Reactions opposing

One complication which frequently exists is that a reaction may proceed to a state of equilibrium which differs appreciably from completion. This situation often arises in biological processes, such as metabolic reactions. For example, the reaction 

In considering the kinetics of such a reaction in one direction, we would have to take into account the reverse reaction. 

The simplest case is when both forward and reverse reactions are of the first order  

Integration of this, subject to the boundary condition that x = 0 when t == 0, gives 

The amount of x present at equilibrium, Xe, can be measured directly. A procedure for obtaining ki is therefore to obtain values of x at various values of t, and then to plot 

Perhaps the simplest example of a reaction mechanism which is more complicated than that of equation (1) is the one-step reversible reaction 

For reaction (21), equations (10), (11), and (14) (with the subscript k omitted) imply that 

If Cj = Cjo at f = 0 for all i, then equation (22) shows (through appropriate integrations) that 

Thus the problem of determining the time dependence of the concentration of any species reduces to the problem of evaluating the quadrature on the left-hand side of equation (23). 

A classical example of a process that follows the mechanism of (21) is the hydrogen-iodine reaction H2 -I- I2 2HL If Cjj, = 0 and = C[ = Cg at t = 0 for this bimolecular process, then equation (22) becomes 

Perhaps the simplest example of a reaction mechanism which is more 

Summing together the terms on the right-hand side of the equation, substituting k-i + i)[Bg] for i[Ao], and integrating for the boundary conditions B = 0 at t = 0 and B = B at time t. 

A plot of ln([Bg]/[Bg — B]) versus time results in a straight line with positive slope (ki + k-i) (Fig. l.lb). 

The half-Hfe is another useful measure of the rate of a reaction. A reaction half-life is the time required for the initial reactant(s) concentration to decrease by Useful relationships between the rate constant and the half-life can be derived using the integrated rate equations by substituting 5A0 for At. 

The resulting expressions for the half-life of reactions of different orders (n) are as follows  

The half-life of an nth-order reaction, where n 1, can be calculated from the expression 

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Consecutive reactions are those in which the product of one reaction is the reactant in the next reaction. These are also called series reactions. Reversible (opposing) reactions, autocatalytic reactions, and chain reactions can be viewed as special types of consecutive reactions. 

What conditions might alter the equilibrium state Concentration and temperature These are factors that affect the rate of reaction. Equilibrium is attained when the rates of opposing reactions become equal. Any condition that changes the rate of one of the reactions involved in the equilibrium may affect the conditions at equilibrium. 

The Law of Chemical Equilibrium Derived from Rates of Opposing Reactions... 

Chemists picture equilibrium as a dynamic balance between opposing reactions. An understanding of the Law of Chemical Equilibrium can be built upon this basis. 

These reactions can be viewed as a competition between two kinds of atoms (or molecules) for electrons. Equilibrium is attained when this competition reaches a balance between opposing reactions. In the case of reaction (3), copper metal reacting with silver nitrate solution, the Cu(s) releases electrons and Ag+ accepts them so readily that equilibrium greatly favors the products, Cu+2 and Ag(s). Since randomness tends to favor neither reactants nor products, the equilibrium must favor products because the energy is lowered as the electrons are transferred. If we regard reaction (5) as a competition between silver and copper for electrons, stability favors silver over copper. 

It is found that after the elapse of a sufficient time interval, all reversible reactions reach a state of chemical equilibrium. In this state the composition of the equilibrium mixture remains constant, provided that the temperature (and for some gaseous reactions, the pressure also) remains constant. Furthermore, provided that the conditions (temperature and pressure) are maintained constant, the same state of equilibrium may be obtained from either direction of a given reversible reaction. In the equilibrium state, the two opposing reactions are taking place at the same rate so that the system is in a state of dynamic equilibrium. 

To this point we have focused on reactions with rates that depend upon one concentration only. They may or may not be elementary reactions indeed, we have seen reactions that have a simple rate law but a complex mechanism. The form of the rate law, not the complexity of the mechanism, is the key issue for the analysis of the concentration-time curves. We turn now to the consideration of rate laws with additional complications. Most of them describe more complicated reactions and we can anticipate the finding that most real chemical reactions are composites, composed of two or more elementary reactions. Three classifications of composite reactions can be recognized (1) reversible or opposing reactions that attain an equilibrium (2) parallel reactions that produce either the same or different products from one or several reactants and (3) consecutive, multistep processes that involve intermediates. In this chapter we shall consider the first two. Chapter 4 treats the third. 

In each of these, 8 is defined by 8, = [A], - [A]tf. These expressions are applicable to any equilibration experiment on the system shown, whether by dilution, or mixing, or otherwise. These simplified forms are the result of using the equilibrium concentrations rather than the starting concentrations as the point of reference. The fourth system of opposing reactions, A P + Q as in Eq. (3-18), is left to Problem 3-7. 

A special case of opposing reactions is the one in which chemical equilibrium has been attained, but not isotopic equilibrium. Isotopic equilibration reactions are termed exchange reactions. They occur with virtually no net driving force i.e., AG 4 is very nearly zero, save for that provided by the entropy of isotopic mixing. 

Opposing reactions. D. S. Martin has considered the following reversible reaction ... 

Opposing reactions. Consider the kinetics of the N02-catalyzed isomerization of olefins. Derive the expression shown for kCV]. where KTC = k,mns/kCiS, in experiments starting only with m-olefin,11... 

Opposing reactions. Use the data on the right side of Table 3-2, concerning the triphenyl methyl radical, to calculate ki. This experiment refers to the concentration-jump method in which the parent solution was diluted with solvent to twice its initial volume. 

Opposing reactions. Derive a kinetic equation for the system A P + Q that expresses the time dependence of 8, the shift in a concentration-jump experiment. Could 8 also be regarded as the difference between the timed value of [A] and the equilibrium value of [A] If so, what are the limitations on the ways in which A, P, and Q might be mixed ... 

Opposing reactions. Calculate half-times for equilibration in the triphenyl methyl system starting with all A or with the equilibrated mixture, for the conditions given in Table 3-3. Use algebraic equations, not the tabulated numerical values. Compare the latter with the t fi from the approximate solution given in Eq. (3-39). Compare the values of 4AT-15o and a, to assess whether Eq. (3-39) provides an adequate representation. 

Opposing reactions. The reversible interconversion of d.y-[Cr(cn)2(OH)2 + to trans is characterized by an equilibrium constant of 0.16 and a forward rate constant of 3.30 X 10 4 s"1.In an experiment starting with the pure cis isomer, how long does it take to form half the equilibrium concentration of trans Starting with pure trans, how long does it take for half the equilibrium concentration of trans to form ... 

Obviously if one knows the complete form of the rate expression in one direction and the order of the opposing reaction with respect to one species, this information is sufficient to determine n uniquely and thus determine the complete form of the rate expression for the opposing reaction. 

Reaction complexities include reversible or opposing reactions, reactions occurring in parallel, and reactions occurring in series. The description of a reacting system in terms of steps representing these complexities is called a reaction network. The steps involve only species that can be measured experimentally. 

Examples of reversible reacting systems, the reaction networks of which involve opposing reactions, are ... 

A reaction network, as a model of a reacting system, mas7 consist of steps involving same ar all of opposing reactions, which may or may not be considered to be at equilibrium, parallel reactions, and series reactions. Some examples ate dted in Section 5.1. 

Reversible reactions are also termed equilibrium or opposing reactions. 

The only reactions considered so far have been those that proceed to all intents and purposes (>95%) to completion. The treatment of revers/We reactions is analogous to that given above, although now it is even more important to establish the stoichiometry and the thermodynamic characteristics of the reaction. A number of reversible reactions are reduced to pseudo first-order opposing reactions when reactants or products or both are used in excess... 

Two opposing reactions thus likely occur in caries, namely the decrease in physiological cross-links and the formation of new crosslinks in the advanced stages of the Maillard reaction. It is conceivable that the latter will prevail after a prolonged period. 

In a system at chemical equilibrium, there are always two opposing reactions, one endothermic and the other exothermic. 

Please realize that the effect of temperature on the equilibrium constant depends on which of the two opposing reactions is exothermic and on which is endothermic. You must have information on the heat of a reaction before you can apply Le Chateliers principle to judge how temperature alters the equilibrium. 

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