The Born–Mayer equation

Key points: The Born–Mayer equation is used to estimate lattice enthalpy for an ionic lattice. The Madelung constant reflects the effect of the geometry of the lattice on the strength of the net Coulombic interaction. To calculate the lattice enthalpy of a supposedly ionic solid we need to take into account several contributions, including the Coulombic attractions and repulsions between the ions and the repulsive interactions that occur when the electron densities of the ions overlap. This calculation yields the Born–Mayer equation for the lattice enthalpy at T=0:

where d=r₊ + r_- is the distance between centres of neighbouring cations and anions, and hence a measure of the ‘scale’ of the unit cell (for the derivation, see Further information 3.1). In this expression NA is Avogadro’s constant, z_A and z_B the charge numbers of the cation and anion, e the fundamental charge, ε0 the vacuum permittivity, and d* a constant (typically 34.5 pm) used to represent the repulsion between ions at short range. The quantity A is called the Madelung constant, and depends on the structure (specifically, on the relative distribution of ions, Table 3.8). The Born–Mayer equation in fact gives the lattice energy as distinct from the lattice enthalpy, but the two are identical at T 0 and the difference may be disregarded in practice at normal temperatures.

■ A brief illustration. To estimate the lattice enthalpy of sodium chloride, we use z(Na⁺) 1, z (Cl^-)=^-1, from Table 3.8, A=1.748, and from Table 1.4 d=r_Na++r_Cl- 283 pm; hence (using fundamental constants from inside the back cover):

or 756 kJ mol 1. This value compares reasonably well with the experimental value from the Born Haber cycle, 788 kJ mol^-1.The form of the Born–Mayer equation for lattice enthalpies allows us to account for their dependence on the charges and radii of the ions in the solid. Thus, the heart of the equation is

Therefore, a large value of d results in a low lattice enthalpy, whereas high ionic charges result in a high lattice enthalpy. This dependence is seen in some of the values given in Table 3.7. For the alkali metal halides, the lattice enthalpies decrease from LiF to LiI and from LiF to CsF as the halide and alkali metal ion radii increase, respectively. We also note that the lattice enthalpy of MgO (|z_A z_B|=4) is almost four times that of NaCl (|z_A z_B|= 1) due to the increased charges on the ions for a similar value of d, noting that the Madelung constant is the same.

The Madelung constant typically increases with coordination number. For instance, A 1.748 for the (6,6)-coordinate rock-salt structure but A 1.763 for the (8,8) coordinate caesium-chloride structure and 1.638 for the (4,4)-coordinate sphalerite structure. This dependence reflects the fact that a large contribution comes from nearest neighbours, and such neighbours are more numerous when the coordination number is large. However, a high coordination number does not necessarily mean that the interactions are stronger in the caesium-chloride structure because the potential energy also depends on the scale of the lattice. Thus, d may be so large in lattices with ions big enough to adopt eightfold-coordination that the separation of the ions re verses the effect of the small increase in the Madelung constant and results in a smaller lattice enthalpy.

اخر الاخبار

اخبار العتبة العباسية المقدسة

قسم الصيانة ينفذ أعمال إدامة لإحدى المدارس في محافظة النجف الأشرف

المجمع العلمي يستأنف الختمة القرآنية الأسبوعية في محافظة الديوانية

حملة تطوير وتأهيل تشهدها مدينة الفردوس الترفيهية استعدادًا لاستقبال العوائل

العتبة العباسية المقدسة تطلق فعاليات مهرجان (ترتيل الرأس) السنوي الثاني في بغداد

المزيد

الآخبار الصحية

لون أسنانك يتحدث عن صحتك.. إليك ما يقول

عادة يومية بسيطة تخفض الكوليسترول وتحمي القلب

الزنجبيل.. دواء سحري لحماية القلب

المشروبات شديدة السخونة.. "خطر خفي" يهدد صحتك

المزيد

الآخبار التكنلوجية

كيف يمكن لعادات القيادة السيئة أن تؤثر على قيمة سيارتك

في طوفان الذكاء الاصطناعي.. نصائح لحجز "تذكرة النجاة"

هكذا يسرق "قراصنة الذكاء الاصطناعي" المليارات بطرق بسيطة

فيديو لانفجار كرة نارية في سماء اليابان.. ما تفسير ذلك؟

المزيد

مواضيع ذات صلة

Structure maps

The radius ratio

Ionic radii

Characteristic structures of ionic solids

Atomic radii of metals

Nonclose-packed structures

Polytypism

The close packing of spheres

Lattices and unit cells

The orbitals

Bonding and antibonding orbitals

Hybridization