论文标题
基于库仑力的分析，用于离子活性系数的扩展Bjerrum方程的推导
Derivation of an extended Bjerrum equation for the activity coefficient of ions based on an analysis of Coulombic forces
论文作者
论文摘要
Bjerrum（1916）提出了溶液中离子的活性系数，以依赖于浓度的立方根，因为与低和中度盐浓度的数据非常吻合。但是，德比和赫克尔（DH）后来发展了一种盛行的理论，该理论具有平方根的依赖性。 DH方程的推导并不容易或直观，并且具有各种不确定的元素，并且对数据的拟合对于具有非合理价值的盐并不是很好。我们根据离子和最近电荷离子之间的库仑力（包括这些离子之间的分离分布）开发了活性系数的模型。对于对称盐，我们只需要分析一个阴离子和一种阳离子之间的相互作用，并且我们可以得出稀释极限的表达，该稀限取决于Bjerrum长度和盐浓度的立方根，与Bjerrum提出的表达相同。我们的理论还分析地描述了较高的盐浓度和非零离子大小，不仅在1：1，而且对于2：2和3：3盐。对于1：1盐，立方根缩放定律在低和中间的盐浓度下很好地描述了活性系数的数据，而扩展的Bjerrum方程式则更好地描述了数据甚至更高浓度，还包括离子大小的效果。对于非对称盐（2：1和3：1盐），可以使用数值程序，该程序可以考虑在相反电荷的中央多价离子周围的两个或三个单价离子的所有可能的轨道，这也准确地拟合了数据。
The activity coefficient of ions in solution was proposed by Bjerrum (1916) to depend on the cube root of concentration, because of a good fit with data for low and moderate salt concentration. However, Debye and Hückel (DH) later developed a theory that prevailed, and this theory has a square root dependence. The derivation of the DH equation is not easy or intuitive, and has various uncertain elements, and the fit to data is not very good for salts with non-unity valencies. We develop a model for the activity coefficient based on the Coulombic forces between an ion and the nearest ions of opposite charge, including the distribution of separations between these ions. For a symmetric salt, we only have to analyse the interactions between one anion and one cation, and we can derive an expression for the dilute limit that depends on the Bjerrum length and the cube root of salt concentration, the same as an expression put forward by Bjerrum. Our theory also analytically describes higher salt concentrations and a non-zero ion size, not only for 1:1 but also for 2:2 and 3:3 salts. For a 1:1 salt, the cube root scaling law describes data of the activity coefficient very well, both at low and intermediate salt concentrations, while the extended Bjerrum equation describes data even better and up to higher concentrations, and also includes the effect of ion size. For asymmetric salts (2:1 and 3:1 salts), a numerical procedure can be used which considers all possible orbits of two or three monovalent ions around a central multivalent ion of opposite charge, which also fits data accurately.
