Debye function equation

Figure 5.3 The Debye function, Equation (5.31), for a random coil chain is plotted and compared with the independent scattering intensity function for a thin rod and a thin circular disk.

Form Factors The plot of Si(k) as a function of k at small k gives the radius of gyration for any conformation, but, beyond that range, Si(k) depends on the conformation. For a Gaussian chain, SiCk) follows the Debye function. Equations 2.59 and 2.76 allow us to calculate SiCk) for other conformations. Let us first define a form factor P(k) by... 

From the dielectric function, the relaxation time can be extracted as a fitting parameter based on empirical functions. In the simplest case, the Debye function Equation 1.5 is applicable. 

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... 

Geochemical modelers currently employ two types of methods to estimate activity coefficients (Plummer, 1992 Wolery, 1992b). The first type consists of applying variants of the Debye-Hiickel equation, a simple relationship that treats a species activity coefficient as a function of the species size and the solution s ionic strength. Methods of this type take into account the distribution of species in solution and are easy to use, but can be applied with accuracy to modeling only relatively dilute fluids. 

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).

Here, i, j, and k are subscripts representing the various species in solution and /dh is a function of ionic strength similar in form to the Debye-Hiickel equation. The terms Xy and Hijk are second and third virial coefficients, which are intended to account for short-range interactions among ions the second virial coefficients vary with ionic strength, whereas the third virial coefficients do not. 

Activity coefficients can be determined by experimental observations. Since they are functions of ionic strength, temperature and pressure, marine scientists typically estimate values at the environmental conditions of interest from semi-empirical equations. In dilute solutions, the activity coefficient of a monoatomic ion can be calculated from the Debye-Hiickel equation ... 

The standard emf E° of the cell was determined by means of an extrapolation technique involving a function of the measured emf E (which was measured experimentally), taken to the limit of zero ionic strength /. A linear function of I was observed when the Debye-Hiickel equation (in its extended form) (12) was introduced for the activity coefficient of hydrobromic acid over the experimental range of molalities m. With this type of mathematical treatment, the adjustable parameter became a0, the ion-size parameter, and a slope factor / . This procedure is essentially the same as that used in our earlier determinations (7,10) although no corrections of E° for ion association were taken into account (e = 49.5 at 298.15°K). 

The solution of eqn. (44) for a coulomb potential with boundary conditions (45) and (46) for either initial conditions (48) or (49) has only been developed in recent years. Hong and Noolandi [72] showed that the solution of the Debye—Smoluchowski equation is related to the Mathieu equation. Many of the details of their analysis are discussed in the Appendix A, Sect. 4, and the Appendix eqn. (A.21) is the Green s function (fundamental solution), which is the probability that a reactant B is at r given that it was initially at r0. This equation is developed as the Laplace transform. To obtain the density of interest p(r, ), with either condition, the Green s function has to be averaged over the initial distribution, as in eqn. (A.12), and the Laplace transform inverted. Alternatively, the density p(r, ) can be found from the inverse Laplace transform of the linear combination of independent solutions (A.17) which satisfy the boundary and initial conditions. This is shown in Fig. 10. For a Boltzmann initial condition, Hong and Noolandi [72] found... 

The variational principle has not been widely used in diffusion kinetic problems. Nevertheless, it is such a powerful technique that it is suitable for discussing the many-body problems which have still to be tackled. Wherever approximate methods are necessary, the variational principle should be considered. The trial function(s) should be chosen with care, based on a good idea of the nature of the trial function from its behaviour in certain asymptotic limits. The only application known to the author of the variation principle to a numerical study of a diffusion kinetic problem on a molecular system is that of Delair et al. [377]. They used the variational principle to generate an implicit finite difference scheme for solving the Debye—Smoluchowski equation. Interesting comments have been made by Brykalski and Krason more in the context of heat diffusion [510]. 

E3 Extended Debye-Hiickel equation. Use Equation 8-6 to calculate the activity coefficient (-y) as a function of ionic strength... 

Equation (13.24) shows that q2i 2 > 1 for / bounded necessarily implies q2 Nr 1, so that we can replace the Debye function by the asymptotic... 

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 ... 

This form for PG(an) is the widely used Debye function. Note that for homopolymer chains, n is large (large degree of polymerization) and a is small (SANS instruments do not see monomer chemistry) so that these last expressions can be used. However, for short block copolymers, n is not necessarily large and the more general equations are more appropriate to use. 

WATEQ2 consists of a main program and 12 subroutines and is patterned similarly to WATEQF ( ). WATEQ2 (the main program) uses input data to set the bounds of all major arrays and calls most of the other procedures. INTABLE reads the thermodynamic data base and prints the thermodynamic data and other pertinent information, such as analytical expressions for effect of temperature on selected equilibrium constants. PREP reads the analytical data, converts concentrations to the required units, calculates temperature-dependent coefficients for the Debye-HKckel equation, and tests for charge balance of the input data. SET initializes values of individual species for the iterative mass action-mass balance calculations, and calculates the equilibrium constants as a function of the input temperature. MAJ EL calculates the activity coefficients and, on the first iteration only, does a partial speciation of the major anions, and performs mass action-mass balance calculations on Li, Cs, Rb, Ba, Sr and the major cations. TR EL performs these calculations on the minor cations, Mn, Cu, Zn, Cd, Pb, Ni, Ag, and As. SUMS performs the anion mass... 

A more rigid but laborious method, for deriving transference numbers from E.M.p. data, makes use of the fact that the activity coefficient of an electrolyte can be expressed, by means of an extended form of the Debye-Hiickel equation, as a function of the concentration and of two empirical constants.When applied to the same data, however, this procedure gives results which are somewhat different from those obtained by the method just described. Since the values are in better agreement with the transference data derived frorq moving boundary and other measurements, they are probably more reliable. 

This chapter shows that more thermodynamic information can be obtained for enzyme-catalyzed reactions for which Af H° is known for all the species in addition to Af G°. When Af H° is independent of temperature, Af G° of the species can be expressed as a function of temperature. Since the temperature dependence of the parameter in the Debye-Huckel equation is known, the Af G ° for a reactant can be derived as a function of temperature, pH, and ionic strength by use of derivetr-GibbsT, As shown in Chapter 3, this means that Af G Af H Af 5. and derivatives of these properties of reac-... 

Now it has been proved that this formula in combination with equation (13) does reproduce the main features of the variation of the thermal resistance with temperature for arbitrary specimens of a metal.f But systematic deviations were immediately observed to occur, the Debye function C Q/T) falling ofT too slowly at very low temperatures with the value of 0 obtained from the atomic heat. Thus the empirical formula cannot well be taken as a proof of Peierls law. 

The particular application of the Debye-Hiickel equation to be described here refers to the determination of the true equilibrium constant K from values of the equilibrium function K at several ionic strengths the necessary data for weak acids and bases can often be obtained from conductance measurements. If the solution of the electrolyte MA is sufficiently dilute for the limiting law to be applicable, it follows from equation (40.12), for the activity coe cient of a single ionic species, that... 

Some ion activity coefficients at 25°C computed with the Debye-Huckel equation as a function of ionic strength, ion size, and charge, are shown in Table 4.2. Debye-Huckel ion activity coefficients up to 0.1 mol/kg ionic strength, are plotted in Fig. 4.3 for some monovalent and divalent ions. The Debye-Huckel equation can be used to compute accurate activity coefficients for monovalent ions up to about / = 0.1 mol/kg, for divalent ions to about / = 0.01 mol/kg, and for trivalent ions up to perhaps / = 0.001 mol/kg. 

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