The Mass Action Law

What is commonly known as the mass action law as defined by Guldberg and Waage is a consequence of a combination of the mass actions of individual substances participating in a reaction. Let us, once again, consider a reaction in a 

Equilibrium is established when there is no longer any potential gradient and the drive JL disappears. Therefore  

By solving the equation for A and applying —ln(x/y) = In(y/r), we obtain When we divide by RT and take the antilogarithm, we finally obtain 

Equation (6.17) characterizes the relationship between concentrations in equilibrium which has been referred to by the index eq. and shows a possible form of the mass 

7C is a number and it is not constant but dependent upon temperature, pressure, solvent, etc. The index °, which is actually superfluous, has been inserted in order to 

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The Langmuir-Hinshelwood picture is essentially that of Fig. XVIII-14. If the process is unimolecular, the species meanders around on the surface until it receives the activation energy to go over to product(s), which then desorb. If the process is bimolecular, two species diffuse around until a reactive encounter occurs. The reaction will be diffusion controlled if it occurs on every encounter (see Ref. 211) the theory of surface diffusional encounters has been treated (see Ref. 212) the subject may also be approached by means of Monte Carlo/molecular dynamics techniques [213]. In the case of activated bimolecular reactions, however, there will in general be many encounters before the reactive one, and the rate law for the surface reaction is generally written by analogy to the mass action law for solutions. That is, for a bimolecular process, the rate is taken to be proportional to the product of the two surface concentrations. It is interesting, however, that essentially the same rate law is obtained if the adsorption is strictly localized and species react only if they happen to adsorb on adjacent sites (note Ref. 214). (The apparent rate law, that is, the rate law in terms of gas pressures, depends on the form of the adsorption isotherm, as discussed in the next section.)... 

The mass action law or Saha equation for thermal ionisation of seed atoms is... 

This was first demonstrated ia 1862 by Berthelot and Saint-Gibes (32), who found that when equivalent quantities of ethyl alcohol and acetic acid were abowed to react, the esterification stopped when two-thirds of the acid had reacted. Sinularly, when equal molar proportions of ethyl acetate and water were heated together, hydrolysis of the ester stopped when about one-third of the ester was hydroly2ed. By varyiag the molar ratios of alcohol to acid, yields of ester >66% were obtained by displacement of the equbibrium. The results of these tests were ia accordance with the mass action law shown ia equation 5. 

Ion Exchange A useful tool is provided by the mass action law for describing the general exchange equilibrium in fully ionized exchanger systems as... 

The inaccuracy seems not to prohibit study of the structural properties of associating fluids, at least at low values of the association energy. However, what is most important is that this difficulty results in the violation of the mass action law, see Refs. 62-64 for detailed discussion. To overcome the problem, one can apply thermodynamical correspondence between a dimerizing fluid and a mixture of free monomers of density p o = P/30 = Po/2 and dimer species [12]. The equation of state of the corresponding mixture... 

For both cases, the assumption is valid that only one helical sequence exists and that products with the same number of hydrogen bonds have the same stability. Considering the statistical weights of the possible intermediates, the whole measurable degree of conversion, t0,ai, is computed by the mass-action law and can be derived from Eq. (6)149. ... 

Connection between the reversible target binding and the change in fluorescence intensity can be easily established based on the mass action law. In the simplest case... 

This type of defect equilibrium treatment has been used extensively to model the defect chemistry and non-stoichiometry of inorganic substances and has the great advantage that it easily takes several simultaneous defect equilibria into account [22], On the other hand, the way the mass action laws are normally used they are focused on partial thermodynamic properties and not on the integral Gibbs energy. The latter is often preferred in other types of thermodynamic analyses. In such cases the following solid solution approach is an alternative. 

Calculations of the composition (112/ns) of symmetric nuclear matter (np = nn, no Coulomb interaction) are shown in Fig. 3 [7], At low densities, the contribution of bound states becomes dominant at low temperatures. At fixed temperature, the contribution of the correlated density 112 is first increasing with increasing density according to the mass action law, but above the Mott line it is sharply decreasing, so that near nuclear matter density (ns = ntot = 0.17 fm-3) the contribution of the correlated density almost vanishes. Also, the critical temperature for the pairing transition is shown. 

In the case of classic chemical kinetics equations, one can get in a few cases analytical solution for the set of differential equations in the form of explicit expressions for the number or weight fractions of i-mcrs (cf. also treatment of distribution of an ideal hyperbranched polymer). Alternatively, the distribution is stored in the form of generating functions from which the moments of the distribution can be extracted. In the latter case, when the rate constant is not directly proportional to number of unreacted functional groups, or the mass action law are not obeyed, Monte-Carlo simulation techniques can be used (cf. e.g. [2,3,47-52]). This technique was also used for simulation of distribution of hyperbranched polymers [21, 51, 52],... 

DiCera s book starts with the construction of the PF of the system, then switches to the BP based on the mass action law, but still refers to it as the PF of the system. Much of the remainder of the book contains lengthy lists of mass-action-law equations for binding reactions and the corresponding equilibrium constants. This is followed by lengthier lists of contracted BPs (referred to as contracted PFs). The contracted BPs (or PFs) do not provide any new information that is not contained in, or can be extracted from, the PF of the binding system, nor do they possess any new interpretive power. 

We may begin the examination of ionic micelle formation by reviewing the main theories already presented. First of all, the mass action law is extended to ionic micelle formation( 14---16) as... 

Although Freudenberg s hypothesis that complex-formation occurred by inclusion within the cavity was generally accepted, there was no direct evidence for this, either in solution or in the solid state. Broser and Lautsch had found by spectrophotometric titration that the complexes of a series of dyes with the cyclodextrins in solution obeyed the mass action law with a stoichiometry of 1 1. They suggested that association on the outside of the ring might not have a defined stoichiometric composition, and they thus interpreted their results as being consistent with inclusion by cyclodextrin. Their results were not conclusive, however. 

Applying the mass action law and assuming unitary activity for the water solvent and the condensed phase, we obtain... 

The network formation theories are based mainly on the assumption of the validity of the mass action law and Arrhenius dependence of the rate constants. However, diffusion control can be taken into account by some theories in which whole molecules appear as species developing in time. 

A mathematical description based on the mass action law was derived in Chapter 2. In this ideal system, the change in drug mobility observed when the CD is introduced into the capillary is caused solely by the com-plexation of the drug with the CD ... 

In many cases it is assumed that the rate of any elementary reaction is governed by the Mass-Action-Law (MAL). We will make the same assumption as well. 

The Jacobian matrix for the mass action law kinetic equation (77) is ... 

The oxidation is thought to proceed by reaction (8.9) at interface II, followed by subsequent diffusion of copper vacancies and holes from II to I movement of copper vacancies determines the rate. Applying the mass-action law we obtain... 

Ion-exchange equilibrium can be considered to be analogous to chemical equilibrium. From that point of view, the mass-action law can be used to express the state of equilibrium despite the fact that this law is defined exclusively for homogeneous systems. Derived this way, the so-called pseudo-equilibrium constant Ke is not really a constant, since it depends on the total concentration ... 

Thus the partial derivative required for Scc(0) can be obtained in terms of x, T, z and the corresponding derivative of z. The latter can be obtained in terms of x, T, and z also, as shown immediately below. Therefore for a given x and T and values of the interaction parameters, z is obtained numerically from Eq. (80) and the value of Scc(0) is then calculated. The required composition derivative of z can be obtained starting with the mass action law given by Eq. (63). Eliminating and y3 with Eq. (57) and rearranging, this is... 

Consider the mass action law given by Eq. (63). In the A-C binary at atom fraction the effective equilibrium constant can be obtained from Eq. (82) with / 14 = -/ 34 to write the mass action law as... 

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