Activity coefficient added

Another approach to matrix matching, which does not rely on knowing the exact composition of the sample s matrix, is to add a high concentration of inert electrolyte to all samples and standards. If the concentration of added electrolyte is sufficient, any difference between the sample s matrix and that of the standards becomes trivial, and the activity coefficient remains essentially constant. The solution of inert electrolyte added to the sample and standards is called a total ionic strength adjustment buffer (TISAB). 

Quantitative Analysis Using the Method of Standard Additions Because of the difficulty of maintaining a constant matrix for samples and standards, many quantitative potentiometric methods use the method of standard additions. A sample of volume, Vx) and analyte concentration, Cx, is transferred to a sample cell, and the potential, (ficell)x) measured. A standard addition is made by adding a small volume, Vs) of a standard containing a known concentration of analyte, Cs, to the sample, and the potential, (ficell)s) measured. Provided that Vs is significantly smaller than Vx, the change in sample matrix is ignored, and the analyte s activity coefficient remains constant. Example 11.7 shows how a one-point standard addition can be used to determine the concentration of an analyte. 

Because extractive solvents work by altering the relative volatihty between the components to be distilled, a good solvent causes a substantial change in relative volatihty when present at moderate compositions. As seen from equation 3, the solvent modifies the relative volatihties by affecting the ratio of the hquid-phase activity coefficients yWhereas it is possible to find solvents which increase or decrease this ratio, it is usually preferable to select a solvent which accentuates the natural difference in vapor pressures between the components to be separated that is, a solvent that increases relative to when is favored, over one that increases relative to y. In the latter case, adding small amounts of the solvent actually makes the separation... 

It is important to note that the solubility product relation applies with sufficient accuracy for purposes of quantitative analysis only to saturated solutions of slightly soluble electrolytes and with small additions of other salts. In the presence of moderate concentrations of salts, the ionic concentration, and therefore the ionic strength of the solution, will increase. This will, in general, lower the activity coefficients of both ions, and consequently the ionic concentrations (and therefore the solubility) must increase in order to maintain the solubility product constant. This effect, which is most marked when the added electrolyte does not possess an ion in common with the sparingly soluble salt, is termed the salt effect. 

It should, however, be noted that as the concentration of the excess of precipitant increases, so too does the ionic strength of the solution. This leads to a decrease in activity coefficient values with the result that to maintain the value of Ks more of the precipitate will dissolve. In other words there is a limit to the amount of precipitant which can be safely added in excess. Also, addition of excess precipitant may sometimes result in the formation of soluble complexes causing some precipitate to dissolve. 

For this calculation it is assumed that both the acid and the base are completely dissociated and the activity coefficients of the ions are unity in order to obtain the pH values during the course of the neutralisation of the strong acid and the strong base, or vice versa, at the laboratory temperature. For simplicity of calculation consider the titration of 100 mL of 1M hydrochloric acid with 1M sodium hydroxide solution. The pH of 1M hydrochloric acid is 0. When 50 mL of the 1M base have been added, 50 mL of unneutralised 1M acid will be present in a total volume of 150 mL. 

As it happens, the product Ba is near unity. In particular, it is about one in water if a (the interaction distance) is 3 X 10 10 m. This is not an unreasonable distance. Even if a is somewhat different, the simplification that Ba is nearly unity is a fair approximation, and one that is often made. This is allowed because Ba occurs as a multiplier of fju in the denominator of Eq. (9-39), where this product is invariably less than the unity to which it is added. And the equations for the activity coefficients are in any event reliable only at low ionic strength. For these reasons, and because of the resulting simplification, we shall approximate the expression for the activity coefficient as... 

If one of the partners in a second-order reaction is not an ion, then in ideal solutions there will be little effect of added salts on the rate. The activity coefficient of a nonelectrolyte does not depend strongly on ionic strength the way that the activity coefficients of ions do. In a reaction with only one participating ion, it and the transition... 

When a solute is added to an acidic solvent it may become protonated by the solvent. If the solvent is water and the concentration of solute is not very great, then the pH of the solution is a good measure of the proton-donating ability of the solvent. Unfortunately, this is no longer true in concentrated solutions because activity coefficients are no longer unity. A measurement of solvent acidity is needed that works in concentrated solutions and applies to mixed solvents as well. The Hammett acidity function is a measurement that is used for acidic solvents of high dielectric constant. For any solvent, including mixtures of solvents (but the proportions of the mixture must be specified), a value Hq is defined as... 

The most widely used method of assessing vitamin Bg status is by the activation of erythrocyte aminotransferases by pyridoxal phosphate added in vitro, expressed as the activation coefficient. 

These relationships are termed the Harned rules and have been verified experimentally up to high overall molality values (e.g. for a mixture of HC1 and KC1 up to 2 mol-kg-1). If this linear relationship between the logarithm of the activity coefficient of one electrolyte and the molality of the second electrolyte in a mixture with constant overall molality is not fulfilled, then a further term is added, including the square of the appropriate molality ... 

The secondary salt effect is important when the catalytically active ions are produced by the dissociation of a weak electrolyte. In solutions of weak acids and weak bases, added salts, even if they do not exert a common ion effect, can influence hydrogen and hydroxide ion concentrations through their influence on activity coefficients. 

For protonation-dehydration processes, such as trityl cation formation from triphenylcarbinols, equation (24), the water activity has to be included if the formulation of the activity coefficient ratio term is to be the same as that in equation (7), which it should be if linearity in X is to be expected see equation (25). The excess acidity expression in this case becomes equation (26) this is a slightly different formulation from that used previously for these processes,36 the one given here being more rigorous. Molarity-based water activities must be used, or else the standard states for all the species in equation (26) will not be the same, see above. For consistency this means that all values of p/fR listed in the literature will have to have 1.743 added to them, since at present the custom... 

By analogy with equation (12), the assumption made regarding the linearity of activity coefficient ratios is equation (45) (slope parameter j), and the resulting Bunnett-Olsen equations that apply to kinetic measurements are equations (46) and (47) for unprotonated and protonated substrates, respectively.156 These apply to the A1 and A-SE2 mechanisms for the A1 and A2 mechanisms they may require correction for partial substrate protonation as in equations (25) and (26) above. For A2 reactions an additional term such as the log water activity has to be added as in equation (33). These equations have been widely tested and work quite well.155-160 The difference between the Bunnett-Olsen and the excess acidity kinetic methods (discussed below) is that the Bunnett-Olsen method features an additive combination of the slope parameters e and , whereas the excess acidity method features a multiplicative one. There seems to be no theoretical justification for the former. Also the Bunnett-Olsen method still uses H0, whereas acidity functions are not needed for the excess acidity approach see above. 

The second method of minimizing the junction potential is to employ a swamping electrolyte S. We saw in Section 4.1 how diffusion occurs in response to entropy effects, themselves due to differences in activity. Diffusion may be minimized by decreasing the differences in activity, achieved by adding a high concentration of ionic electrolyte to both half-cells. Such an addition increases their ionic strengths I, and decreases all activity coefficients y to quite a small value. 

A second approach is the standard addition method, which is commonly employed when the sample is unknown. The potential of the electrode is measured before and after addition of a small volume of a standard to the known volume of the sample. The small volume is used to minimize the dilution effect. The change in the response is related only to the change in the activity of the primary ion. This method is based on the assumptions that the addition does not alter the ionic strength and the activity coefficient of the analyte. It also assumes that the added standard does not significantly change the junction potential. 

Fig. 9. Logarithm of the cation activity coefficient versus the square root of the concentration for the system of manganese ions and cation vacancies in sodium chloride at 500°C. Filled-in circles represent the association theory with Rq = 2a, and open circles the association theory with R = 6/2. Crosses represent the present theory with cycle diagrams plus diagrams of two vertices, and triangles represent the same but with triangle diagram contributions added.

Activity coefficient vary with the concentration especially in the presence of added electrolyte. Lewis and Randall introduced the quantity called ionic strength which is a measure of the intensity of the electric field due to the ions in a solution. It is defined as the sum of the terms obtained by multiplying the molarity (concentration) of each ion present in solution by the square of its valence... 

The local compostion model is developed as a symmetric model, based on pure solvent and hypothetical pure completely-dissociated liquid electrolyte. This model is then normalized by infinite dilution activity coefficients in order to obtain an unsymmetric local composition model. Finally the unsymmetric Debye-Huckel and local composition expressions are added to yield the excess Gibbs energy expression proposed in this study. 

For applications where the ionic strength is as high as 6 M, the ion activity coefficients can be calculated using expressions developed by Bromley (4 ). These expressions retain the first term of equation 9 and additional terms are added, to improve the fit. The expressions are much more complex than equation 9 and require the molalities of the dissolved species to calculate the ion activity coefficients. If all of the molalities of dissolved species are used to calculate the ion activity coefficients, then the expressions are quite unwieldy. However, for the applications discussed in this paper many of the dissolved species are of low concentration and only the major dissolved species need be considered in the calculation of ion activity coefficients. For lime or limestone applications with a high chloride coal and a tight water balance, calcium chloride is the dominant dissolved specie. For this situation Kerr (5) has presented these expressions for the calculation of ion activity coefficients. 

Calculate the temperatures and vapor compositions from the vapor-liquid equilibrium data, using the subroutine BUfiPT. Raoult s law is used in the example, but nonideality can be included by adding activity coefficient equations. Newton-Raphson convergence is used. 

In Chapter 3, we looked at the way the activity coefficients can be more or less equalized if there is a swamping electrolyte in solution (see Section 3.4.4, SAQ 3.9 and Figure 3.8). By the nature of the species studied in a chronoamperometric experiment, (a) a swamping electrolyte is added to the solution in order to minimize migration effects, and... 

For an ionic solute dissociating into v ions, the temperature coefficient is 1/v times the right-hand side of Eq. (2.60). Again, it is assumed that the solubility is sufficiently low for the mean ionic activity coefficient to be effectively equal to unity and independent of the temperature. When this premise is not met, then corrections for the heat of dilution from the value of the solubility to infinite dilution must be added to Asoi //°b in Eq. (2.60). 

The Debye-Hiickel term, which is the dominant term in the expression for the activity coefficients in dilute solution, accounts for electrostatic, nonspecific long-range interactions. At higher concentrations, short-range, nonelectrostatic interactions have to be taken into account. This is usually done by adding ionic strength dependent terms to the Debye-Hiickel expression. This method was first outlined by Bronsted [5,6], and elaborated... 

Usnally, only very dilute solutions can be considered ideal. In most aqueous solutions, ions are stabilized because they are solvated by water molecules. As the ionic strength is increased, ions interact with each other. Thus, when calculating the chemical potential of species i, a term that takes into account the deviation from ideal conditions is added. This term is called an excess term and can be either positive or negative. The term usually is written as 7 riny., where y. is the activity coefficient of component i. The complete expression for the chemical potential of species i then becomes... 

Temperature was set at 35 °C. Brine-rock mass ratio was set to 0.4 10, which corresponds to a porosity of approximately 10%. The mineral content and brine compositions were set to measured values (Table 1). Debyc-Hiickel equations were used to correct activity coefficients for saline solutions. The brine was allowed to come to equilibrium with the C02, then the 10 kg of sandstone was added and equilibrium assemblages were computed a second time. 

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