The ionic atmosphere

The Debye-Huckel theory is very complicated mathematically and a detailed treatment is outside the scope of this book. Here, an account will be given of the theoretical treatment of the ionic atmosphere, and this account will be followed by a brief qualitative discussion of the way in which the theory of the ionic atmosphere explains conductivity behavior. 

An ion in solution of charge z e. with an element of volume dV situated at a distance r. 

Suppose that the average electric potential in the volume element dV is 4 . The work required to bring a positive ion of charge z+e from infinity up to this point is then z+e. (Note that in this derivation z+ and z- are taken to be the numerical values of the valences of the ions e.g., for Na , z+ == 1 for CP, z- = 1.) According to the Boltzmann principle, the time-average numbers of positive and negative ions present in the volume element are (see Section 5.15, especially equation (5.216)). 

For a univalent electrolyte (one in which both ions are univalent an example is NaCl) and z are unity, and n+ and n must be equal because the entire solution is neutral equation (6.17) then becomes 

In the more general case, in which there are a number of different types of ions, 

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Inorganic Ions. Because of electrostatic attraction, positive ions are attracted to negatively charged surfaces and have a higher concentration near the surface than in the bulk. Negative ions are repeUed from the negative surface and have a lower concentration near that surface. Ions which are very strongly bound (// ds Stem layer, whereas those that can move into and out of the ionic atmosphere < kT) are in the Helmholtz... 

It is shown that solute atoms differing in size from those of the solvent (carbon, in fact) can relieve hydrostatic stresses in a crystal and will thus migrate to the regions where they can relieve the most stress. As a result they will cluster round dislocations forming atmospheres similar to the ionic atmospheres of the Debye- Huckel theory ofeleeti oly tes. The conditions of formation and properties of these atmospheres are examined and the theory is applied to problems of precipitation, creep and the yield point."... 

Oppositely charged ions are attracted to each other by electrostatic forces and so will not be distributed uniformly in solution. Around each ion or polyion there is a predominance of ions of the opposite charge, the counterions. This cloud of counterions is the ionic atmosphere of the polyion. In a dynamic situation, the distribution of counterions depends on competition between the electrostatic binding forces and the opposing, disruptive effects of thermal agitation. 

Not all ions are mobile within the ionic atmosphere of the polyion. A proportion are localized and site-bound-a concept apparently first suggested by Harris Rice (1954). Localized ion binding is equivalent to the formation of an ion-pair in simple electrolytes. Experimental evidence comes mainly from studies on monovalent counterions. 

The ionic atmosphere has a blurred (diffuse) structure. Because of thermal motion, one cannot attribute precise locations to its ions relative to the central ion one can only dehne a probability to find them at a certain point or define a time-average ionic concentration at that point (the charge of the ionic atmosphere is smeared out around the centraf ion). In DH theory, the interaction of the central ion with specific (discrete) neighboring ions is replaced by its interaction with the ionic atmosphere (i.e., with a continuum). 

The most important parameters of the ionic atmosphere are the charge density Qv r) and the electrostatic potential /(r) at the various points. Each of these parameters is understood as the time-average value. These values depend only on distance r from the central ion, not on a direction in space. For such a system it is convenient to use a polar (spherical) coordinate system having its origin at the point where the central ion is located then each point can be described by a single and unique coordinate, r. 

The total charge, of the ionic atmosphere can be calcnlated by integrating the charge density over its total volnme. Since the system is electroneutral, the total charge of the ionic atmosphere mnst be eqnal in absolnte valne and opposite in sign to the central ion s charge <2m- The charge density is constant in an elementary volnme dV=4nr dr enclosed between two concentric spherical snrfaces with radu r and r + dr. Therefore,... 

The electrostahc potential /(r) at each point is reckoned relahve to the solution s constant average potential the latter is assnmed to be zero. The total value of potential /o(r) can be written as the sum of two components, one dne to the central ion / (r), and one due to the ionic atmosphere ... 

The energy of interaction of the central ion with its ionic atmosphere depends on the potential of this atmosphere t /atm(0) at a point where the central ion is located (r = 0). Therefore, it is the main task of the physical theory of ion-ion interaction to calculate the potential of the ionic atmosphere, j/a,ni. 

In the first version of DH theory it was shown that the potential of the ionic atmosphere can be represented by the equation... 

We can see from this equation that the potential / at the point r = 0 has the value that would exist if there were at distance 1/k a point charge -zj or, if we take into account the spherical symmetry of the system, if the entire ionic atmosphere having this charge were concentrated on a spherical surface with radius 1/k around the central ion. Therefore, the parameter = 1/k with the dimensions of length is called the ejfective thickness of the ionic atmosphere or Debye radius (Debye length). This is one of the most important parameters describing the ionic atmosphere under given conditions. 

The expression for the distribntion of potential of the ionic atmosphere becomes... 

Within a spherical space of radius a, by definition Qy = 0, so that the value of potential of the ionic atmosphere here is constant and equal to that at point r = a ... 

In practical applications of this equation, one must pick values for constant a. To a first approximation it can be regarded as equal to the sum of the radii of two solvated ions. It is not clear, however, whether the solvation sheaths of approaching ions would not be deformed. Moreover, in deriving Eq. (7.43) it was assnmed without sufficient reasoning that the constant a for a given central ion will be the same for different ions present in the ionic atmosphere. 

Ideas concerning the ionic atmosphere can be used for a theoretical interpretation of these phenomena. There are at least two effects associated with the ionic atmosphere, the electrophoretic effect and the relaxation effect, both lowering the ionic mobilities. Formally, this can be written as... 

The electrophoretic (cataphoretic) ejfect arises because the central ion and its ionic atmosphere, which differ in the sign of the charge, will move in opposite directions in an electric field (Fig. 7.6). The countermovement of the ionic atmosphere (as the surrounding medium) slows down the motion of the central ion. Usually, the value of the ionic atmosphere s own conductivity, A jj is adopted as the value of... 

AA-gpjj. Conditionally, the ionic atmosphere is regarded as a sphere with radius r. The valnes of approach the size of colloidal particles, for which Stokes s law applies (i.e., the drag coefficient 9 = where r is the liquid s viscosity) when they... 

The relaxation effect arises because a certain time, is required for the formation or collapse of an ionic atmosphere around the central ion. When an ion moves in an electric held, its ionic atmosphere lags somewhat behind, as it were its center (Fig. 7.7, point B) is at a point where the central ion had been a little earlier. The conhgurahon of the ionic atmosphere around the central ion (point A) will no longer be spherical but elongated (ovoid). Because of this displacement of the charges, the ionic atmosphere has an electrostahc effect on the central ion which acts in a direction opposite to the ion s motion. A rigorous calculation of this effect was made in 1927 by Lars Onsager. His solution was... 

According to the basic ideas concerning ionic atmospheres, the ions contained in them are in random thermal motion, uncoordinated with the displacements of the central ion. But at short distances between the central ion m and an oppositely charged ion j of the ionic atmosphere, electrostatic attraction forces will develop which are so strong that these two ions are no longer independent but start to move together in space like one particle (i.e., the ion pair). The total charge of the ion pair... 

We can find the potential of the ionic atmosphere by subtracting from the overall value of potential /(r) in accordance with Eq. (7.32) the value of potential of the central ion ... 

The theory of Debye and Hiickel started from the assumption that strong electrolytes are completely dissociated into ions, which results, however, in electrical interactions between the ions in such a manner that a given ion is surrounded by a spherically symmetrical distribution of other ions mainly of opposite charges, the ionic atmosphere. The nearer to the central ions the higher will be the potential U and the charge density the limit of approach to the central ion is its radius r = a. 

The Debye-Hiickel limiting law is the least accurate approximation to the actual situation, analogous to the ideal gas law. It is based on the assumption that the ions are material points and that the potential of the ionic atmosphere is distributed from r = 0 to r->oo. Within these limits the last equation is integrated by parts yielding, for constant k, the value ezk/Aite. Potential pk is given by the expression... 

The final approximate form of Eq. (1.3.14) was again obtained by expanding the exponential in a series and retaining only the linear term. Obviously, the potential pk can be expanded to give two terms, the first of which ezk Alter) describes the contribution of the central ion and the second ([Pg.43]

The ionic atmosphere can thus be replaced by the charge at a distance of Lu = k 1 from the central ion. The quantity LD is usually termed the effective radius of the ionic atmosphere or the Debye length. The parameter k is directly related to the ionic strength I... 

The first value is valid for basic SI units, the second is more practical I in moles per cubic decimetre, LD in nanometres.) The radii of the ionic atmosphere for various solution concentrations of a single binary electrolyte (for which / = — z+z vc) are listed in Table 1.2. 

In view of this equation the effect of the ionic atmosphere on the potential of the central ion is equivalent to the effect of a charge of the same magnitude (that is — zke) distributed over the surface of a sphere with a radius of a + LD around the central ion. In very dilute solutions, LD a in more concentrated solutions, the Debye length LD is comparable to or even smaller than a. The radius of the ionic atmosphere calculated from the centre of the central ion is then LD + a. 

A particular case of electrolyte mixtures occurs if one electrolyte is present in a large excess over the others, thus determining the value of the ionic strength. In this case the ionic atmospheres of all the ions are formed almost exclusively from these excess ions. Under these conditions, the activities of all the ions present in the solution are proportional to their concentrations, the activity coefficient being a function of the concentration of the excess electrolyte alone. 

Fig. 2.6 Electrophoretic effect. The ion moves in the opposite direction to the ionic atmosphere... 

For very dilute solutions, the motion of the ionic atmosphere in the direction of the coordinates can be represented by the movement of a sphere with a radius equal to the Debye length Lu = k 1 (see Eq. 1.3.15) through a medium of viscosity t] under the influence of an electric force ZieExy where Ex is the electric field strength and zf is the charge of the ion that the ionic atmosphere surrounds. Under these conditions, the velocity of the ionic atmosphere can be expressed in terms of the Stokes law (2.6.2) by the equation... 

The electrolytic mobility of the ionic atmosphere around the ith ion can then be defined by the expression... 

This quantity can be identified with deceleration of the ion as a result of the motion of the ionic atmosphere in the opposite direction, i.e. 

Fig. 2.7 Time-of-relaxation effect. During the movement of the ion the ionic atmosphere is renewed in a finite time so that the position of the ion does not coincide with the centre of the ionic atmosphere... 

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