Consider a hypothetical X^+-Y^- ion pair for which the...

Consider a hypothetical $X^{+}-Y^{-}$ ion pair for which the equilibrium interionic spacing and bonding energy values are $0.38 \mathrm{nm}$ and $-5.37 \mathrm{eV},$ respectively. If it is known that $n$ in Equation 2.11 has a value of $8,$ using the results of Problem 2.14 , determine explicit expressions for attractive and repulsive energies $E_{A}$ and $E_{R}$ of Equations 2.8 and 2.9

Key Concepts

Potential Energy Modeling in Ionic Solids

This concept involves the construction of a total potential energy function that is a sum of contributions from long-range electrostatic attractions and short-range repulsions. By incorporating experimentally determined parameters such as equilibrium spacing and bonding energy, and using appropriate mathematical forms (e.g., 1/r for attraction and 1/r? or exponential forms for repulsion), one can derive explicit expressions for each energy term. Such models are central to understanding the stability, structure, and properties of ionic materials.

Ionic Equilibrium and Energy Minimization

At the equilibrium interionic spacing, the attractive and repulsive forces exactly balance, leading to a minimum in the total potential energy of the system. By setting the derivative of the total energy (sum of attractive and repulsive terms) with respect to the ionic separation to zero, explicit expressions for these energy components can be formulated. This equilibrium condition is essential in deriving models such as the Born–Landé equation, which quantitatively describes the bonding in ionic compounds.

Repulsive Interaction (Born or Born–Mayer Repulsion)

Repulsive energy accounts for the short-range force that prevents ions from collapsing into each other when their electron clouds overlap. Generally described by an exponential or power law term (with an exponent n), this repulsive component increases rapidly at short distances and balances the attractive Coulomb forces. This concept is pivotal in models of ionic bonding, as it defines the steep energy rise when ions are brought too close together.

Electrostatic (Coulomb) Attraction

This concept refers to the attractive force between ions of opposite charge as described by Coulomb’s law. The associated potential energy is inversely proportional to the distance between the ions, typically expressed as -kq?q?/r, where k is a constant and q? and q? are the ionic charges. This term is fundamental in calculations of bonding energy in ionic systems, as it represents the long?range attraction that stabilizes the structure.

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