Thermodynamic functions activity equilibrium constant

The true thermodynamic equilibrium constant is a function of activity rather than concentration. The activity of a species, a, is defined as the product of its molar concentration, [A], and a solution-dependent activity coefficient, Ya. 

Equation (5-43) has the practical advantage over Eq. (5-40) that the partition functions in (5-40) are difficult or impossible to evaluate, whereas the presence of the equilibrium constant in (5-43) permits us to introduce the well-developed ideas of thermodynamics into the kinetic problem. We define the quantities AG, A//, and A5 as, respectively, the standard free energy of activation, enthalpy of activation, and entropy of activation from thermodynamics we now can write... 

Considering only the aqueous phase of the biocatalytic system, the equilibrium constant for the reaction is given as a function of thermodynamic activities of the components shown ... 

Thus, rate constant for a reaction can be given by equation (4.34) in terms of equilibrium constant for the formation of activated complex K which can be expressed either in terms of partition or thermodynamic functions. 

Therefore, the physical meaning of the solubility curve of a surfactant is different from that of ordinary substances. Above the critical micelle concentration the thermodynamic functions, for example, the partial molar free energy, the activity, the enthalpy, remain more or less constant. For that reason, micelle formation can be considered as the formation of a new phase. Therefore, the Krafft Point depends on a complicated three phase equilibrium. 

If the heat capacity functions of the various terms in the reaction are known and their molar enthalpy, molar entropy, and molar volume at the 2) and i). of reference (and their isobaric thermal expansion and isothermal compressibility) are also all known, it is possible to calculate AG%x at the various T and P conditions of interest, applying to each term in the reaction the procedures outlined in section 2.10, and thus defining the equilibrium constant (and hence the activity product of terms in reactions cf eq. 5.272 and 5.273) or the locus of the P-T points of univariant equilibrium (eq. 5.274). If the thermodynamic data are fragmentary or incomplete—as, for instance, when thermal expansion and compressibility data are missing (which is often the case)—we may assume, as a first approximation, that the molar volume of the reaction is independent of the P and T intensive variables. Adopting as standard state for all terms the state of pure component at the P and T of interest and applying... 

The kinetic constants, calculated in the previous section, can immediately by applied to the investigation of the energetics of the flow-equilibrium, especially to the calculation of the activation enthalpy AH and entropy AS related to the coil relaxation in the actual gel phase as mentioned above, and to the coil release from segment-segment contacts with the gel, before the retarded rediffusion of these coils from the gel into the sol sets on. These thermodynamic functions can then be compared with those of the reversible polymer transfer gel - sol calculated in Section 3.1. 

The thermodynamic reaction equilibrium constant K, is only a function of temperature. In Equation 4.18, m, the activity of the guest in the vapor phase, is equal to the fugacity of the pure component divided by that at the standard state, normally 1 atm. The fugacity of the pure vapor is a function of temperature and pressure, and may be determined through the use of a fugacity coefficient. The method also assumes that an, the activity of the hydrate, is essentially constant at a given temperature regardless of the other phases present. 

The pseudothermodynamic equilibrium constant of the activated complex (A q) is related to the thermodynamic state functions by... 

In each case these parameters represent differences between the state function of the activated complex in a particular standard state and the state function of the reactants referred to in the same standard state. One is giving all the characteristics of a thermodynamic equilibrium constant, although it should be multiplied by a transitional partition function. For ideal systems the magnitude of AH° does not depend on the choice of standard state, and for most of the nonideal systems that are encountered the dependence is slight. For all systems, the magnitudes of AG° and AS0 depend strongly on the choice of standard state, so it is not useful to... 

Correlations for the determination of the dissociation equilibrium constants and solubility values for SO2 and CO2 as functions of temperature as well as the equations for activity coefficients are given in Ref. [70], Thermodynamic non-idealities are taken into account depending on whether species are charged, or not. For uncharged species, a simple relationship from Ref. [102] is applied, whereas for individual ions, the extended Debye-Hiickel model is used according to Ref. [103]. 

The thermodynamic information is normally summarized in a Pourbaix diagram7. These diagrams are constructed from the relevant standard electrode potential values and equilibrium constants and show, for a given metal and as a function of pH, which is the most stable species at a particular potential and pH value. The ionic activity in solution affects the position of the boundaries between immunity, corrosion, and passivation zones. Normally ionic activity values of 10 6 are employed for boundary definition above this value corrosion is assumed to occur. Pourbaix diagrams for many metals are to be found in Ref. 7. 

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... 

From all that has been said about activity and activity coefficients, it is apparent that whenever precise results are to be expected, activities should be used when expressing equilibrium constants or other thermodynamic functions. In the present text however we shall be using simply concentrations. For the dilute solutions of strong and weak electrolytes that are mainly used in qualitative analysis, errors introduced into calculations are not considerable. 

WATEQ2 consists of a main program and 12 subroutines and is patterned similarly to WATEQF ( ). WATEQ2 (the main program) uses input data to set the bounds of all major arrays and calls most of the other procedures. INTABLE reads the thermodynamic data base and prints the thermodynamic data and other pertinent information, such as analytical expressions for effect of temperature on selected equilibrium constants. PREP reads the analytical data, converts concentrations to the required units, calculates temperature-dependent coefficients for the Debye-HKckel equation, and tests for charge balance of the input data. SET initializes values of individual species for the iterative mass action-mass balance calculations, and calculates the equilibrium constants as a function of the input temperature. MAJ EL calculates the activity coefficients and, on the first iteration only, does a partial speciation of the major anions, and performs mass action-mass balance calculations on Li, Cs, Rb, Ba, Sr and the major cations. TR EL performs these calculations on the minor cations, Mn, Cu, Zn, Cd, Pb, Ni, Ag, and As. SUMS performs the anion mass... 

The equilibrium constant can also be determined through studies of the dissociation of the exchanger in its H -form during neutralization with standard base (pH titration) by the method of Argersinger et al. [203]. However, numerical values of thermodynamic functions include the difference in hydration between the two cations in the solid and solution phases and the change in water activity cannot be ignored. The changes... 

The thermodynamic quantities could be calculated from measurements of the equilibrium constant as a function of temperature. KlSpffer (9) estimated a value of -940 cal/mol for AG in bls-carbazolylpropane. Recently, Johnson (14b) made a very elegant and detailed study of 1,3-biscarbozolylpropane in 2 Me THE using steady state and dynamic measurements. They found an activation energy for exclmer formation of 4.24 kcal/mole. The enthalpy of formation was observed to be -2.75 kcal/mole and the entropy of formation equals -7.2 cal mole deg. The entropy loss in the Intramolecular exclmer formation is smaller than in the inter-molecular case (16). Therefore Intramolecular excimer formation is possible with chromophores that do not show Intermolecular excimer formation in the monofunctional models, as is the case in biscarbazolylpropane relative to N-isopropylcarbazole. 

Equation (5.14) describes the relationship between real (thermodynamic) equilibrium constant and concentrations of the reagents. Equations for activity coefficients of aqueous ionic species as the function of concentrations of all ionic species in solution (at least at ionic strengths up to 0.1 mol dm are well known and generally accepted. It should be emphasized that these equations apply only to the solution species. When E Vy log 7, - log 7, in Eq. (5.14) is constant for each i over the entire data set, one can simply use Eqs. (5.7) and (5,9) to calculate AT, and then calculate using the following relationship... 

Summary This chapter has presented a set of variations on the theme of free energy changes. We have seen how accurate values of AG° may be derived at any temperature, how this data may be summarized either as free energy functions or in terms of Ellingham diagrams, and how the data may be applied in a few instances. In virtually all cases, we have seen that activity can simplify the calculations of equilibrium constants and that allowances can always be made for non-ideal behaviour, assuming that activity coefficient data are available. Complete thermodynamic data have been published for relatively few compounds, however, and there are for example many common organic compounds for which only an enthalpy of formation has been determined. As more complete information is circulated, the number of applications of chemical... 

In order that the equilibrium constant be properly related to the other thermodynamic functions for the substance, such as free energy Fy heat content Hy and entropy S, it should be expressed in terms of the activity rather than the concentration. For the type of system with which we will be most concerned, namely, reactions... 

The relationship between thermodynamics and kinetics in chemical reactions is usually expressed by the Bronsted equation (eq. 3.52 in chapter 3.4) k = gKa, where k is the rate constant, K is the equilibrium constant of the elementary stage, and g and a (Polanyi parameter) are constant values for a serious of reactions. These constants are determined by parameters characterizing the elementary mechanism (composition and structure of the activated complexes, etc.) thus allowing for the existence of an optimum catalyst, on which the rate of catalytic reaction per unit of surface has a maximum value. Equations of the type (3.52) were used for the explanation of "volcano-curves", when catalytic activity as a function of thermodynamic characteristics follows a curve with a maximum. An example for a volcano curve in methanation of CO is given in Figure 7.6. 

The equilibrium constant for the generalized reaction may be written in terms of partial pressure p if the components are behaving as ideal gases or thermodynamic functions as activity a or fugacity / may be used for nonideal systems. 

Now we know from thermodynamics that the concentration equilibrium constant is not the proper one in the sense that it can be a function of concentrations in addition to temperature, especially for liquids and for gases at high pressure. Thus, in thermodynamics, the proper variable of activity is introduced ... 

According to the limitation stated above, our standard functions /u. = pI, T) depend only on temperature and therefore also equilibrium constants depend on temperature only and by (4.474) give restrictions on the values of activities a° in chemical equilibrium (denoted by superscript cf. Sect.4.7). Equations (4.473) and (4.474) permit calculations of chemical equilibria KLp may be calculated from the right-hand side of (4.473) (e.g. from thermodynamic data for pure constituents if they are taken as the standard state) and composition of equilibrium mixture is restricted by (4.474) if we know the relation of activities to composition simple results follow for important case (4.469), which will be used below (4.475). 

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