Berechnung der Madelungkonstante von Aragonit

Zusammenfassung

Die MadelungkonstanteM _a des Aragonits wird unter Berücksichtigung der nichtsphärischen Gestalt der CO ²⁻₃ -Ionen durch eine Multipolentwicklung als Funktion des C−O-Abstandesd und der Ladung am Sauerstoffatomq _o berechnet:

$$\begin{gathered} M_a = \frac{1}{4}\left\{ {10,4446---\left[ {0,65849 + \sum\limits_{n = 1}^{10} {A_n \left( {\frac{{d---0,8}}{a}} \right)^n } } \right]} \right\} \cdot q_o \hfill \\ \left. \begin{gathered} \hfill \\ ---\left[ {0,11066 + \sum\limits_{n = 1}^{12} {B_n \left( {\frac{{d---0,8}}{a}} \right)} ^n } \right] \cdot q_o^2 \hfill \\ \end{gathered} \right\}. \hfill \\ \end{gathered}$$

Mit den der Literatur entnommenen Wertend=1,30 kXE undq _o =−0,66 Elementarladungen ergibt sich:M _a =2,75148 bzw.U _M =739,38 kcal/Mol.

Abstract

Madelung's coefficientM _a of aragonite has been calculated considering the non-spherical shape of the CO ²⁻₃ -ions. As a result of the multipole expansionM _a has been found as a function of the C−O-distanced and the charge on the oxygen atomq _o to:

$$\begin{gathered} M_a = \frac{1}{4}\left\{ {10,4446---\left[ {0,65849 + \sum\limits_{n = 1}^{10} {A_n \left( {\frac{{d---0,8}}{a}} \right)^n } } \right]} \right\} \cdot q_o \hfill \\ \left. \begin{gathered} \hfill \\ ---\left[ {0,11066 + \sum\limits_{n = 1}^{12} {B_n \left( {\frac{{d---0,8}}{a}} \right)} ^n } \right] \cdot q_o^2 \hfill \\ \end{gathered} \right\}. \hfill \\ \end{gathered}$$

Using the valuesd=1.30 kXE andq _o =−0.66 elementary charges, which are taken from literature, the results are:M _a =2.75148 andU _M =739.38 kcal/mole respectively.