Kapustinskii equation

Kapustinskii equation For an ionic crystal composed of cations and anions, of respective charge and z, which behave as hard spheres, the lattice energy (U) may be obtained from the expression... 

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

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. 

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. 

Table 5.2 Experimental and calculated (Kapustinskii equation) lattice energies of selected crystalline halides. All energies are in kJ mol ...

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,... 

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