The Kapustinskii equation calculates the

lattice energy In chemistry, the lattice energy is the energy change upon formation of one mole of a crystalline ionic compound from its constituent ions, which are assumed to initially be in the gaseous state. It is a measure of the cohesive forces that bin ...

''U_L'' for an

ionic crystal In chemistry, an ionic crystal is a crystalline form of an ionic compound. They are solids consisting of ions bound together by their electrostatic attraction into a regular lattice. Examples of such crystals are the alkali halides, including ...

, which is experimentally difficult to determine. It is named after Anatoli Fedorovich Kapustinskii who published the formula in 1956. :

U_ =  \cdot \frac \cdot \biggl( 1 - \frac \biggr)

: The calculated lattice energy gives a good estimation for the Born–Landé equation; the real value differs in most cases by less than 5%. Furthermore, one is able to determine the

ionic radii Ionic radius, ''r''ion, is the radius of a monatomic ion in an ionic crystal structure. Although neither atoms nor ions have sharp boundaries, they are treated as if they were hard spheres with radii such that the sum of ionic radii of the cation ...

(or more properly, the thermochemical radius) using the Kapustinskii equation when the lattice energy is known. This is useful for rather complex ions like

sulfate The sulfate or sulphate ion is a polyatomic anion with the empirical formula . Salts, acid derivatives, and peroxides of sulfate are widely used in industry. Sulfates occur widely in everyday life. Sulfates are salts of sulfuric acid and many ...

(SO) or

phosphate In chemistry, a phosphate is an anion, salt, functional group or ester derived from a phosphoric acid. It most commonly means orthophosphate, a derivative of orthophosphoric acid . The phosphate or orthophosphate ion is derived from phosph ...

(PO).

Derivation from the Born–Landé equation

Kapustinskii originally proposed the following simpler form, which he faulted as "associated with antiquated concepts of the character of repulsion forces". :

U_ =  \cdot \frac

Here, ''K''' = 1.079 J·m·mol⁻¹. This form of the Kapustinskii equation may be derived as an approximation of the Born–Landé equation, below. :

U_L =- \frac\left(1-\frac\right)

Kapustinskii replaced ''r''₀, the measured distance between ions, with the sum of the corresponding ionic radii. In addition, the Born exponent, ''n'', was assumed to have a mean value of 9. Finally, Kapustinskii noted that the

Madelung constant The Madelung constant is used in determining the electrostatic potential of a single ion in a crystal by approximating the ions by point charges. It is named after Erwin Madelung, a German physicist. Because the anions and cations in an ionic sol ...

, ''M'', was approximately 0.88 times the number of ions in the empirical formula. The derivation of the later form of the Kapustinskii equation followed similar logic, starting from the quantum chemical treatment in which the final term is where ''d'' is as defined above. Replacing ''r''₀ as before yields the full Kapustinskii equation.

References

Literature

* * A. F. Kapustinskii; ''Zhur. Fiz. Khim.'' Nr. 5, 1943, pp. 59 ff. Chemical bonding Crystallography Soviet inventions {{crystallography-stub

Derivation from the Born–Landé equation

See also

References

Literature