Debye-Huckel equation extended

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... 

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. 

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. 

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 ... 

This is referred to as the extended Debye-Huckel equation. It is an approximation that gives a good fit of data at low ionic strengths (Goldberg and Tewari, 1991) when B= 1.6 L1/2 mol 1/2. Better fits can be obtained with more complicated equations with more parameters, but these parameters are not known for solutions involved in studying biochemical reactions. The way that thermodynamic properties vary with the ionic strength is discussed in more detail in Section 3.6. 

The BMREP and SDM currently use the Davies technique for activity coefficient prediction. The Davies technique is a combination of the extended Debye-Huckel equation (6) and the Davies equation (7). The Davies technique (and hence both equilibrium models) is accurate up to ionic strengths of 0.2 molal and may be used for practical calculations up to ionic strengths of 1 molal (8). Ion-pair equilibria are incorporated for species that associate (e.g., 1-2 and 2-2 electrolytes). The activity coefficients (y ) are calculated as a simple function of ionic strength (I) and are represented as ... 

The second term in the DAVIES and extended DEBYE-HUCKEL equations forces the activity coefficient to increase at high ionic strength. This is owed to the fact, that ion interactions are not only based on Coulomb forces any more, ion sizes change with the ionic strength, and ions with the same charge interact. 

TABLE 2.2. Values of the Parameter a Used in the Extended Debye-Huckel Equation (Eq. 2.3a) for Selected Ions... 

TABLE 2.3. Single-Ion Activity Coefficients Calculated from the Extended Debye-Huckel Equation at 25°C... 

Cupric ion activities and cupric ion concentrations were determined using the Nernst equation from the differences in potential between the test solutions and a standard solution consisting of 10-5 M CUSO4 and 0.01 M KNO3 at pH 5.4 + 0.3. Values of cupric ion activity in test solutions were based on a cupric ion activity coefficient of 0.68 in the standard solution as calculated from the extended Debye-Huckel equation. For measurements in defined solutions containing 0.01 M KNO3 or 0.01 M NaHC03 cupric ion concentrations could be directly computed via the Nernst equation because activity coefficients were the same in both test and standard solutions. 

According to the extended Debye-Huckel equation, the value of log 7h oh may be represented by — + C i, where A is a known constant for... 

Now we are in a position to calculate the standard transformed thermodynamic properties of reactants from the standard properties of the species that make them up. In this chapter the transformed thermodynamic properties are calculated only at 298.15 K. Caleulations at other temperatures are presented in the next chapter. The first step is to adjust the properties at zero ionic strength to the desired ionic strength in the range 0-0.35 M. Equations 1.3-5 and 1.3-6 Chapter 1 show how these calculations can be made using the extended Debye-HUckel equation. Substituting equation 1.3-5 in equation 3.5-3 in two places yields... 

Either (2-20) or (2-21) is referred to as the extended Debye-Huckel equation (EDHE) this pair of equations gives results appreciably different from the DHLL when H > 0.01 (that is, V/i > 0.1). For comparison, some ionic activity coefficients calculated from (2-15) and (2-20) are listed in Table 2-1. 

TABLE 4.2. Individual ion activity coefficients at 25°C for different ion sizes (a ) in angstroms (1 A = 10 cm) as a function of ionic strength, computed using the extended Debye-Huckel equation with A = 0.5091 and B = 0,3286... 

Rearrange the extended Debye-Huckel equation so that the experimental data may be plotted as a straight line depending on the square root of the ionic strength whose slope is proportional to the average ionic radius. 

The extended Debye-Huckel equation, which applies for ionic strengths of less than 0.1, is given by... 

In most soil solutions, the ionic strength I is low (<0.01), so that the extended Debye-Huckel equation is applicable for the correction of ionic concentrations to the more thermodynamically meaningful activities. Typically, conductivity measurements are used to estimate the ionic strength, a much less laborious procedure than measurement of each cation and anion present in solution. More problematic, however, is the detection and measurement of complexing anions and molecules (ligands). As will be shown later in this chapter, their presence can result in activities of metal ions in soil solution being much lower than measured concentrations would suggest. 

The distinction between "pure" electrostatic effects and ion association is somewhat arbitrary for example many workers have used an extended Debye-Huckel equation with fixed ion-size parameters to compute the activity coefficients, and assigned all other nonideality to ion-pairing equilibria (3, 13, 14). Recent reviews have surveyed more elaborate solvation-based thermodynamic theories of electrolytes (15, 16). 

The difference between the observed and calculated values of y is now much smaller. Ion association would be expected to be greater at these high ionic strengths and the difference between the y values would be expected to be even greater than at the low ionic strengths. The fact that this is not so can only be attributed to the inadequacy of the Debye-Huckel equation at these high ionic strengths and as evidence for the necessity of a bl term as is postulated in the extended Debye-Huckel equation ... 

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