Net rates of formation

Free radicals are molecular fragments having one or more unpaired electrons, usually short-lived (milhseconds) and highly reaclive. They are detectable spectroscopically and some have been isolated. They occur as initiators and intermediates in such basic phenomena as oxidation, combustion, photolysis, and polvmerization. The rate equation of a process in which they are involved is developed on the postulate that each free radical is at equihbrium or its net rate of formation is zero. Several examples of free radical and catalytic mechanisms will be cited, aU possessing nonintegral power law or hyperbohc rate equations. 

To eliminate the concentration of the intermediate N202 we proceed as follows. First, we write the expression for its net rate of formation. According to the mechanism, N202 is formed in the forward reaction in step 1, removed in the reverse reaction, and removed in step 2. Therefore the net rate of its formation is... 

Now we make the steady-state approximation, that any intermediate remains at a constant, low concentration. The justification for this approximation is that the intermediate is so reactive that it reacts as soon as it is formed. Because the concentration of the intermediate is constant, its net rate of formation is zero, and the previous equation becomes... 

To eliminate the O atom concentration from this expression, we consider the net rate of formation of O atoms and use the steady-state approximation to set that net rate equal to zero ... 

The most common states of a pure substance are solid, liquid, or gas (vapor), state property See state function. state symbol A symbol (abbreviation) denoting the state of a species. Examples s (solid) I (liquid) g (gas) aq (aqueous solution), statistical entropy The entropy calculated from statistical thermodynamics S = k In W. statistical thermodynamics The interpretation of the laws of thermodynamics in terms of the behavior of large numbers of atoms and molecules, steady-state approximation The assumption that the net rate of formation of reaction intermediates is 0. Stefan-Boltzmann law The total intensity of radiation emitted by a heated black body is proportional to the fourth power of the absolute temperature, stereoisomers Isomers in which atoms have the same partners arranged differently in space, stereoregular polymer A polymer in which each unit or pair of repeating units has the same relative orientation, steric factor (P) An empirical factor that takes into account the steric requirement of a reaction, steric requirement A constraint on an elementary reaction in which the successful collision of two molecules depends on their relative orientation. 

The component balance for a batch reactor. Equation (1.21), still holds when there are multiple reactions. However, the net rate of formation of the component may be due to several different reactions. Thus,... 

If the data are sufficiently accurate, one may use a general differential approach in which the expression for the net rate of formation of a stable intermediate is postulated and tested against experimental data. The difference between the rate of formation by the initial reaction for which the kinetics are known and the actual net rate of formation is tested against the proposed rate expression for the reaction responsi-... 

The net rate of formation of products will be the difference between the two, i.e. 

As discussed in Chapter 15, the size distribution of particles in an agglomeration process is essentially determined by a population balance that depends on the kinetics of the various processes taking place simultaneously, some of which result in particle growth and some in particle degradation. In a batch process, an equilibrium condition will eventually be established with the net rates of formation and destruction of particles of each size reaching an equilibrium condition. In a continuous process, there is the additional complication that the residence time distribution of particles of each size has an important influence. 

We can similarly show that the net rate of formation of -mers, i.e., polymer containing n monomers, is... 

However, A2B is a common intermediate in these two steps. If A2B is an intermediate, rather than a product, the concentration of A2B must be essentially constant, i.e the net rate of formation of A2B must be essentially zero ri+ - - r2+ + r2- = 0... 

In the steady state approximation the net rate of formation for all intermediates is set to zero. 

Assuming that the net rate of formation for intermediates is zero does not imply that the coverage by the intermediates is small. 

A complete kinetic description of the gas phase reactions leading to the formation of a ceramic material is a set of microscopic reactions and the corresponding rate coefficients. The net rate of formation of species j, rj by chemical reactions is the sum of the contributions of the various reactions in the set of elemental steps called the mechanism ... 

In this expression the concentrations of O and N atoms are unknown and must be expressed in terms of known quantities. This is done using two important simplifying assumptions. The first assumption is the steady-state assumption. This assumption applies for very reactive species such as radicals, where it can be assumed that the rate of consumption is equal to the rate of formation, meaning that the net rate of formation is equal to zero. Applying the steady assumption to the nitrogen atom, we get... 

At this point, we make the steady-state approximation, and set the net rate of formation of any intermediate equal to 0. The justification for this approximation is that the reactions in step 1 are so fast that they keep the concentrations of intermediates constant. When we set this expression equal to 0, we can rearrange it and obtain an expression for the concentration of N202 ... 

Further, from its definition, a deactivation function is connected to one reaction only, not to the net rate of formation of a certain product, which may involve rates of formation and... 

The relevant kinetic model for competition experiments with a radiolabeled drug [D] and an unlabeled competitor [I] is shown in the two equations in (19.15). When both sets of reactions have proceeded to equilibrium, the net rate of formation of both (DR) and (IR) are zero, and the following Eq. (19.16) can be derived from mass-action principles. 

Another evident mechanism for energy transfer to activated ions may be by bimolecular collisions between water molecules and solvated ion reactants, for which the collision number is n(ri+ r2)2(87tkT/p )l/2> where n is the water molecule concentration, ri and r2 are the radii of the solvated ion and water molecule of reduced mass p. With ri, r2 = 3.4 and 1.4 A, this is 1.5 x 1013 s"1. The Soviet theoreticians believed that the appropriate frequency should be for water dipole librations, which they took to be equal 10n s 1. This in fact corresponds to a frequency much lower than that of the classical continuum in water.78 Under FC conditions, the net rate of formation of activated molecules (the rate of formation minus rate of deactivation) multiplied by the electron transmission coefficient under nonadiabatic transfer conditions, will determine the preexponential factor. If a one-electron redox reaction has an exchange current of 10 3 A/cm2 at 1.0 M concentration, the extreme values of the frequency factors (106 and 4.9 x 103 cm 2 s 1) correspond to activation energies of 62.6 and 49.4 kJ/mole respectively under equilibrium conditions for adiabatic FC electron transfer. 

If a reaction intermediate X is so unstable that it decomposes practically as soon as it is formed, its concentration necessarily remains quite small. The same must be true for its net rate of formation rx If that rate were large and positive, the concentration would rise to large values, which it is known not to do if that rate were large and negative, the concentration would have to become negative, which it cannot. Accordingly, provided the intermediate is and remains at trace level, its net formation rate rx is small compared separately with its rates of formation and decay ... 

The net rate of formation of an intermediate that is and remains at trace level is negligible compared with its contributing formation and decay rates. 

Granted the invariance of the rate coefficients for propagation, kp, and catalyst split-off, trm, the net rate of formation of an adduct Pj, with j + 1 monomer units is... 

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