The Kapustinskii Equation

In this series, the terms neither lead to a recognizable series nor converge very rapidly. In fact, it is a rather formidable process to determine the sum, but the value obtained is 1.74756. Note that this is approximately equal to the value given earlier for the ratio of the energy released when a crystal forms to that when only ion pairs form. As stated earlier, the Madelung constant is precisely that ratio. 

Details of the calculation of Madelung constants for all of the common types of crystals are beyond the scope of this book. When the arrangement of ions differs from that present in NaCl, the number of ions surrounding the ion chosen as a starting point and the distances between them may be difficult to determine. They will most certainly be much more difficult to represent as a simple factor of the basic distance between a cation and an anion. Therefore, each arrangement of ions (crystal type) will have a different value for the Madelung constant. The values for several common types of crystals are shown in Table 7.3. 

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. 

This expression has its origins in the Bom-Lande equation, with a value of 9 for the Bom exponent (the value for NaCl) and half the value of the Madelung constant for NaCl. The inclusion of the factor v shows why half of A is included. Although the Kapustinskii equation is useful, it is a gross approximation and values obtained in this way must be treated with caution. 

A species disproportionates if it undergoes simultaneous oxidation and reduction. 

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Lattice energies may be derived from the Born-Haber cycle or calculated using the Kapustinskii equation. ... 

By means of appropriate thermochemical cycles, it is possible to calculate proton affinities for species for which experimental values are not available. For example, using the procedure illustrated by the two foregoing examples, the proton affinities ofions such as HC03-(g) (1318 k J mol-1) and C032-(g) (2261 kj mol-1) have been evaluated. Studies of this type show that lattice energies are important in determining other chemical data and that the Kapustinskii equation is a very useful tool. 

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 Kapustinskii equation is unreliable for trifluorides (Table 5.2) but the lattice energy of BaF3 can be estimated to be a little more than twice that of BaF2, for which an experimental value is given in Table 5.2. Our estimate of —4800 100 kJ mol-1 is in line with the experimental value for LaF3 (Table 5.2). 

As an example, let us pose the question why does BF3 adopt a molecular structure, while A1F3 is apparently ionic As shown in Table 5.2, the ionic model (using the Kapustinskii equation) gives a fair approximation to the thermochemistry of formation of A1F3. Let us estimate the enthalpy of formation of a hypothetical ionic substance BF3(s), having a structure similar to that of A1F3. The lattice energy can be estimated by means of the Kapustinskii equation. We require the... 

A useful application of the Kapustinskii equation is the prediction of the existence of previously unknown compounds. From Table 4.2.5, it is seen that all dihalides of the alkali metals with the exception of CsF2 are unstable with respect to their formation from the elements. However, CsF2 is unstable with respect to disproportionation the enthalpy of the reaction CsF2 CsF+l/2F2 is -405 kJ mol-1. 

In cases where the lattice energy is known from the Born-Haber cycle, the Kapustinskii equation can be used to derive the ionic radii of complex anions such as S042- and P043-. The values determined in this way are known as thermochemical radii some values are shown in Table 4.2.6. 

It is possible to obtain other semiempirical expressions that express the lattice energy in terms of ionic radii, charges on the ions, and so on. One of the most successful of these is the Kapustinskii equation,... 

Suppose you could make an aqueous solution containing Na+, Ag+, F , and I- without a precipitate forming. If the solution were evaporated to dryness, what crystals would form Use calculations from the Kapustinskii equation to support your conclusions. 

The results obtained by using Equation 1 are summarized in Table I. The heats of formation given for the isolated anions were computed from the standard heat of formation of the corresponding potassium salt by means of the Kapustinskii approximation. The value thus derived from F (using a radius of 1.33 A.) falls within 5 kcal./mole of that from the heat of formation and measured electron affinity of F. The value for BF4 differs by 19 kcal./mole from that obtained in a detailed lattice-energy calculation by Altschuller (I) his value is very close to that derived by Kapustinskii and Yatsimirskii (11) by modifying the Kapustinskii equation. 

Kapustinskii s equation, of course, cannot be expected to produce values for the lattice energies as exact as those produced by extended calculation. Essentially the Kapustinskii formula ignores all contributions to the lattice energy save Us and Uu. If the ions are nonspherical there is considerable difficulty in choosing a repulsion envelope for the ion and assigning a radius. This difficulty also arises in the extended calculation of the lattice energy but only in calculation of Ur, which is an order of magnitude less than Um- In the Kapustinskii equation this difficulty arises in Um as well. 

The values of Huggins are probably, the most accurate lattice energies obtainable and agree with the Born-Haber cycle values to within the experimental accuracy of the cycle terms. The values given by the Kapustinskii equation will be seen to be rather low. The Bom cycle values are obtained from the values of AH/ M+(g) and aH/ MX(s) given by the U. S. Bureau of Standards, circular 500, and the values of AH/ X (g) decided upon by Pritchard 108), as a result of a review of all the experimental data. 

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