Mass-action-like equations

This expression is known as the law of mass action. Like the equilibrium constant K, Q is subscripted with a c to indicate that the quotient is defined in terms of concentrations. For the N2O4-NO2 system, this approach gives the same equilibrium expression as we got from our kinetics approach [N02]eq/[N204leq. But the law of mass action was developed empirically from countless observations of many different reactions—long before the principles of kinetics were developed. Additionally, it applies not only to elementary reactions, but also to more complex reactions that occur via a series of steps. Furthermore, the law of mass action enables us to write the equilibrium expression for any reaction for which we know the balanced equation. Knowing the equilibrium expression for a reaction, we can use equilibrium concentrations to calculate the value of the equilibrium constant. 

Like all formulations of the multicomponent equilibrium problem, these equations are nonlinear by nature because the unknown variables appear in product functions raised to the values of the reaction coefficients. (Nonlinearity also enters the problem because of variation in the activity coefficients.) Such nonlinearity, which is an unfortunate fact of life in equilibrium analysis, arises from the differing forms of the mass action equations, which are product functions, and the mass balance equations, which appear as summations. The equations, however, occur in a straightforward form that can be evaluated numerically, as discussed in Chapter 4. 

In the absence of an enzyme, the reaction rate v is proportional to the concentration of substance A (top). The constant k is the rate constant of the uncatalyzed reaction. Like all catalysts, the enzyme E (total concentration [E]t) creates a new reaction pathway, initially, A is bound to E (partial reaction 1, left), if this reaction is in chemical equilibrium, then with the help of the law of mass action—and taking into account the fact that [E]t = [E] + [EA]—one can express the concentration [EA] of the enzyme-substrate complex as a function of [A] (left). The Michaelis constant lknow that kcat > k—in other words, enzyme-bound substrate reacts to B much faster than A alone (partial reaction 2, right), kcat. the enzyme s turnover number, corresponds to the number of substrate molecules converted by one enzyme molecule per second. Like the conversion A B, the formation of B from EA is a first-order reaction—i. e., V = k [EA] applies. When this equation is combined with the expression already derived for EA, the result is the Michaelis-Menten equation. 

The second equation applies the law of mass action with rate constant iifp. The equilibrium density of pairs is proportional to the phosphorus concentration. Therefore, the square root law for the defects in Fig. 5.9 is inconsistent with the formation of pairs, if the equilibrium model is valid. The different concentration dependences for paired and isolated states imply that pairing is most likely at the highest phosphorus or boron concentrations. 

Cato Guldberg (1836-1902) and Peter Waage (1833-1900) were Norwegian chemists whose primary interests were in the field of thermodynamics. In 1864, these workers were the first to propose the law of mass action, which is expressed in Equation 9-7. If you would like to learn more about Guldberg and Waage and read a translation of their original paper on the law of mass action, use your Web browser to connect to http //cheniistry.brookscoIe com/skoogfac/. From the Chapter Resources menu, choose Web Works, find Chapter 9, and dick on the link to the paper. 

Note that the stoichiometric coefficients in a balanced chemical equation like eq. (2.5) bear no necessary relationship to the orders that appear in the empirical (i.e., experimentally derived) rate law. This statement becomes obvious if one considers that the chemical equation can be multiplied on both sides by any arbitrary number and remain an accurate representation of the stoichiometry even though all the coefficients will change. However, the orders for the new reaction will remain the same as they were for the old one. There are cases in which the rate law depends only on the reactant concentrations and in which the orders of the reactants equal their molecularity. A reaction in which the order of each reactant is equal to its molecularity is said to obey the Law of Mass Action or to behave according to mass action kinetics. 

Equation (15.2) is the mathematical form of the law of mass action. It relates the concentrations of reactants and products at equilibrium in terms of a qrrantity called the equilibrium constant The equilibritun constant is defined by a quotient. The ntunera-tor is obtained by mrrltiplying together the eqtrilibrium concentrations of the products, each raised to a power equal to its stoichiometric coefficient in the balanced equation. The same procedrrre is applied to the equilibrium concentrations of reactants to obtain the denominator. This formulation is based on purely empirical evidence, such as the study of reactions like NO2-N2O4. 

With kinetics assumed to be of the mass action type, two main characteristics remain to be determined by the kinetic analysis 1) the rate coefficient, k, 2) the reaction order, global a + b oi partial a with respect to A h with respect to B. The order of a reaction is preferably determined from experimental data. It only coincides with the molecularity for elementary processes that actually occur as described by the stoichiometric equation (1.1.1-1). When the latter is only an "overall" equation for a process that really consists of several steps, the order cannot be predicted on the basis of this equation. Only for elementary reactions does the order have to be 1, 2, or 3. The order may be a fraction or even a negative number. In Section 1.6 examples will be given of reactions whose rate cannot be expressed as a simple product like (1.2.1-1). 

As a direct consequence of the particular role of Dynamics, as such,in the study of non-equilibrium behaviour of chemical systems, two classes of models are to be considered, depending on which aspect one is insisting on. Formal models, of mathematical or chemical-like nature, are designed to exhibit specific dynamical behaviours, without too much concern about chemical significance. Their aim is to provide examples of evolution equations of chemical reacting systems, as described by mass action kinetics, that are able to produce those exotic behaviours, such as bistability or multistability, between various types of attractors, like steady states, oscillations or deterministic chaos. A typical historical model of that kind is the "Brusselator ... 

This chapter is meant as a brief introduction to chemical kinetics. Some central concepts, like reaction rate and chemical equilibrium, have been introduced and their meaning has been reviewed. We have further seen how to employ those concepts to write a system of ordinary differential equations to model the time evolution of the concentrations of all the chemical species in the system. The resulting equations can then be numerically or analytically solved, or studied by means of the techniques of nonlinear dynamics. A particularly interesting result obtained in this chapter was the law of mass action, which dictates a condition to be satisfied for the equilibrium concentrations of all the chemical species involved in a reaction, regardless of their initial values. In the forthcoming chapters we shall use this result to connect different approaches like chemical kinetics, thermodynamics, etc. 

Example 4.3 In elementary chemistry we learn the law of mass action, which says that for a reaction like the iso-butane-normal-butane isomerization the concentrations at equilibrium can be represented by an equation of the form... 

For ideal gas reactions in which the number ofmols does not change (like this one) the P and the pressure dimension in the mass action equation always cancel, and we find the expression in terms of mol fractions. We will see that for reactions in which the number of mols changes, this cancellation does not occur, and the formulae are more complex. 

As with divalent eation adsorption at the aqueous S—MO interfaee, numerous eation—surfaee site stoiehiometries are possible. Sinee there is no means to predict which stoichiometries are predominant and which are insignificant, the trivalent cation adsorption equation derivation will consider six of the more likely complexes. Modification of the resulting adsorption equation can then proceed as experimental evidence identifies which stoichiometries are in fact more prevalent. Since the derivation of mass action and mass balance equations will proceed as for other cations, the derivations will be given without much discussion. 

Equations (400) and (401) are written as complexation or formation reactions not as acid ionization reactions like Eqs (14) and (19). The unfortunate aspect of these different conventions is that the same symbol was used in the literature to represent both the mass action constant for the acid ionization chemical reactions, Eqs (14) and (19), and the formation reactions, Eqs (400) and (401), as depicted in Table 17. 

Prom this survey it appears that there seem to be two criteria for distinguishing chemical laws from chemical equations as law-like generalities. The chemical laws cited above describe relationships that are thought to hold good for any chemically relevant substance. Compare Berthollet s Law of Mass Action with a chemical equation, for example Ca(OH)2 + 2HNO3 Ca(N03)2 + 2H2O. The equation is general over all samples of calcium hydroxide and all samples of nitric acid. But it has no application to any other chemical substances, a fortiori. However, the Law of Mass Action applies to all chemical substances when suitably juxtaposed. 

Let us start with the action of Young-Laplace law (Equation 9.6), which determines the equilibrium configuration of the fluids (liquid and liquid-like phases) and the driving force of mass transfer that cause the spontaneous formation of equilibrium configurations. 

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