufficient for equilibrium swelling of the particles [298]. As a result, the rate of polymerization becomes zero order with respect to time. 

Conduritol-B epoxide, a compound structurally related to substrates for yeast (Saccharomyces sp.) p-D-fructofuranosidase, is an active-site inhibitor of the enzyme. Inactivation is first-order with respect to time and the concentration of the inhibitor, and does not occur in the presence of substrates. The dependence of inactivation on pH revealed the presence of two dissociable groups (pK values 3.05 and 6.8) in the enzyme, one of which, a carboxylic acid group of pK 3.05, appears to react with conduritol-B epoxide. 

The Peaceman-Rachford ADI method is second-order with respect to time, and performs similarly to Crank-Nicolson. Indeed, Lapidus and Finder write [224, p. 246] ... is a variation of the Crank-Nicolson approximation . It is known to be unconditionally stable [225]. As with CN, ADI may show some error oscillations, as also evidenced by the fact that some habitually use expanding time intervals when employing ADI [226-231], although some of these same workers on occasion also use equal time intervals [232,233]. 

At temperatures higher than 57°C the experimental points are well fitted by a single straight line and the denaturation reaction is first order with respect to time. At temperatures lower than 57°C deviations from the linear behavior are evident and the order of the reaction with respect to time is appreciably greater than unity. 

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