Question

Based on data in Table 8.2 , estimate (within \(30 \mathrm{~kJ} / \mathrm{mol}\) ) the lattice energy for (a) LiBr, (b) CsBr, (c) \(\mathrm{CaCl}_{2}\)

Short answer

By using the Born-Lande equation, \(E_l = -\frac{N_AMz^+z^-e^2}{4\pi \epsilon_0r_0}(1-\frac{1}{n})\), the lattice energies for LiBr, CsBr, and CaCl2 can be calculated after extracting the values of Madelung constant (M), ion charges (z⁺ and z⁻), ionic radii (r₀), and the Born exponent (n) for each compound from Table 8.2. Once the values are plugged into the equation and calculated, you can estimate the lattice energies for LiBr, CsBr, and CaCl2 within 30 kJ/mol.

Step-by-step solution

Write down the Born-Lande equation

The Born-Lande equation allows us to calculate the lattice energy, which is given by: \[E_l = -\frac{N_AMz^+z^-e^2}{4\pi \epsilon_0r_0}(1-\frac{1}{n})\] Where: - \(E_l\) is the lattice energy - \(N_A\) is Avogadro's number (approximately \(6.022\times10^{23}\) mol⁻¹) - \(M\) is the Madelung constant, a dimensionless constant specific to the crystal structure - \(z^+\) and \(z^-\) are the charges of the positive and negative ions, respectively - \(e\) is the elementary charge (approximately \(1.602\times10^{-19}\) C) - \(\epsilon_0\) is the vacuum permittivity constant (approximately \(8.854\times10^{-12}\) C²/(N·m²)) - \(r_0\) is the sum of the ionic radii of the positive and negative ions - \(n\) is the Born exponent, typically in the range of 5-12, depending on the compound.

Extract relevant data

From Table 8.2 of the given problem, extract the values of Madelung constant (M), charges of the ions (z⁺ and z⁻), ionic radii (r₀), and the Born exponent (n) for LiBr, CsBr, and CaCl2.

Calculate lattice energy for LiBr

Using the data extracted in Step 2, plug the values for LiBr into the Born-Lande equation and calculate the lattice energy for LiBr: \[E_l = -\frac{N_AMz^+z^-e^2}{4\pi \epsilon_0r_0}(1-\frac{1}{n})\]

Calculate lattice energy for CsBr

Using the data extracted in Step 2, plug the values for CsBr into the Born-Lande equation and calculate the lattice energy for CsBr: \[E_l = -\frac{N_AMz^+z^-e^2}{4\pi \epsilon_0r_0}(1-\frac{1}{n})\]

Calculate lattice energy for CaCl2

Using the data extracted in Step 2, plug the values for CaCl2 into the Born-Lande equation and calculate the lattice energy for CaCl2: \[E_l = -\frac{N_AMz^+z^-e^2}{4\pi \epsilon_0r_0}(1-\frac{1}{n})\] Once you have completed Steps 3-5, you will have estimated the lattice energy for LiBr, CsBr, and CaCl2 within 30 kJ/mol.

Key concepts

Born-Lande Equation

The Born-Lande Equation is a fundamental tool in predicting and understanding the lattice energy of ionic compounds. Lattice energy refers to the energy required to separate one mole of a solid ionic compound into gaseous ions. The equation accounts for various physical properties that influence this energy.

Here’s how it breaks down:

• The term \(N_A\) stands for Avogadro’s number, a constant representing the number of constituent particles in one mole of a substance.
• \(M\), known as the Madelung constant, varies with the specific geometric arrangement of ions in the crystal lattice.
• The charges of the ions \(z^+\) and \(z^-\) denote both the magnitude and sign of the respective ions.
• \(e\) is the elementary charge, a fixed value crucial for electrostatic calculations.
• \(\epsilon_0\), the vacuum permittivity constant, explains how electric fields interact in a vacuum.
• \(r_0\), the sum of ionic radii, gives a fundamental measurement of the size and distance between these ions.
• The Born exponent \(n\) reflects the compressibility of the ionic lattice, which can change depending on the crystalline structure.

By organizing these components, the Born-Lande Equation reveals how ionic sizes, charges, and crystal geometry collaboratively impact lattice energy. This equation serves as a bridge to quantifying electrostatic forces in solid ionic compounds.

Madelung Constant

In the realm of chemistry and physics, the Madelung Constant is pivotal in understanding crystal lattices. This dimensionless factor characterizes the electrostatic interactions in a highly structured ionic crystal. It describes how each ion within a lattice is influenced by its neighboring ions.

• The Madelung constant is specific to the type of crystal structure, requiring precise calculation for different forms like cubic, hexagonal, or complex geometries.
• It combines mathematical summations that account for the distances between each pair of charges in the lattice. Consequently, it offers a comprehensive measure of the total electrostatic interaction within a crystal.
• This constant is integral in calculating lattice energy through its role in the Born-Lande Equation, where it helps predict how much energy is needed to disassemble the lattice into separate ions.

Understanding the Madelung Constant is critical for insights into the stability and properties of ionic compounds. It illuminates how the distribution and interaction of charges within a lattice determine its overall lattice energy.

Ionic Compounds

Ionic compounds are formed by the electrostatic attraction between oppositely charged ions. These compounds have unique properties that distinguish them from molecular compounds and other chemical structures.

• Their formation typically involves a metal losing electrons to become a cation and a non-metal gaining those electrons to become an anion. This electron transfer establishes a strong bond, leading to highly structured crystalline arrangements.
• A common feature of ionic compounds is their high melting and boiling points, as significant energy is needed to overcome the electrostatic forces between ions.
• They usually dissolve in water and other polar solvents, where the ionic bonds break and disperse into the solvent, making these compounds excellent conductors of electricity when dissolved or molten.
• Ionic compounds are often brittle; applying stress can shift the lattice layers, causing like-charged ions to repel and shatter the structure.

These properties arise from the strong electrostatic forces holding the ions together, making understanding crystal structures essential to predicting and explaining the behavior of ionic compounds.

Crystal Structure

Crystal structure refers to the organized, repeating pattern of atoms or ions within a solid. It defines the arrangement and symmetry of the particles, influencing a range of physical and chemical properties.

• In ionic compounds, the crystal structure is pivotal for determining how cations and anions arrange to minimize repulsion and maximize attractive forces between opposite charges.
• Common types of crystal structures include face-centered cubic, body-centered cubic, and hexagonal close-packed arrangements, each with distinct geometric and spatial characteristics.
• The choice of crystal structure affects properties like density, conductivity, and reactivity, making it essential for engineers and chemists to understand a material’s crystalline form for applications.
• The lattice parameters, such as cell dimensions and angles between axes, are also crucial in defining crystal structures.

The understanding of crystal structures is not only crucial for the identification and prediction of material properties but also for the exploitation in technology and industry, where specific crystalline forms can be engineered to enhance performance.