Radius univalent

The flow can be radial, that is, in or out through a hole in the center of one of the plates [75] the relationship between E and f (Eq. V-46) is independent of geometry. As an example, a streaming potential of 8 mV was measured for 2-cm-radius mica disks (one with a 3-mm exit hole) under an applied pressure of 20 cm H2 on QT M KCl at 21°C [75]. The i potentials of mica measured from the streaming potential correspond well to those obtained from force balance measurements (see Section V-6 and Chapter VI) for some univalent electrolytes however, important discrepancies arise for some monovalent and all multivalent ions. The streaming potential results generally support a single-site dissociation model for mica with Oo, Uff, and at defined by the surface site equilibrium [76]. 

From this equation we can calculate the actual crystal radius (for use in normal sodium chloride type crystals) from Ru the univalent crystal radius. A knowledge of the repulsion exponent n is needed, however. This can be derived from the experimental measurement of the compressibility of crystals. Bom11 and Herzfeld,12 give the values in Table III, obtained in this way. 

The radii found in this way are given in Table II, the univalent radii being also included, in parentheses. The dependence of the univalent crystal radius and the crystal radius on the atomic number is shown graphically in Fig. 3.14 The effect of the valence in causing the crystal radius to deviate from the regular dependence on the atomic number shown by the univalent crystal radius is clearly evident. 

Fig. 3.—The crystal radius (solid circles) and the univalent crystal radius (open circles) for a number of ions.

In deriving theoretical values for inter-ionic distances in ionic crystals the sum of the univalent crystal radii for the two ions should be taken, and corrected by means of Equation 13, with z given a value dependent on the ratio of the Coulomb energy of the crystal to that of a univalent sodium chloride type crystal. Thus, for fluorite the sum of the univalent crystal radii of calcium ion and fluoride ion would be used, corrected by Equation 13 with z placed equal to y/2, for the Coulomb energy of the fluorite crystal (per ion) is just twice that of the univalent sodium chloride structure. This procedure leads to the result 1.34 A. (the experimental distance is 1.36 A.). However, usually it is permissible to use the sodium chloride crystal radius for each ion, that is, to put z = 2 for the calcium... 

It is to be expected that the relative values of the univalent crystal radius would be of significance with respect to physical properties involving atomic sizes. That this is true for the viscosity of the rare gases is seen from the radii evaluated by Herzfeld (Ref. 12, p. 436) from viscosity data He, 0.04 (0.93) Ne, 1.18 (1.12) Ar, 1.49 (1.54) Kr, 1.62 (1.69) Xe, 1.77 (1.90). (The values in parentheses are the univalent crystal radii.)... 

In this discussion, two mutually canceling simplifications have been made. For the transition value of the radius ratio the phenomenon of double repulsion causes the inter-atomic distances in fluorite type crystals to be increased somewhat, so that R is equal to /3Rx-5, where i has a value of about 1.05 (found experimentally in strontium chloride). Double repulsion is not operative in rutile type crystals, for which R = i M + Rx- From these equations the transition ratio is found to be (4.80/5.04)- /3i — 1 = 0.73, for t = 1.05 that is, it is increased 12%. But Ru and Rx in these equations are not the crystal radii, which we have used above, but are the univalent crystal radii multiplied by the constant of Equation 13 with z placed equal to /2, for M++X2. Hence the univalent crystal radius ratio should be used instead of the crystal radius ratio, which is about 17% smaller (for strontium chloride). Because of its simpler nature the treatment in the text has been presented it is to be emphasized that the complete agreement with the theoretical transition ratio found in Table XVII is possibly to some extent accidental, for perturbing influences might cause the transition to occur for values a few per cent, higher or lower. 

Since the repulsive forces are determined by the true sizes of ions, and not their crystal radii, the radius ratios to be used in this connection are the ratios of the univalent cation radii to univalent anion radii.12 Values of this ratio for small ions are given in Table II, together with predicted and observed coordination numbers, the agreement between which is excellent. 

Ahrens (1952) proposed the first extended tabulation of ionic radii, partially modifying the univalent radii in the Vl-fold coordination of Pauling (1927a) on the basis of the observed correlation between ionic radius (r) and ionization potential (/), which can be expressed in the forms... 

Most commonly, metal ions M2+ and M3+ (M = a first transition series metal), Li+, Na+, Mg2+, Al3+, Ga3+, In3+, Tl3+, and Sn2+ form octahedral six-coordinate complexes. Linear two coordination is associated with univalent ions of the coinage metal (Cu, Ag, Au), as in Ag(NH3)2+ or AuCL Three and five coordination are not frequently encountered, since close-packing considerations tell us that tetrahedral or octahedral complex formation will normally be favored over five coordination, while three coordination requires an extraordinarily small radius ratio (Section 4.5). Coordination numbers higher than six are found among the larger transition metal ions [i.e., those at the left of the second and third transition series, as exemplified by TaFy2- and Mo(CN)g4 ] and in the lanthanides and actinides [e.g., Nd(H20)93+ as well as UC Fs3- which contains the linear uranyl unit 0=U=02+ and five fluoride ligands coordinated around the uranium(VI) in an equatorial plane]. For most of the metal complexes discussed in this book a coordination number of six may be assumed. 

When the ratio of the radii increases, then a new type of crystal structure occurs, in which each divalent ion is surrounded by six univalent ions, and each univalent by three divalent ones. This arrangement, shown in Figure 14, is called the rutile type, named after the mineral Ti02. While CaF2 still has the structure with 8 4 coordination, the rutile structure is observed in MgF2 because of the smaller radius of the Mg2+ ion. After the rutile type there is a further reduction of the coordination to 4 2 this type of structure occurs in BeF2 and Si02 in the different modifications of silica, each silicon ion is surrounded by 4 O2" ions, and each O2 ion is between two Si4+ ions. The ionic ratio r+/r can also be decreased if,... 

It is possible to extend the treatment of this section, based on Equation 13-9, to crystals other than the alkali halogenides, with suitable choice of standard radii. It is found that the standard radii differ somewhat in general from the univalent radii of Table 13-3, because of the different choice of values of n. An approximate value for the standard radius of a multivalent ion can be obtained by multiplying its crystal radius by z f z being the magnitude of the valence of the ion this is the correction factor from crystal radius to univalent radius qorresponding to n = 9. 

Tt is found empirically, indeed, that these complicated and unreliable calculations need not be made in general even for substances of un-symmetrical valence type the interionic distances are very closely approximated by the sum of the crystal radii. For fluorite this sum is 2.35 A, which agrees very well with the observed value. The reason for this is apparent in the crystal radius of Ca++ a correction for bivalence of cation and anion is made, and this has nearly the same magnitude as the correction for bivalence of cation alone made for the sum of the univalent radii of calcium and fluorine. 

The substances to which the rules apply are those in which the bonds are largely ionic in character rather than largely covalent, and in which all or most of the cations are small (with radius less than 0.8 A) and multivalent, the anions being large (greater than 1.35 A in radius) and univalent or bivalent. The anions that are most important are those of oxygen and fluorine. 

Fig. 14. Coion partition coefficients SN for uni-univalent electrolyte, CF/w = 21.7 mmol/dm3, Kpn = w, calculated by the EVM (line A) or the ESM for slit-shaped pores (lines B-F) (i) Single pores of radius r = 6 (line B) or 40 (line F) nm. (ii) Assemblies of two pores of equal surface charge density of mean radius 6 nm and individual radii (in nm) 5.306,40 (line C) 5.454,80 (line D) 4.583,40 (line E) where 98 % (C), 99 / (D), or 96 % (E) of the total pore wall surface area is in narrow pores 121). Note that the power law of Eq. (46) is obeyed over a considerable range in cases C and E...

An approximate value of the radius of the ionic atmosphere r as a function of concentration, for a uni-univalent (1-1) electrolyte at 25 °C and water as solvent, considering 79 as the dielectric constant value, may be obtained from the relationship... 

Recently Cartledge5 has found the equivalent heat of formation of metal halides to be a linear function of (in calculating in this paper, the effective, univalent, ionic radii for cations with coordination numbers of six, as estimated by Zachariasen, are used). Though properties may generally be estimated qualitatively from charge and radius considerations, the following data illustrate the type of... 

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