Chemical Equilibrium and the Law of Mass Action

In this section we shall study chemical equilibrium in detail. At equilibrium the pressures and temperatures are uniform moreover affinities and the corresponding velocities of reaction vanish. For a reaction such as 

The condition that the thermodynamic force, affinity A equals zero implies that the corresponding thermodynamic flow, i.e. the reaction velocity d /dt, also equals zero. The condition A = 0 means that at equilibrium the stoichiometric sums for the chemical potentials of the reactants and products are equal, as in (9.3.3). It is easy to generalize this result to an arbitrary chemical reaction of the form 

Such equalities of chemical potentials are valid for all reactions, changes of phase, chemical, nuclear and elementary particle reactions. Just as a difference in temperature drives the flow of heat until the temperatures difference vanishes, a nonzero affinity drivfes a chenfical reaction until the affinity vanishes. 

To understand the physical meaning of the mathematical conditions such as (9.3.3) or (9.3.5), we express the chemical potential in terms of experimentally measurable quantities. We have seen in (5.3.5) that a general chemical potential can be expressed as 

For ideal solutions = 1. For nonideal solutions Yjt is obtained by various means, depending on the type of solution. The chemical potential can also be written in terms of the concentrations by appropriately redefining Pj o-We can now use (9.3.6) to express the condition for equilibrium (9.3.3) in terms of the activities, which are experimentally measurable quantities  

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The fundamental law of chemical equilibrium is the law of mass action, formulated in 1864 by Cato Maximilian Guldberg and Peter Waage. It has since been redefined several times. Consider the equilibrium between the four chemical species A, B, C and D ... 

We have introduced the equilibrium constant in terms of a ratio of rate constants, but the original research on chemical equilibrium was developed many years before the principles of kinetics. In 1864, two Norwegian chemists, Cato Guldberg and Peter Waage, observed that at a given temperature, a chemical system reaches a state in which a particular ratio of product to reactant concentrations has a constant value. This is a statement of the law of chemical equilibrium, or the law of mass action. 

There are two main ideas in the foundation of classical chemical thermodynamics and in chemical kinetics the notion of dynamic equilibrium and the law of mass action. Historically, formulating the mass action law Van t Hoff proposed that the reaction rate was determined by the concentrations of reacting molecules [5]. The elementary acts of chemical transformations in forward and backward reactions can proceed independently. According to the notion of dynamic equilibrium, the chemical equilibrium is established when the rates of forward and backward reactions become equal. 

These two new quantities have the merit that they are dimensionless (which the fugacity is not) and, as we will see later, they lead to very useful correlations of liquid-phase fugacities. We will also see in Chapter 12 that the normal chemical equilibrium statement, the law of mass action, is given in terms of activities. 

Listed after the reactions are the corresponding equilibrium quotients. The law of mass action sets the concentration relations of the reactants and products in a reversible chemical reaction. The negative log (logarithm, base 10) of the quotients in Eqs. (3.1)—(3.4) yields the familiar Henderson-Hasselbalch equations, where p represents the operator -log ... 

The equilibrium concentrations of point defects can be derived on the basis of statistical mechanics and the results are identical to those obtained by a less fundamental quasi-chemical approach in which the defects are treated as reacting chemical species obeying the law of mass action. The latter, and simpler, approach is the one widely followed. 

A chemical reaction is normally not described by the linear relation in the third line of eq 14.17. The rate is a non-linear function of hfijT on the macroscopic level, and the law of mass action is used. We explain in subsection 14.3.3 how chemical reactions far from equilibrium also can be included into the scheme of non-equilibrium thermodynamics, cf. also point e of section 14.1.1. 

Complex chemical mechanisms are written as sequences of elementary steps satisfying detailed balance where tire forward and reverse reaction rates are equal at equilibrium. The laws of mass action kinetics are applied to each reaction step to write tire overall rate law for tire reaction. The fonn of chemical kinetic rate laws constmcted in tliis manner ensures tliat tire system will relax to a unique equilibrium state which can be characterized using tire laws of tliennodynamics. 

Guldberg and Waage (1867) clearly stated the Law of Mass Action (sometimes termed the Law of Chemical Equilibrium) in the form The velocity of a chemical reaction is proportional to the product of the active masses of the reacting substances . Active mass was interpreted as concentration and expressed in moles per litre. By applying the law to homogeneous systems, that is to systems in which all the reactants are present in one phase, for example in solution, we can arrive at a mathematical expression for the condition of equilibrium in a reversible reaction. 

The deduction adopted is due to M. Planck (Thermodynamik, 3 Aufl., Kap. 5), and depends fundamentally on the separation of the gas mixture, resulting from continuous evaporation of the solution, into its constituents by means of semipermeable membranes. Another method, depending on such a separation applied directly to the solution, i.e., an osmotic process, is due to van t Hoff, who arrived at the laws of equilibrium in dilute solution from the standpoint of osmotic pressure. The applications of the law of mass-action belong to treatises on chemical statics (cf. Mel lor, Chemical Statics and Dynamics) we shall here consider only one or two cases which serve to illustrate some fundamental aspects of the theory. 

To construct such a diagram, a set of defect reaction equations is formulated and expressions for the equilibrium constants of each are obtained. The assumption that the defects are noninteracting allows the law of mass action in its simplest form, with concentrations instead of activities, to be used for this purpose. To simplify matters, only one defect reaction is considered to be dominant in any particular composition region, this being chosen from knowledge of the chemical attributes of the system under consideration. The simplified equilibrium expressions are then used to construct plots of the logarithm of defect concentration against an experimental variable such as the log (partial pressure) of the components. The procedure is best illustrated by an example. 

Every chemical reaction reaches after a time a state of equilibrium in which the forward and back reactions proceed at the same speed. The law of mass action describes the concentrations of the educts (A, B) and products (C, D) in equilibrium. The equilibrium constant K is directly related to the change in free enthalpy G involved in the reaction (see p.l6) under standard conditions (AG° = - R T In K). For any given concentrations, the lower equation applies. At AG < 0, the reaction proceeds spontaneously for as long as it takes for equilibrium to be reached (i.e., until AG = 0). At AG > 0, a spontaneous reaction is no longer possible (endergonic case see p.l6). In biochemistry, AG is usually related to pH 7, and this is indicated by the prime symbol (AG° or AG ). 

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. 

Intuitively, the uniqueness of the chemical equilibrium state of a mixture of reacting gases is more or less obvious. However, it may be of some interest to rigorously prove that the system of equations of the law of mass action (LMA), together with the imposed conditions of conservation of matter for given T and v or T and p, has one and only one real-valued and positive solution. 

Historically, the state of reaction at chemical equilibrium was evaluated for fairly simple reactions, with only a few species, from the "Law of Mass Action. 1 In recent years, high-temperature reactions, including many possible species (as many as 20 or more), have become of interest and newer techniques suitable for numerical solution on high-speed digital computers have been developed.2 Initially, we will discuss chemical equilibrium from the vantage point of the "Law of Mass Action." It states that the rate at which a chemical reaction proceeds is proportional to the "active" masses of the reacting substances. The active mass for a mixture of ideal gases is the number density of each react-... 

Process (VIA) is the net reaction in excess arsenite ([H3As03]o/[IOJ]0 > 3) it is equivalent to (VIB) + 3 (VIC). Process (VIB), the Dushman reaction, is normally rate determining. Therefore, the net process (VIA) is autocatalytic in [I-], which causes a dramatic color change to arise at the stoichiometric point due to the sudden appearance of I2. Bognar and Sarosi exploited this fact to devise a chronometric technique for the determination of traces of iodide43. Process (VIC), the Roebuck reaction44, has played an important role in the development of chemical kinetics and teaching of physical chemistry, as it was the first multi-step reaction for which it was shown that the quotient of the independently determined forward and reverse rate laws equals the equilibrium quotient obtained from the law of mass action. 

Many chemical and biocatalytic conversions involve reactions with an unfavorable equilibrium such as condensations.8 In both cases the Law of Mass Action will apply such that the removal of one species from the reaction mixture will shift the equilibrium position. This is particularly useful for biocatalytic conversions where the alternative approach of using a reactant in excess may have deleterious effects on the biocatalyst. 

The last equation, one of the most important physicochemical equations, expresses exactly the law of mass action, formulated for the first time by Guldberg and Waage in a less exact form. The equation enables the calculation of the equilibrium composition of a reaction mixture or determination of theoretically possible yields of chemical processes starting from the known value of the equilibrium constant K which can be determined by thermodynamic methods. 

ACTIVITY AND ACTIVITY COEFFICIENTS In our deduction of the law of mass action we used the concentrations of species as variables, and deduced that the value of the equilibrium constant is independent of the concentrations themselves. More thorough investigations however showed that this statement is only approximately true for dilute solutions (the approximation being the better, the more dilute are the solutions), and in more concentrated solutions it is not correct at all. Similar discrepancies arise when other thermodynamic quantities, notably electrode potentials or chemical free energies are dealt with. To overcome these difficulties, and still to retain the simple expressions derived for such quantities, G. N. Lewis introduced a new thermodynamic quantity, termed activity, which when applied instead of concentrations in these thermodynamic functions, provides an exact fit with experimental results. This quantity has the same dimensions as concentration. The activity, aA, of a species A is proportional to its actual concentration [A], and can be expressed as... 

The equations of heterogeneous isotope exchange are simpler than ion-exchange equations because the two ions are chemically the same. In the treatment by the law of mass action, it means that the equilibrium constant is equal to 1. The selectivity coefficients at X = 0 and X = 1 can be determined by measuring heterogeneous isotope exchange in which the concentration of the radioactive isotopes is very low and approaches zero (carrier-free radioactive isotope). 

K is defined in terms of the chemical potentials of the standard states and so is a constant of the reaction, independent of the concentrations of the species. According to the definition of the chemical potential, A/f is the difference in formation energies of the species on the two sides of Reaction (6.15). Eq. (6.19) is the law of mass action, which gives the equilibrium concentrations of the different species in terms of the reaction constant K. 

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