Net rate of reaction

Having written the mole balances, the key point for multiple reactions is to write the net rate of t ormaijon of each species (e.g., A. B). That is. we have to sum up the rate.s of formation for each reaction in order to obtain the net rate  

We note the reaction rate constants, k. in Reactions I and 2 are defined with respect to A. while k in Reaction 3 is defined with respect to B. 

The net rates of reaction of A and B are found by summing up the rates of formation of A and B for every reaction that species A and B occur. 

When we sum the rates of the individual reaction for a species, we note that for those reactions in which a species (e.g., A. B) does not appear, the rate is zero. For the first three reactions above, = 0. / iq = 0, and = 0. 

In general the net rate of reaction for species j is the sum of all rates of the reactions in which spede.s j appears. For q reactions taking place, the net rate of formation of species j. 

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V. For reversible reactions the net rate of reaction can be expressed as the difference between forward and reverse reactions ... 

Given the postulated reaction scheme, the net rate of reaction often takes a simple form when it is expressed in terms of the concentration of the intermediate. Such an expression is algebraically correct, and is the form one needs so as to propose and interpret the mechanism. This form is, however, usually not useful for the analysis of the concentration-time curves. In such an expression the reaction rate is given in terms of the concentration of the intermediate, which is generally unknown at the outset. To eliminate the concentration term for the intermediate, one may enlist certain approximations, such as the steady-state approximation. This particular method is applicable when the intermediate remains at trace levels. 

The time dependence of the composition of a system in which a reversible reaction is occurring is governed by the mathematical form of the rate expressions for the forward and reverse reactions, the net rate of reaction being the difference between these two quantities. 

The first two terms on the right represent the net rate of reaction at equilibrium which must be zero. Hence... 

The analysis is very similar to that employed in proceeding from equation 5.1.12 to equation 5.1.16, but the physical situation is somewhat different. The reaction is first allowed to come to equilibrium with Ae and Be representing the equilibrium concentrations of species A and 5, and the equilibrium extent of reaction per unit volume in a constant volume system. Under these conditions the net rate of reaction is zero. 

Now suppose that the temperature of the system is suddenly altered slightly so that it is no longer at equilibrium. The net rate of reaction is now given by... 

For reversible reactions one normally assumes that the observed rate can be expressed as a difference of two terms, one pertaining to the forward reaction and the other to the reverse reaction. Thermodynamics does not require that the rate expression be restricted to two terms or that one associate individual terms with intrinsic rates for forward and reverse reactions. This section is devoted to a discussion of the limitations that thermodynamics places on reaction rate expressions. The analysis is based on the idea that at equilibrium the net rate of reaction becomes zero, a concept that dates back to the historic studies of Guldberg and Waage (2) on the law of mass action. We will consider only cases where the net rate expression consists of two terms, one for the forward direction and one for the reverse direction. Cases where the net rate expression consists of a summation of several terms are usually viewed as corresponding to reactions with two or more parallel paths linking reactants and products. One may associate a pair of terms with each parallel path and use the technique outlined below to determine the thermodynamic restrictions on the form of the concentration dependence within each pair. This type of analysis is based on the principle of detailed balancing discussed in Section 4.1.5.4. 

At equilibrium we require that the net rate of reaction be zero. If we postulate a net rate expression of the general power function form... 

Since the rate limiting step in the overall process is the rate of adsorption of species A, the net rate of reaction is equal to the difference between the rates of adsorption and desorption. 

The dissolution rate, according to the theory, does not depend on the mineral s saturation state. The precipitation rate, on the other hand, varies strongly with saturation, exceeding the dissolution rate only when the mineral is supersaturated. At the point of equilibrium, the dissolution rate matches the rate of precipitation so that the net rate of reaction is zero. There is, therefore, a strong conceptual link between the kinetic and thermodynamic interpretations equilibrium is the state in which the forward and reverse rates of a reaction balance. 

To account for reverse as well as forward reaction, the Monod (and dual Monod) equation can be modified by appending to it a thermodynamic potential factor, as shown by Jin and Bethke (2005), in which case the equation predicts the net rate of reaction. The thermodynamic factor Ft, which can vary from zero to one, is given... 

The canonical form of equation 1.4-10, or its corresponding conventional form, is convenient for relating rates of reaction of substances in a complex system, corresponding to equation 1.4-8 for a simple system. This convenience arises because the rate of reaction of each noncomponent is independent. Then the net rate of reaction of each component can be related to a combination of the rates for the noncomponents. 

We assume that the experimental (net) rate of reaction, r, is the difference between the forward rate, rf, and the reverse rate, rr ... 

At equilibrium, the rate of forward reaction is equal to backward reaction and, therefore, net rate of reaction will be zero. Thus, at equilibrium when x = xe (concentration of B, measured separately)... 

In the foregoing discussion, we have used the basic assumption that chemical reactions go to completion, or until at least one reactant is completely used up — that they are not reversible. Many reactions do go to completion, or so nearly so as to make no difference. But a huge number of reactions are reversible, and to such an extent that the products form and accumulate and then react with each other to re-form the reactants. The reaction ultimately goes to a position of dynamic equilibrium far from completion where the rate of the forward reaction is the same as the rate of the reverse reaction, and the reaction appears to have ceased. Under these conditions the experimenter observes the net rate of reaction, which is simply the difference between the rates of the forward and reverse reactions ... 

At a given net rate of reaction in either direction, the electrical current in the external circuit is... 

The rate of the forward reaction expressed with respect to A, 9 is given by 9tM = kfCACB, and the rate of the reverse reaction (again expressed with respect to A and written 9LJ is given by = krCMCN. The net rate of reaction in the direction left to right is thus ... 

The concentration data given can be used to determine net rates of reaction by the material-balance expressions. These rates must be analyzed in terms of the stoichiometry to get the individual rates of reaction. Thus, for the trimer,... 

If one measures the net rate of reaction near equilibrium and plots the results against AF/RTj the points should fall on a straight line whose slope is equal to sRb- If one now repeats such measurements with different starting concentrations of the reacting species, one can deduce the dependence of Rb on concentrations and hence the form of the rate laws for the system. Such measurements have been made for the catalytic hydrogenation of benzene by Prigogine et al. with confirmation of the relationships shown. 

By again carrying out the approximate steady-state treatment utilized for unimolecular reactions (Sec. XI.5) we find for the net rate of reaction... 

Next, we (toermine the net rate of reaction for each species by using the appropiiate stoichiometric coefficients and then summing the rates of the individual reaction.s. ... 

Table 6-1 gives the forms of the mole balances we shall use for complex reactions where and fg are the net rates of reaction. 

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