Radii thermochemicals

There is another use of the Kapustinskii equation that is perhaps even more important. For many crystals, it is possible to determine a value for the lattice energy from other thermodynamic data or the Bom-Lande equation. When that is done, it is possible to solve the Kapustinskii equation for the sum of the ionic radii, ra + rc. When the radius of one ion is known, carrying out the calculations for a series of compounds that contain that ion enables the radii of the counterions to be determined. In other words, if we know the radius of Na+ from other measurements or calculations, it is possible to determine the radii of F, Cl, and Br if the lattice energies of NaF, NaCl, and NaBr are known. In fact, a radius could be determined for the N( )3 ion if the lattice energy of NaNOa were known. Using this approach, which is based on thermochemical data, to determine ionic radii yields values that are known as thermochemical radii. For a planar ion such as N03 or C032, it is a sort of average or effective radius, but it is still a very useful quantity. For many of the ions shown in Table 7.4, the radii were obtained by precisely this approach. 

The charge distribution in complex fluoroanions is more difficult to treat. An overall thermochemical radius can be assigned by inverting the Kapustinskii method (138), but more recent energy minimization calculations have claimed to arrive at charge distributions (125). 

For correlation of the coefficients we have found two parameters for which values are known for all or most of the single-charged anions in the data base and also for bicarbonate and hydrosulfide. These are the entropy of the ion and the thermochemical radius of the ion as calculated by Yatsimirskii(37,38). 

Concentration of indicated ion, mol/kg H2O Thermochemical radius of anion and cation, A Gas constant... 

As we end this section, let us reconsider ionic radii briefly. Many ionic compounds contain complex or polyatomic ions. Clearly, it is going to be extremely difficult to measure the radii of ions such as ammonium, NH4, or carbonate, COs, for instance. However, Yatsimirskii has devised a method which determines a value of the radius of a polyatomic ion by applying the Kapustinskii equation to lattice energies determined from thermochemical cycles. Such values are called thermochemical radii, and Table 1.17 lists some values. 

The cyanide ion is sometimes described as a pseudo-halide ion, i.e. it behaves much like a halide ion. It has a radius of 187 pm, not very different from the 181 pm radius of the chloride ion, yet HC1 is a very strong acid and HCN is very weak, the difference between the two pKA values being about 16 units. The thermochemical cycle shown in Figure 3.1 is also applicable to the HC1/HCN comparison, and appropriate data are given in Table 3.4. 

The most stable state of nitrogen in acidic solution is the ammonium ion, NH4(aq), which is isoelectronic with CH4 and H30+. It is a tetrahedral ion with strong N-H bonds. The mean N-H bond enthalpy in NH4(aq) is 506 kJ mol 1 (that of the O-H bonds in H30 + is 539 kJ mol" ). The enthalpy of hydration of the ammonium ion is — 345 kJ mol V This value placed into the Born equation (3.32) gives an estimate of the radius of the ammonium ion of 135 pm, a value insignificantly different from its thermochemical radius of 136 pm. The value is comparable to that estimated for the smaller H30+ ion (99 pm) from its more negative enthalpy of hydration (— 420 kJ mol -see Section 2.6.1). The proton affinity of the ammonia molecule is of interest in a comparison of its properties with those of the water molecule. The proton affinity is defined as the standard enthalpy change for the reaction ... 

The first values necessary are some estimates of the ionic radii of O and BF4. For the latter we may use the value obtained thermochemically by Yauimirskii, 218 pm. An educated guess hus to be made for 02. since if we arc attempting to make it for the first tune (as was assumed ahove), we will not have any experimental data available for this species. However, we note that the CN ion, a diatomic ion which should be similar in size, has a thermocheinical radius of 177 ppm. Furthermore, an estimate based on covalent and van dec Waals radii (see Chapter 8) gives a similar value. Because OJ has lost one electron and is positively charged, it will probably be somewhat smaller than this. We can thus take 177 pm as a conservative estimate if the cation is smaller than this, the compound will be more stable than our prediction and even more likely to exist. Adding the radii we obtain an estimate of 395 pm for the interionic distance. 

The distance between the fractionally-positive H atom and the anion will be rather smaller than that between the O atom and a cation, other things being equal. A similar expression works reasonably well for polyatomic anions, with the denominator set equal to (r + 30) pm, r in this case being the thermochemical radius of the anion. 

The weakness of F-O bonds compared with the bond in the 02 molecule is clearly important, as for F03(0H). But an additional factor is likely to be the high lattice energy of NaF compared with NaF04 the thermochemical radius of F04 is expected to be larger than the crystal radius of F. Note, however, that NaC104 - a well-known substance - is likewise thermodynamically unstable. For the decomposition ... 

Liquid-water clouds (5) represent a potentially important medium for atmospheric chemical reactions in view of their high liquid water content [104 to 105 times that associated with clear-air aerosol (6)] and high state of dispersion (typical drop radius 10 pm). Clouds are quite prevalent in the atmosphere (fractional global coverage 50%) and persistent (lifetimes of a few tenths of an hour to several hours). The presence of liquid water also contributes to thermochemical driving force for production of the highly soluble sulfuric and nitric acids. 

Figure 2. Correlation of thermochemical radius Rk with sum of bond distance R(B—X) and van der Waals radius R(X) in tetrahedral ions...

This expression has turned out to be remarkably useful in correlating the heats of formation of the salts of tetrahedral ions, provided suitable values are assumed for the ionic radii R+ and R. Kapustinskii and his coworkers recognized that these quantities are not necessarily equal to the packing radii of the ions in the actual structure of the crystal consequently, they have come to be known as thermochemical radii. The thermochemical radius and heat of formation for a tetrahedral ion are normally determined from Equation 1 and the known heats of formation of two of its salts. 

In order to apply Equation 1 to the hypothetical salts of NF4+ it is necessary to estimate a thermochemical radius for that ion. We have found that a fairly good correlation exists for a number of symmetrical tetrahedral ions BX between the thermochemical radius Rk and the sum of (a) the intemuclear distance R(B—X) between the central atom of the ion and one of its ligands and (b) the van der Waals radius, R, (X) of the ligand. This correlation, shown in Figure 2, is described approximately by ... 

Yatrimirskii has provided an ingwious method for estimating the radii of polyatomic ions. A Bom-Haber calculation utilizing the enthalpy of formation and related data can provide an estimate of the lattice energy, ft is then possible to find what value of the radius of the ion in question is consistent with this lattice encroy. These values are thus termed thermochemical radii. The most recent set of such values is given in Table 4.5. In many cases the feet that the ions (such a COf", CNS", CHjCOO") are markedly nonspherical limits the use of these radii. Obviously they... 

The lattice energies of the alkali metal thiocyanates can be calculated from Yatsimirskii s (133) thermochemical radius of 1.95 A or, in the case of NaCNS and KCNS, from the hydration enthalpies obtained from the Buchner interpolation by Waddington (1 6). The results are given in Table XXIII. There is a considerable discrepancy between them. Waddington s results lead to a value of —25.9 kcal/mole for AH/CNS (g) and Yatsimirskii s to a value of —11.5 kcal/mole. 

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