The law of mass action

When the reaction is at equilibrium, the quantity AjG is equal to zero. The relationship between partial pressures at equilibrium can be deduced from equations (36) and (37)  

This equilibrium constant is a pure number, and is a function of the temperature only. Example 5 

Calculate the standard equilibrium constants of the reactions given in Example 4 at 1000 K. 

FIGURE 11.16 Comparison of (a) predicted and (b) measured size distributions as a function of time for July 27. 2001 in Pittsburgh, PA. Particle number concentration (z axis) is plotted against time of day (jt axis) and particle diameter (y axis) (Gaydos et al. 2005). 

The rate of an irreversible chemical reaction can generally be written in the form r — kF(cj), where k is the rate constant of the reaction and F(ct is a function that depends on the composition of the system as expressed by concentrations c,. The rate constant k does not depend on the composition of the system—hence the term rate constant. Frequently, the function F(c, ) can be written as 

This form is termed the law of mass action. When a reaction is reversible, its rate can generally be expressed as the difference between the rates in the forward and reverse directions. The net rate can be written as 

Now the ratio kf/kr depends only on temperature. The equilibrium constant Kc, defined by 

This relation must hold regardless of the value of the concentrations. This is so if 

Now using MNH4Ci(g) = MNH,(g + MHCKg) and MNH4CKS) = MNH4CI together with (10.115), (10.126) becomes 

We find that the Kelvin equation for the critical cluster size depends on the ratio of the product of the partial pressures of NH3 and HCl to the equilibrium constant Kp. The product pNH,f HCi must exceed Kp for a critical cluster to exist. Physically this condition merely means that the product of the gas-phase partial pressures must exceed the equilibrium partial pressure product at that temperature for a solid phase to exist. It is equivalent to the condition that 5 1 for a critical cluster to exist for homogeneous-homomolecular nucleation. 

Let us compute the critical cluster size and the number of NH4CI molecules in a specific case. Countess and Heicklen (1973) studied the growth of NH4CI particles from the reaction of NH3 and HCl in a flow reactor. The experimental procedure involved mixing 60 ppm of NH 3 and 60 ppm of HCl in 1 atm of nitrogen at 293 K. The critical cluster size can be computed from (10.128). First, we need to know Kp at 293 K. We may use 

The conservation of the atoms involved in chemical reactions is stated in expressions of the type 

The last equality defines the rate v of the chemical reaction. In order to turn the relationships (3.2) into differential equations for the concentrations, the dependence of the rate v = n([A], [B].) on the concentrations should be elucidated, which is commonly done experimentally. There are reactions for which the concentration dependence of v is particularly simple, namely the product of the reactant concentrations raised to some powers, v = k[A]a[B]b. In such cases, the sum of the exponents a + b + is called the order of the reaction. In fact, molecular collision arguments (Kreuzer, 1981) imply that if the expression (3.1) really describes the elementary molecular reaction process, then the reaction rate is 

The reaction constant k, fixing the time scale, may depend on external conditions such as temperature. Expression (3.3) is commonly called the law of mass action, although this name is also given to an important relationship among product and reactant concentrations at chemical equilibrium. When the molecular mechanism leading to the reaction (3.1) involves intermediate molecular steps, expression 

In general, if we have M chemicals A, A2. Am which can appear both as reactants and as products in N coupled reactions (with reaction constants ka, a = 1. N)  

In this way we can establish the kinetic equations for the most elementary types of reactions, that will be the building blocks of our further modelling developments. Unless otherwise stated, the chemical reactions written in the following are assumed to be elementary, so that the law of mass action can be applied. In the following we will denote the molar concentrations of the chemical species with the same symbol as the species itself, i.e. [A] = A. 

Chemical equilibria for many reactions have been studied. In each case it has been found that at equilibrium the concentrations of reactants and products remained constant. As long ago as 1864, Guldberg and Waage were working on equilibrium reactions they claimed that each chemical taking part had an active mass , which was the force that controlled the progress of a reaction. They concluded that the force was proportional to the masses of the chemicals involved. 

We now know that the active masses are the concentrations of reactants and products. 

The rate of a chmiical reaction is proportional to the concentrations (in mol dm ) of the reacting substances. 

For a reacting system it is possible to withdraw samples from the equilibrium mixture and measure the concentrations of the reactants and products. By definition, we know that at equilibrium the concentrations of the reactants and products remain constant, so the ratio of their values will be an indicator of the extent of the reaction. 

Once again the reaction between ethanol and ethanoic acid provides a clear example  

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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. 

In the early twentieth century, the law of mass action was appHed to the basic pathway of dmg—receptor interaction. Assuming that response, R, is proportional to the concentration of the dmg—receptor complex, ITT, and that maximum response, occurs when all receptors are occupied (2,6,10),... 

In an undoped, intrinsic semiconductor the equiHbrium concentrations of electrons, and holes,/), are described by a lever rule derived from the law of mass action (eq. 3) ... 

The law of mass action, the laws of kinetics, and the laws of distillation all operate simultaneously in a process of this type. Esterification can occur only when the concentrations of the acid and alcohol are in excess of equiUbrium values otherwise, hydrolysis must occur. The equations governing the rate of the reaction and the variation of the rate constant (as a function of such variables as temperature, catalyst strength, and proportion of reactants) describe the kinetics of the Hquid-phase reaction. The usual distillation laws must be modified, since most esterifications are somewhat exothermic and reaction is occurring on each plate. Since these kinetic considerations are superimposed on distillation operations, each plate must be treated separately by successive calculations after the extent of conversion has been deterrnined (see Distillation). 

The rates of many reactions are not represented by application of the law of mass action on the basis of their overall stoichiometric relations. They appear, rather, to proceed by a sequence of first- and second-order processes involving short-lived intermediates which may be new species or even unstable combinations of the reaclants for 2A -1- B C, the sequence could be A -1- B AB followed by A -1- AB C. 

The two basic laws of kinetics are the law of mass action for the rate of a reac tion and the Arrhenius equation for its dependence on temperature. Both of these are strictly empirical. They depend on the structures of the molecules, but at present the constants of the equations cannot be derived from the structures of reac ting molecules. For a reaction, aA + hE Products, the combined law is... 

Power law type, based directly on the law of mass action, say,... 

The interpretation of kinetic data is largely based on an empirical finding called the Law of Mass Action In dilute solution the rate of an elementary reaction is... 

Note the rate constant symbolism denoting the forward (fc,) and backward (/c i) steps.] The differential rate equation is written, according to the law of mass action, as... 

According to the law of mass action the differential rate equation is... 

From the law of mass action it is straighforward to show that the concentrations satisfy the following three equations ... 

The subscript i labels the six concentrations and energies defined previously. It is straightforward to show that by minimising G with respect to the c, subject to the three constraints, we recover the three equations for the q predicted by the law of mass action. 

If n is the concentration of defects (cation vacancies or positive holes) at equilibrium, then, applying the law of mass action to equation 1.157... 

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. 

In the deduction of the Law of Mass Action it was assumed that the effective concentrations or active masses of the components could be expressed by the stoichiometric concentrations. According to thermodynamics, this is not strictly true. The rigorous equilibrium equation for, say, a binary electrolyte ... 

This is the rigorously correct expression for the Law of Mass Action as applied to weak electrolytes. 

If the acid is a weak electrolyte, the Law of Mass Action may be applied, and the following expressions obtained ... 

Applying the Law of Mass Action to this equation, we obtain, for any given temperature ... 

By applying the Law of Mass Action along the lines of Case 1, the following equations are obtained ... 

The equilibrium concentration of defects is obtained by applying the law of mass action to Eq. (7) or (8). This leads in the case of Frenkel disorder to... 

If the substance shared between two solvents can exist in different molecular states in them, the simple distribution law is no longer valid. The experiments of Berthelot and Jungfleiscli, and the thermodynamic deduction show, however, that the distribution law holds for each molecular state separately. Thus, if benzoic acid is shared between water and benzene, the partition coefficient is not constant for all concentrations, but diminishes with increasing concentration in the aqueous layer. This is a consequence of the existence of the acid in benzene chiefly as double molecules (C6H5COOH)2, and if the amount of unpolymerised acid is calculated by the law of mass action (see Chapter XIII.) it is found to be in a constant ratio to that in the aqueous layer, independently of the concentration (cf. Nernst, Theoretical Chemistry, 2nd Eng. trans., 486 Die Verteilnngssatz, W. Hertz, Ahrens h annulling, Stuttgart, 1909). 

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. 

The proportion of polymersed substance, or of the compound, may be calculated by the law of mass action. 

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