Equations Born-Mayer equation

The binding energy of a solid is the energy required to disperse a solid into its constituent atoms, against the forces of cohesion. In the case of ionic crystals, it is given by the Born-Mayer equation. See Crystal. 

Born-Mayer equation — This equation predicts the lattice energy of crystalline solids [i]. It is based on a simple model, in which the attraction and repulsion for a given arrangement of ions is calculated. The Born-Mayer energy (EB-m) is given by [i] ... 

Substituting this value for B in Eq. (3.6) and changing signs because U is defined as the energy required to separate a mole of the crystal into gaseous ions (so it has a positive sign) yields the Born-Mayer equation,... 

For application to nonmolecular solids, the bond description is similar but certain modifications are needed. First, the covalent energy must be multiplied by the equivalent number n of two electron covalent bonds per formula unit that must be broken for atomization. The evaluation of n will be discussed in detail presently. Second, the ionic energy must be evaluated as the potential energy over the entire crystal, corrected for the repulsions among adjacent electronic spheres. This is done by using the Born-Mayer equation for lattice energy, multiplying this expression by an empirical constant, a, which is 1 for the halides and less than 1 for the chal-cides, as follows ... 

In this particular example, if we had assumed the simple ionic model without a covalent contribution, the Born-Mayer equation would have provided a lattice energy (5, 15) ... 

If we had assumed the simple ionic model here, the Born-Mayer equation would have given a lattice energy of 178.4 kcal. per mole. The ionization energy of silver is 176.2 and the electron affinity of bromine (2) is — 79.1, from which the atomization energy of AgBr is 81.3 kcal. per mole, in error by nearly 40 kcal. Efforts to modify the Born-Mayer equation to take other factors into account (5) have not produced satisfactory results for such compounds. 

Later workers have used the Born-Mayer equation and the extended Mayer-Huggins method, allowing for the Van der Waals and zero-point energies. The computations made and their comparison with thermochemical data, utilizing the most recent data, either from the U. S. Bureau of Standards Circular 500 or from later sources, will be considered below. 

E3.28 If all other parameters for cubic and hexagonal ZnS are kept constant, then according to the Born-Mayer equation, the difference is only in the Made lung constants for the two polymorphs. From Table 3.8 the constants are 1.641 and 1.638 for hexagonal and cubic ZnS respectively. Since the A is larger for hexagonal ZnS, the hexagonal ZnS should be a more stable polymorph. 

For the case of completely ionic bonding, the Born-Mayer equation will be used ... 

The Born-Mayer equation is an alternative (and possibly more accurate) form based on the assumption of an exponential form for the repulsive energy. Both equations predict lattice energies for compounds such as alkali halides that are in reasonably close agreement with the experimental values from the Born-Haber cycle. Some examples are shown in Table 1. A strict comparison requires some corrections. Born-Haber values are generally enthalpies, not total energies, and are estimated from data normally measured at 298 K not absolute zero further corrections can be made, for example, including van der Waals forces between ions. 

Using the Born Mayer equation from Section 7.2.1 ... 

Where b and p are constants determined from compressibility measurements. This gives the Born-Mayer equation (Born and Mayer, 1932) for lattice energy ... 

The Born-Mayer equation emphasizes the fact that Eq. 4.1 is designed only to match the observed phenomenon. It is not a fundamental truth like the Coulomb interaction. 

At the equilibrium interatomic distance d UIdd = 0, i.e. the attractive and the repulsive forces are equal. From here we obtain the well known Born-Land6 and Born-Mayer equations. 

Reddy and Murphy showed that Pauling s equation is valid only for a limited range of molecules where Ax is small, and substitution of the arithmetical mean by the geometrical mean makes little improvement. A better correlation is found if the extra ionic energy (EIE) is expressed as feAx rather than as kA. The EIE may be represented by a quasi-Coulombic expression based on the Born-Mayer equation, thus Eq. 2.74 transforms into... 

Equations 1.113 and 1.114 are the most frequently used in geochemical studies. However, the most physically sound equation of state for solids is perhaps the Born-Mayer form ... 

Equation 5 is often used to decribe the interaction between the incoming ion and the target atoms. The interaction between two target atoms generally occurs at low energy where the Thomas-Fermi potential overestimates the interaction. Under this situation a Born-Mayer potential is more appropriate , i.e. ... 

The interaction between dissolved ions and the dipole molecules of water in aqueous solutions is called - hydration. (See also - Born equation, -> Born-Haber cycle, -> Born-Mayer equation, -> hydrated ion, -> hydration number.)... 

We present how to treat the polarization effect on the static and dynamic properties in molten lithium iodide (Lil). Iodide anion has the biggest polarizability among all the halogen anions and lithium cation has the smallest polarizability among all the alkaline metal cations. The mass ratio of I to Li is 18.3 and the ion size ratio is 3.6, so we expect the most drastic characteristic motion of ions is observed. The softness of the iodide ion was examined by modifying the repulsive term in the Born-Mayer-Huggins type potential function in the previous workL In the present work we consider the polarizability of iodide ion with the dipole rod method in which the dipole rod is put at the center of mass and we solve the Euler-Lagrange equation. This method is one type of Car-Parrinello method. 

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