First-order rate expression

A strict kinetic limitation based on the gas-phase reactant can be modeled using a variable value for h although experience shows that a first order rate expressions with n=l often provides an excellent fit to experimental data regardless of the underlying reaction mechanism. A site-competition model such as Equation (10.12) can also be used. 

Note here that at high pressures of M, fe[M]kj. and Eq. (9.3) reduces to the first-order rate expression v (A i 3/ 2)[A] = A [A], whereas at low pressures 2[1V[] -C kj. and the expression becomes v A i[A][M], the normal second-order form. (Approximations such as these are commonly used in many areas of science and mathematics.)... 

A first-order rate expression results when the termination process involves the dissimilar radicals, Rl and R2. That is, reaction (4b) is the chain breaking step. 

This equation shows that the reaction rate is neither first-order nor second-order with respect to species A. However, there are two limiting cases. At high pressures where [A] is lar e, the bimolecular deactivation process is much more rapid than the unimolecular decomposition (i.e., /c2[A][A ] /c3[A ]). Under these conditions the second term in the denominator of equation 4.3.20 may be neglected to yield a first-order rate expression. 

This expression can be substituted into the rate expression for the production of dissolved iron(II) yielding a pseudo first-order rate expression written in terms of the surface concentration of ascorbate. 

It is a monotonically decreasing function with time, where the rate of decay is directly proportional to the stress. Using a first-order rate expression to describe the decay of stress we get the following expression ... 

As a first guess try a first-order rate expression. Then for plug flow Eq. 44 gives... 

The amount of catalyst needed is given by the first-order rate expression for plug flow, Eq. 44. Thus... 

These reactions do not satisfy total mass conservahon because the mole of water is omitted as a reactant. We have also redefined a new rate coefficient as k = [H20] by grouping the nearly constant [H2O] with k. After grouping the concentrahon of the solvent [H2O] into the rate coefficient, we say that we have a pseudo-first-order rate expression. 

For such complex reactions, the experimental rate data are fit into power law or even first-order rate expressions for simplification. Unfortunately, these tend to be limited to a specific catalyst, fuel composition, and operating conditions. It would be desirable to develop predictive models to account for variations in these parameters, but meager information is available on the kinetics of liquid... 

For the rate of ammonia oxidation a simple first-order rate expression in the surface concentration of ammonia has been assumed, which also considers the effect of oxygen, i.e. 

If the second term on the r.h.s. can be ignored, either because k-i ki (a so-called irreversible reaction), or because we start with [A] [B] and only observe the system over a time frame where that relationship continues to hold, then we may rearrange Eq. (15.6) to give the first-order rate expression... 

The second simplified form corresponds to the case where A, R and S are all weakly adsorbed, i.e. 1 KApA + KRpR + Ksps. Then eqn. (3) is reduced to a first-order rate expression... 

The acid hydrolysis of (en)2Co(OH)2Co(en)24+ has been studied several times, but with very conflicting results (359, 366-368). Rasmussen and Bjerrum studied the reaction at low acidities ([H + ] = 0.004-0.1 M) and found a pseudo first-order rate expression of the form given in Eq. (61) (367). 

There are many tricks and shortcuts to this process. For example, rather than compiling all of the transformation rate equations (or conducting the actual kinetic experiments yourself), there are many sources of typical chemical half-lives based on pseudo-first-order rate expressions. It is usually prudent to begin with these best estimates of half-lives in air, water, soil, and sediment and perform a sensitivity analysis with the model to determine which processes are most important. One can return to the most important processes to assess whether more detailed rate expressions are necessary. An illustration of this mass balance approach is given in Figure 27.5 for benzol a Ipyrene. This approach allows a first-order evaluation of how chemicals enter the environment, what happens to them in the environment, and what the exposure concentrations will be in various environmental media. Thus the chemical mass balance provides information relevant to toxicant exposure to both humans and wildlife. 

Direct photolysis of chromophores at the low concentrations found in environmental waters obeys a first-order rate expression (Hedlund and Youngson, 1972) ... 

The proposed NSPS can be met by hydrotreating the coal liquids obtained by filtering the product from the coal dissolution stage. The desulfurization kinetics can be presented by two parallel first-order rate expression, and hydrogen consumption kinetics can be presented by a first-order rate expression. A linear relationship exists between total sulfur content and SRC sulfur content of the hydrotreated product. For the Western Kentucky bituminous 9/14 coal studied here, the maximum selectivity and lowest SRC conversion to oil for a fixed SRC sulfur content are obtained using the highest reaction temperature (435°C) and the shortest reaction time 7 min.). ... 

As log, x = 2.303 logjo x) Equations (3) and (4) are known as first order rate expressions. The constant k is called the first order rate constant. 

Reactions which are not unimolecular, but obey the first order rate expression are known as pseudo-unimolecular reactions. For example, hydrolysis of methyl acetate, inversion of cane sugar etc. are pseudo-unimolecular reactions. In general, when the order of reaction is generally less than the molecularity of a reaction, it is said to be a pseudo order reaction. 

This equation is identical with first order rate expression. 

Thus, the amount of N2 evolved will be a measure of diazobenzene chloride decomposed i.e., x. The total N2 evolved at infinite time will thus give the initial concentration i.e., a. Now, substituting the values in the first order rate expression,... 

SoL The reaction will be of the first order or pseudo unimolecular reaction if the data conforms to the first order rate expression... 

As noted earlier, there is considerable evidence from laboratory and field studies that the LEA is often inadequate to describe reality. When the sorption process occurs at a rate that is similar to the rate of advection, the LEA is not appropriate and rate-limited sorption must be accounted for. Rate limited sorption can be modeled using the following first-order rate expression [21] ... 

Rate-limited sorption can also be modeled assuming a kinetic rate expression coupled with a nonlinear equilibrium expression. If we assume a Freundlich isotherm and a first-order rate expression, we can use the following equation to model sorption kinetics [21] ... 

Note that it is the concentration of A at the external surface which appears in the first-order rate expression and not the concentration of A in the bulk of the fluid phase. Solving Eqn. 7.57 for the concentration of A at the external surface gives ... 

Solve for reactor volume using the material-balance expression. The material balance for the continuous-flow stirred-tank reactor may now be used to calculate the reactor volume required for the isomerization. Inserting the first-order rate expression into the material balance,... 

Fit the data obtained in the laboratory by first order rate expressions of the type = kjC- in the temperature range of interest. Check whether the use of first order kinetics is allowed. 

The rate constant of adsorption is determined from the first order rate expression given by Lagergren [6] log (q -q) = logq - k t / 2.303 (1)... 

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