Equilibrium Constants from Tabulated Data

In cases where LG°, and ACp° are not tabulated for all species comprising a reaction, it is still sometimes possible to obtain Kgp values. The requirements are  

It is then possible to use numerical regression to curve fit an equation of suitable form. The form 

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The equilibrium position is dictated by the thermodynamics of the reaction, and the equilibrium constant is directly related to the standard free energy change for the reaction. So we can determine equilibrium constants from tabulated thermodynamic information. Alternatively, experimental measurements of K can also provide a route to thermodynamic data. 

In addition we will review techniques for computing these equilibrium constants from tabulations of selected values of thermodynamic properties and from experimental data. We will also examine the effects of temperature and pressure on these equilibrium constants. 

Examples 7.12 and 7.13 treated the case where the kinetic equilibrium constant had been determined experimentally. The next two examples illustrate the case where the thermodynamic equilibrium constant is estimated from tabulated data. 

The dimensionless quantity K is the thermodynamic equilibrium constant, which Section 14.3 shows can be calculated from tabulated data on the products and reactants, even if the empirical equilibrium constant defined in Equation 14.1b is not known. Therefore, K is the preferred tool for analyzing reaction equilibria in general. The informal argument by which we replaced ICp with K is made rigorous... 

Eq. (B.l) will allow fairly accurate estimates of the aetivity coefficients in mixtures of electrolytes if the ion interaction coefficients are known. Ion interaction coefficients for simple ions can be obtained from tabulated data of mean activity coefficients of strong electrolytes or from the corresponding osmotic coefficients. Ion interaction coefficients for complexes can either be estimated from the charge and size of the ion or determined experimentally from the variation of the equilibrium constant with the ionic strength. 

Here the rate equations are written as if they were for elementary steps (reaction order corresponds to stoichiometry) except for the more complex steps involved in 4 and 5 where first-order is assumed. The equilibrium constants for 2, 6, and 8 can be obtained from tabulated data thus... 

With this equation, we can calculate an equilibrium constant if we know the standard free energy change for the desired reaction. As we learned in Section 10.7, we can often obtain that from tabulated data. Example Problem 12.15 applies Equation 12.7 to the conversion of methane to methanol. 

Note the strangeness of what we are doing here. On the left-hand side of A,.G° = -RTIn K (Equation 9.11) we enter the standard Gibbs energies of the reactants and products, which in this case includes A G of 1148104 at a concentration of one molal (its concentration in its standard state) in a hypothetical ideal solution, and on the right-hand side calculated its equilibrium concentration, only a few ppm. Remember what we said in deriving the equilibrium constant-the left-hand side consists of tabulated reference state data it has nothing to do with real systems or with equilibrium. But from these data, equilibrium activity ratios and sometimes compositions can be calculated. Think about it. 

We have now acquired all the tools with which to perform one of the most practical calculations of chemical thermodynamics determining the equilibrium constant for a reaction from tabulated data. Example 13-10, which demonstrates this application, uses thermodynamic properties of ions in aqueous solution as well as of compounds. An important idea to note about the thermodynamic properties of ions is that they are relative to H" (aq), which, by convention, is assigned values of zero for AfH°, AfG°, and S°. This means that entropies listed for ions are not absolute entropies, as they are for compounds. Negative values of S° simply denote an entropy less than that of H (aq). 

Table 5 lists equilibrium data for a new hypothetical gas-phase cyclisation series, for which the required thermodynamic quantities are available from either direct calorimetric measurements or statistical mechanical calculations. Compounds whose tabulated data were obtained by means of methods involving group contributions were not considered. Calculations were carried out by using S%g8 values based on a 1 M standard state. These were obtained by subtracting 6.35 e.u. from tabulated S g-values, which are based on a 1 Atm standard state. Equilibrium constants and thermodynamic parameters for these hypothetical reactions are not meaningful as such. More significant are the EM-values, and the corresponding contributions from the enthalpy and entropy terms. 

As demonstrated by the association rate constants listed in Table 10, association is relatively fast and has low activation energies. Table 10 also tabulates the equilibrium constants for reaction of a variety of nucleophiles with carbenium ions. Most of the equilibrium constants involving trityl carbenium ions were obtained from UV studies, whereas those of me-thoxymethylium carbenium ions with both dimethyl ether and methylal were calculated using dynamic NMR [64]. Values for isobutoxy alkyl derivatives have been estimated from polymerization kinetics. The data presented in Table 10 demonstrate that the equilibrium constants are lower for weaker nucleophiles and more stable carbenium ions. For example, carbenium ions react faster with diethyl ether than with the less nucleo-... 

To compute equilibrium constants at different temperatures and pressures from tabulated standard thermodynamic data. 

Clearly, the user of any thermodynamic tables must become familiar with the tables and the interrelationships of the data if he plans to make extensive use of the values. Moreover, he must not use them blindly. The actual numbers tabulated for the different thermodynamic functions are not so significant as the final equilibrium constants that are to be calculated from them. These tables are designed to yield equilibrium constants of as high an accuracy as can be obtained from the available data. Thus, the uncertainty of a given heat of sublimation may be considerably smaller in regard to its use for calculation of vapor pressures than in regard to its use for heat balance calculations. 

The reverse rate constants for the elementary reactions used in the present work were caJculated from the forward rate constants and the equilibrium constant by assuming microscopic reversibility. Standard states used in tabulations of thermodynamic data are invariably at 1 atm and the temperature of the system. Since concentration units were required for rate constant calculations, a conversion between Kp and Kc was necessary. Values of Kp were taken from the JANAF Thermochemical tables (1984). Kc was calculated from the expression ... 

The convention we follow in this book, already stated in Section 14.2, is to describe chemical equilibrium in terms of the thermodynamic equilibrium constant K and to emphasize the fact that K can be obtained easily from tabulated thermodynamic data. Following this convention, the mass action law for a general reaction involving ideal gases is written as... 

The values are from the equation which best fit the data of Johnson (1982). His values for koH Kw are reinterpreted as indicated in Emerson (1995). The equilibrium constants necessary to calculate the reverse rate constants are also tabulated. Where two values are presented in column 1 the first is for fresh water (I = 0) and the second is for seawater. The exponential notation in column 1 indicates the order of magnitude the variable is multiplied by to equal the number in the table. (For example, x 10 in column 1 means 3.44 x 10 was multiplied by 1 x 10 before tabulating it as 3.44 in column 2.)... 

J. V. Sinisterra, J. M. Marinas, and A. Llobura [Can. J. Chem., 61, 230 (1983)] employed an extended form of the Hammett equation to correlate data for esterification of ethanol with substituted benzoic acids in the gas phase over a solid AIPO4 catalyst. The focus of their efforts was correlation of values of the adsorption equilibrium constants and apparent rate constants. Use the information tabulated below to prepare a Hammett plot of experimental absorption equilibrium constants versus the corresponding values of the a parameters for the substituted benzoic acids indicated. Employ the a values in Table 7.2 of the text. In principle the intercept of the best straight line through these data should correspond to the adsorption equilibrium constant for benzoic acid. By how much does this value differ from the experimental value below What value of the parameter p characterizes the chemisorption of substituted benzoic acids ... 

The heat capacity C , thermodynamic functions S , -(G -Hj.)/T, H"- Hj., and the equilibrium constant K for the formation of PH as an ideal gas from the elements have been calculated for a standard-state pressure of 1 atm and tabulated for 298.15 K and between 0 and 6000 K at 100 K intervals [2, 4]. The JANAF data with T = 298 K are based on early spectroscopic data [7, 8] and semiempirical estimates for Dq and the electronically excited states [3]. In the third edition of the JANAF Tables [1], the old values were only converted to Joule units and to a standard pressure of 0.1 MPa. In the Russian compilation [4], values for a reference temperature T = 0 K are given that are based on a partition function established by using spectroscopic data for the X a A, A rij states [9,10] and theoretical data for the b and c 11 states [3, 11]. Selected values from [4] are ... 

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