Debye-Huckel equations

For dilute solutions, the Debye-Huckel equation by calculations based on these Coulombic interactions is... 

Provided the ionic strength is not too high, this equation is obeyed as well as (but no better than) the Debye-Huckel equation for activity coefficients. One can expect deviations at higher ionic strength, and they are in general more serious the higher... 

It then also follows that the rate constant for a first-order reaction, whether or not the solvent is involved, is also independent of ionic strength. This statement is true at ionic strengths low enough for the Debye-Huckel equation to hold. At higher ionic strengths, predictions cannot be made about reactions of any order because all of the kinetic effects can be expected to show chemical specificity. 

These representations offer the advantage that one need not argue which of the reagents carries OH or Cl into the transition state. Since that is usually not known, this notation sidesteps the issue. From the Brpnsted-Debye-Huckel equation, we recognize that the concentration of each transition state (and therefore the reaction rate) will vary with ionic strength in proportion to the values of K for the given equation. For the first term we have... 

This is the electrostatic energy arising from ions approaching within a of each other. When subtracted from the free energy functional above the corrected Debye Huckel equation becomes... 

Changes in activity coefficients (and hence the relationship between concentration and chemical activity) due to the increased electrostatic interaction between ions in solution can be nicely modeled with well-known theoretical approaches such as the Debye-Huckel equation ... 

It should be mentioned that the results of the above extrapolation can be improved when its application is extended to concentrations even higher than 0.01 M, by means of an extended Debye-Huckel equation, viz., introduction of the factor 1/(1 + ba) (cf., eqns. 2.60 and 2.60a) into the final expression 2.62 for log f and its general form 2.64 leads to... 

Table 8.3 Constants of the Debye-Huckel Equation from 0 to 100°C 8.5...

The values were calculated from the modified Debye-Huckel equation utilizing the modifications proposed by Robinson and by Guggenheim and Bates ... 

The result of their analysis, known as the Debye-Huckel equation,... 

Variable di in Equation 8.2 is the ion size parameter. In practice, this value is determined by fitting the Debye-Huckel equation to experimental data. Variables A and B are functions of temperature, and I is the solution ionic strength. At 25 °C, given I in molal units and taking a, in A, the value of A is 0.5092, and B is 0.3283. 

As can be seen in Figure 8.1, the Davies equation does not decrease monotoni-cally with ionic strength, as the Debye-Huckel equation does. Beginning at ionic strengths of about 0.1 molal, it deviates above the Debye-Huckel function and at about 0.5 molal starts to increase in value. The Davies equation is reasonably accurate to an ionic strength of about 0.3 or 0.5 molal. 

Fig. 8.2. Values of A, B, and B for the B-dot (modified Debye-Huckel) equation at 0 °C, 25 °C, 60 °C, 100 °C, 150 °C, 200 °C, 250 °C, and 300 °C (squares) and interpolation functions (lines). Values correspond to I taken in molal and a in A. Data from the LLNL database, after Helgeson (1969) and Helgeson and Kirkham (1974).

And (b) the extended Debye-Huckel equation for the approximation of the activity coefficient yj of the j-th ion. It needs the charge Zi and the ionic radius ay... 

Equation 4, which is obviously an extended Debye-Huckel equation, involves only the independent variables I and q. While appearing formidable at first inspection, equation 4 is easily solved for T°, even on a hand computer. Figure 1, of course, represents a graphical solution to this equation. 

An extended form of the Debye-Huckel equation is the hydration one of Robinson and Stokes (11). It contains two adjustable parameters, ap and h, where h is rilated to the hydration number. It can be fitted to y for several electrolytes for concentrations in excess of 1 m. Their equation has the valuable feature of describing not only the salting-in but also the salting-out part of the y+ versus m curve. It should be noted, however, that the... 

Various attempts have been made to increase the valid range of the Debye- Huckel equation to regions of high ionic strength by the use of empirically fitted parameters. Several of these equations are listed in table I. 

Very few generalized computer-based techniques for calculating chemical equilibria in electrolyte systems have been reported. Crerar (47) describes a method for calculating multicomponent equilibria based on equilibrium constants and activity coefficients estimated from the Debye Huckel equation. It is not clear, however, if this technique has beep applied in general to the solubility of minerals and solids. A second generalized approach has been developed by OIL Systems, Inc. (48). It also operates on specified equilibrium constants and incorporates activity coefficient corrections for ions, non-electrolytes and water. This technique has been applied to a variety of electrolyte equilibrium problems including vapor-liquid equilibria and solubility of solids. 

Marshall s extensive review (16) concentrates mainly on conductance and solubility studies of simple (non-transition metal) electrolytes and the application of extended Debye-Huckel equations in describing the ionic strength dependence of equilibrium constants. The conductance studies covered conditions to 4 kbar and 800 C while the solubility studies were mostly at SVP up to 350 C. In the latter studies above 300°C deviations from Debye-Huckel behaviour were found. This is not surprising since the Debye-Huckel theory treats the solvent as incompressible and, as seen in Fig. 3, water rapidly becomes more compressible above 300 C. Until a theory which accounts for electrostriction in a compressible fluid becomes available, extrapolation to infinite dilution at temperatures much above 300 C must be considered untrustworthy. Since water becomes infinitely compressible at the critical point, the standard entropy of an ion becomes infinitely negative, so that the concept of a standard ionic free energy becomes meaningless. 

To leam that an approximate value of y can be calculated for solution-phase analytes by using the Debye-HUckel equations (equations (3.14) and (3.15)). 

PROGRESS CURVE INITIAL RATE CONDITION SUBSTRATE PURITY ENZYME PURITY WATER PURITY SUBSTRATE STABILITY ENZYME STABILITY MIXING TIME INITIAL RATE CONDITION QUENCHING EXPONENTIAL EXPONENTIAL BREAKDOWN Extended Debye-Huckel equation, DEBYE-HUCKEL TREATMENT EXTENSIVE PROPERTY EXTENT OF REACTION RATE OF CONVERSION... 

If you use a spreadsheet for this exercise, compute activity coefficients with the extended Debye-Huckel equation and compute many more points. (You can look up the results of a similar titration to compare with your calculations.1 )... 

Even if we know all reactions and equilibrium constants for a given system, we cannot compute concentrations accurately without activity coefficients. Chapter 8 gave the extended Debye-Huckel equation 8-6 for activity coefficients with size parameters in Table 8-1. Many ions of interest are not in Table 8-1 and we do not know their size parameter. Therefore we introduce the Davies equation, which has no size parameter ... 

Equation 6-33 suggests that extrapolation of equilibrium constants to infinite dilution is done appropriately by plotting log Kc vs-01. For example, Fig. 6-1 shows plots of pK a for dissociation of H2P04-, AMP, and ADP2-, and ATP3 vs fp. The variation of pK a with- p at low concentrations (Eq. 6-35) is derived by application of the Debye-Huckel equation (Eq. 6-33) ... 

In Chapter 11, we saw that 7 could be calculated over a larger concentration range by using the extended form of the Debye-Huckel equation... 

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