Molality equilibrium constants, calculating

It is easily shown that equation 2.65 is a fair approximation only if the ideal solution model is valid and A, B, and C are in very low concentrations. By accepting unity activity coefficients for all the species, equation 2.63 leads one to the equilibrium constant calculated from the molalities (Km) ... 

Equation 2.67 indicates that the standard enthalpy and entropy of reaction 2.64 derived from Kc data may be close to the values obtained with molality equilibrium constants. Because Ar// is calculated from the slope of In AT versus l/T, it will be similar to the value derived with Km data provided that the density of the solution remains approximately constant in the experimental temperature range. On the other hand, the error in ArSj calculated with Kc data can be roughly estimated as R In p (from equations 2.57 and 2.67). In the case of water, this is about zero for most solvents, which have p in the range of 0.7-2 kg dm-3, the corrections are smaller (from —3 to 6 J K-1 mol-1) than the usual experimental uncertainties associated with the statistical analysis of the data. 

In thermodynamic calculation are usually used constants expressed in molality. Nevertheless, the concentration equilibrium constant calculated from molarities always may be converted into constant by molality and... 

The previous summary of activities and their relation to equilibrium constants is not intended to replace lengthier discussions [1,18,25,51], Yet it is important to emphasize some points that unfortunately are often forgotten in the chemical literature. One is that the equilibrium constants, defined by equation 2.63, are dimensionless quantities. The second is that most of the reported equilibrium constants are only approximations of the true quantities because they are calculated by assuming the ideal solution model and are defined in terms of concentrations instead of molalities or mole fractions. Consider, for example, the reaction in solution ... 

Although one can probably find exceptions, most equilibrium calculations involving flue gas slurries are performed with temperature as a known variable. With temperature known, the numerical values of the appropriate equilibrium constants can be immediately calculated. The remaining unknown variables to be determined are the activities, activity coefficients, molalities, and the gas phase partial pressures. The equations used to determine these variables are formulated from among the equilibrium expressions presented in Table 1, the expressions for the activity coefficients, ionic strength, material balance expressions, and the electroneutrality balance. Although there are occasionally exceptions, the solution sequence generally is an iterative or cyclic sequence. 

A. System NH3 H S-H20. The dissociation of water (re-action 9) and the second dissociation of H2S (reaction 6) are neglected at given temperature and total molalities of NHo and H2S there remain four unknown molalities in the liquid phase (e.g. NH3, NH4+, H2S and HS ), the composition of the vapor phase and the total pressure, which are calculated from 8 equations The dissociation constants of ammonia and hydrogen sulfide (eqs.I and III) together with the phase equilibrium for hydrogen sulfide (eq. XII) are combined resulting in a equilibrium constant K 2... 

We can calculate the equilibrium constant of reaction 8.230 under any given P-T condition by computing the standard molal Gibbs free energy values of the various components at the P-T of interest—i.e.,... 

In table 9.14, for the sake of completion, we list the thermodynamic parameters of the HKF model concerning neutral molecules in solution (Shock et al., 1989). Calculation of partial molal properties of solutes (see section 8.11), combined with calculation of thermodynamic properties in gaseous phases (Table 9.5), allows rigorous estimates of the various equilibrium constants at all P and T of interest. 

As with temperature (see above), one can use Pitzer s theoretical approach to estimate activity coefficients at various pressures for the constituents of a reaction (Eqs. 2.13 and 2.16) coupled with experimental measurements of the molalities to estimate activities and the equilibrium constant directly. Alternatively, the equilibrium constant as a function of pressure can be calculated by... 

Because K, depends on concentrations and the product KyKx is concentration independent, Kx must also depend on concentration. This shows that the simple equilibrium calculations usually carried out in first courses in chemistry are approximations. Actually such calculations are often rather poor approximations when applied to solutions of ionic species, where deviations from ideality are quite large. We shall see that calculations using Eq. (47) can present some computational difficulties. Concentrations are needed in order to obtain activity coefficients, but activity coefficients are needed before an equilibrium constant for calculating concentrations can be obtained. Such problems are usually handled by the method of successive approximations, whereby concentrations are initially calculated assuming ideal behavior and these concentrations are used for a first estimate of activity coefficients, which are then used for a better estimate of concentrations, and so forth. A G is calculated with the standard state used to define the activity. If molality-based activity coefficients are used, the relevant equation is... 

Similarly, the spread of the calculated values of the equilibrium constant of the reaction described by the stoichiometry of equation (2.5.68) is very small at all initial molalities of the Lux base and all the partial pressures of water over the melt studied. Hence, the equilibrium (2.5.68) is more correct for the description of the dissociation process for hydroxide ions, at least, in the melts based on alkali metal halides than the generally accepted reaction (1.2.4). 

These weird standard states have one very attractive feature, which is that because they all have the same value of the activity of A would always be the same in all three phases at equilibrium. The three standard states could also coexist at equilibrium, if they could exist at all. As mentioned earlier, there is no reason why other concentrations or pressures could not be chosen for the standard states, that is, other than one molal or one bar, as long as ideal behavior is still part of the definition. But these other concentrations or pressures would then appear in all activity calculations and all equilibrium constants, and we would have to give up the convenience of being able to think of gaseous activities as approximate or thermodynamic pressures, and of aqueous activities as approximate or thermodynamic concentrations. It seems generally more convenient to add a little diversity to standard states, and keep activity expressions simple, as is the present custom. 

Let s suppose that a measurement of quartz solubility has been used to obtain the free energy of formation (standard or apparent) of H4Si04 in the ideal one molal standard state. This number can then be used (with A/G° terms for the minerals) to calculate the equilibrium constant of the albite-nepheline reaction (equation (13.11)), giving the equilibrium silica concentration in a solution that may never have been experimentally determined, or perhaps never existed, and in which quartz is not stable. Thus knowing the solubility of quartz, one could in a similar way calculate the silica concentration in fluids in contact with a variety of mineral assemblages. 

Here we have represented concentration by molality, m, (instead of number of moles, rii) and include an activity coefficient correction, 7. As with equilibrium constant-based equilibrium calculations, the activity coefficients can be computed from successive estimates of the concentrations. 

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. 

At this point we have shown how the HKF model develops expressions for the standard state parameters V and and hence IT, and at high temperatures and pressures. The standard state universally used is the ideal one molal solution, which means that these parameters refer to the properties of ions or electrolytes in infinitely dilute solutions. You might suppose that therefore they would not be of much use to geochemists interested in natural solutions, which are often quite concentrated, but you would be wrong. The standard state properties allow the calculation of the equilibrium constant for reactions involving ions at high T, P, and thus permit the general nature of many important processes to be understood, even in cases where activity coefficients... 

This very detailed review on hydrofluoric acid contains critically evaluated data for the activity coefficient of HF as a function of molality and temperature (0 to 35 °C), equilibrium constants for the ionic association reactions characteristic of HF, calculated pH values, and calculated concentrations of the pertinent ions. 

We have to use these relations for the calculations which are expressed as functions of activities, but previous workers have used the apparent equilibrium constants as functions of mole fraction and molality. 

The equilibrium constant for the ionization of water can be used to calculate the equilibrium molalities of hydrogen and hydroxide ions ... 

An estimate for the association constants of aliphatic alcohols can be derived from the literature, and Table 1 contains the monomer-tetramer equilibrium constants for various systems [31]. The high values of the equilibrium constants show that, even at the low alcohol concentration region used here, a large proportion of the alcohol molecules take part in the assoeiation process. An equilibrium constant of 220 calculated on the molal scale implies that, at alcohol concentrations as low as 0.1 molal, about 26% of the alcohol molecules will be H-bonded. This shows quite clearly that alcohol association should be taken into account. Therefore, all the monolayer isotherms were modified to take into account die self-association of the alcohol molecules in die hydrocarbon solution by suggesting that only the free alcohol monomers adsorb on to the clay surface. 

Calculate the equilibrium molality m (in mol/kg of water) for each aliquot. If the results for the two aliquots from a given run are consistent, the average values of m and AT may be used in further calculations. For monochloroacetic acid, a weak electrolyte, calculate the effective total molality m from Eq. (10-20) using the appropriate constants in Table 10 I. Then calculate a and Tf for each of the two concentrations studied. 

Because of difficulties in precisely calculating the total ion activity coefficient (y) of calcium and carbonate ions in seawater, and the effects of temperature and pressure on the activity coefficients, a semi-empirical approach has been generally adopted by chemical oceanographers for calculating saturation states. This approach utilizes the apparent (stoichiometric) solubility constant (K ), which is the equilibrium ion molal (m) product. Values of K are directly determined in seawater (as ionic medium) at various temperatures, pressures and salinities. In this approach ... 

Investigations devoted to the practical verification of the reversibility of oxygen electrodes in molten ionic liquids at high pO values are scant enough. Usually the electrochemical cell calibration is performed in the pO = 1-4 range, and subsequent calculations of the equilibrium oxide-ion molalities are performed on the assumption that the slope of the E-pO plot at higher pO remains constant. 

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