玻恩–朗德方程

${\displaystyle E=-{\frac {N_{A}Mz^{+}z^{-}e^{2}}{4\pi \epsilon _{0}r_{0}}}\left(1-{\frac {1}{n}}\right)}$

• NA = 阿伏伽德罗常数
• M = 马德隆常数，取决于晶体中的几何排列；
• z+ = 阳离子电荷数；
• z = 阴离子电荷数；
• e = 元电荷，大约1.6022×10−19 C
• ε0 = 真空电容率
ε0 = 1.112×10−10 C2/(J·m)
• r0 = 最近离子的距离
• n = 玻恩指数，通常在5到12之间，可由实验测定压缩性或理论计算得出[3]

推导

静电势

${\displaystyle E_{\text{pair}}=-{\frac {z^{2}e^{2}}{4\pi \epsilon _{0}r}}}$

z = 一个离子所带电荷
e = 元电荷，大约1.6022×10−19 C
ε0 = 真空电容率
ε0 = 1.112×10−10 C2/(J·m)；
r0 = 最近离子的距离

${\displaystyle E_{M}=-{\frac {z^{2}e^{2}M}{4\pi \epsilon _{0}r}}}$

${\displaystyle M}$ = 马德隆常数，取决于晶体中的几何排列；
${\displaystyle r}$ = 最近不同电性离子的距离

排斥理论

${\displaystyle \,E_{R}={\frac {B}{r^{n}}}}$

${\displaystyle B}$ = 表示推斥作用强度的常数
${\displaystyle r}$ = 最近不同电性离子的距离
${\displaystyle n}$ = 玻恩指数，通常在5到12之间，表示某种晶体的压缩性

总能量

${\displaystyle E(r)=-{\frac {z^{2}e^{2}M}{4\pi \epsilon _{0}r}}+{\frac {B}{r^{n}}}}$

{\displaystyle {\begin{aligned}{\frac {\mathrm {d} E}{\mathrm {d} r}}&={\frac {z^{2}e^{2}M}{4\pi \epsilon _{0}r^{2}}}-{\frac {nB}{r^{n+1}}}\\0&={\frac {z^{2}e^{2}M}{4\pi \epsilon _{0}r_{0}^{2}}}-{\frac {nB}{r_{0}^{n+1}}}\\r_{0}&=\left({\frac {4\pi \epsilon _{0}nB}{z^{2}e^{2}M}}\right)^{\frac {1}{n-1}}\\B&={\frac {z^{2}e^{2}M}{4\pi \epsilon _{0}n}}r_{0}^{n-1}\end{aligned}}}

${\displaystyle E(r_{0})=-{\frac {Mz^{2}e^{2}}{4\pi \epsilon _{0}r_{0}}}\left(1-{\frac {1}{n}}\right)}$

计算晶格能

NaCl −756 kJ/mol −787 kJ/mol
LiF −1007 kJ/mol −1046 kJ/mol
CaCl2 −2170 kJ/mol −2255 kJ/mol

参考资料

1. ^ Brown, I. David. The chemical bond in inorganic chemistry : the bond valence model Reprint. New York: Oxford University Press. 2002. ISBN 0-19-850870-0.
2. Johnson, the Open University ; RSC ; edited by David. Metals and chemical change 1. publ. Cambridge: Royal Society of Chemistry. 2002. ISBN 0-85404-665-8.
3. ^ Cotton, F. Albert; Wilkinson, Geoffrey, Advanced Inorganic Chemistry 4th, New York: Wiley, 1980, ISBN 0-471-02775-8