The Kapustinskii equation calculates the

lattice energy In chemistry, the lattice energy is the energy change (released) upon formation of one mole of a crystalline compound from its infinitely separated constituents, which are assumed to initially be in the gaseous state at 0 K. It is a measure of ...

''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 Anatoli Fyodorovich Kapustinskii (; 29 December 1906 – 26 August 1960) was a Soviet chemist. He derived the Kapustinskii equation that allows an estimation of the lattice energy of an ionic crystal. Biography Kapustinskii was born in Zhytomyr, ...

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 Phosphates are the naturally occurring form of the element phosphorus. 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 orthop ...

(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 ioni ...

, ''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

* {{cite journal , last=Kapustinsky , first=A. , title=Allgemeine Formel für die Gitterenergie von Kristallen beliebiger Struktur , journal=Zeitschrift für Physikalische Chemie , publisher=Walter de Gruyter GmbH , volume=22B , issue=1 , date=1933-01-01 , issn=2196-7156 , doi=10.1515/zpch-1933-2220 , page=257, s2cid=202045251 , language=de * A. F. Kapustinskii; ''Zhur. Fiz. Khim.'' Nr. 5, 1943, pp. 59 ff. Chemical bonding Crystallography Eponymous equations of physics Soviet inventions

Derivation from the Born–Landé equation

See also

References

Literature