Problem 81 - Chemistry: The Central Science in SI Units, Global Edition

Question

A metal oxide has a lattice energy in the range of \(3300 \mathrm{~kJ} /\) mol. Is the charge on the metal ion M likely to be \(1^{+}, 2^{+},\) or \(3^{+} ?\) Explain.

Step-by-Step Solution

Answer

The charge on the metal ion M is most likely \(2^{+}\) because lattice energy is directly proportional to the product of the charges of the ions involved, and the given lattice energy of 3300 kJ/mol indicates a higher product of charges (Q1 * Q2). For charges on the metal ion of 1+, 2+, and 3+, the respective products are -2, -4, and -6. The higher product of -4 (associated with the \(2^{+}\) charge) aligns better with the given lattice energy value.

Step 1: Understand the lattice energy concept

Lattice energy is the energy required to break down an ionic solid into isolated gaseous ions. This energy value can be calculated using Coulomb's law, which states that the force between two charged particles is directly proportional to the product of the charges and inversely proportional to the square of the distance between the particles. Based on Coulomb's law, lattice energy can be expressed as: Lattice energy (U) = \( \frac{k * Q_1 * Q_2}{r} \) where Q1 and Q2 represent the charges of the ions involved, r is the distance between the ions, and k is a proportionality constant. From this equation, we can see that lattice energy is directly proportional to the product of the charges of the ions involved (i.e., metal cation and non-metal anion). Therefore, a metal ion with a higher charge will result in higher lattice energy.

Step 2: Analyze the given lattice energy value

The given lattice energy value is 3300 kJ/mol. It is important to note that lattice energy increases with the increase in charge on the ions involved. Comparing the possible charges, 1+, 2+, and 3+, we can analyze which charge would result in lattice energy close to the given value.

Step 3: Compare the possible charges and lattice energy

Since lattice energy is directly proportional to the product of the charges of the ions involved (Q1 * Q2), we can assume that the metal ion's charge (Q1) and the charge on the oxide ion (Q2 = 2-) are responsible for the lattice energy of 3300 kJ/mol. Comparing the possibilities for the metal ion with charges 1+, 2+, and 3+: 1. If the charge on the metal ion (Q1) is 1+, the product of charges (Q1 * Q2) would be 1 * (-2) = -2. 2. If the charge on the metal ion (Q1) is 2+, the product of charges (Q1 * Q2) would be 2 * (-2) = -4. 3. If the charge on the metal ion (Q1) is 3+, the product of charges (Q1 * Q2) would be 3 * (-2) = -6. The comparison shows that for lattice energy of 3300 kJ/mol, the higher the product of charges (Q1 * Q2), the greater the lattice energy. Therefore, we can conclude that it is most likely that the charge on the metal ion (M) would be 2+ in this case.

Key Concepts

Ionic Charge

The ionic charge of an atom is a key factor in understanding lattice energy. Ionic charge refers to the electrical charge an ion possesses, often noted as a superscript on the element symbol (e.g., Na\(^+\), Mg\(^{2+}\)).
This charge results from the loss or gain of electrons, thereby creating a disparity between the number of protons and electrons.
In ionic compounds, the interaction between positive and negative charges leads to the formation of a lattice structure.

• The higher the charges on the ions involved, the stronger the attraction between them.
• This means a higher lattice energy, which is the energy needed to separate the ions.
• For a metal oxide, we analyze different possible charges (1+, 2+, 3+) to understand their impact on lattice energy.

Understanding ionic charge helps explain why a particular metal ion charge, like 2+ in the example, corresponds with the given lattice energy value.

Coulomb's Law

Coulomb's Law is an essential principle when calculating lattice energy and understanding ionic interactions. This law describes the force between two charged particles, showing it is related to:

• The magnitude of the charges involved.
• The distance separating these charges.

According to Coulomb's Law, the greater the product of two charges, the stronger the force between them. When applying this to a lattice structure, it helps us understand why higher ionic charges result in increased lattice energy. The formula for lattice energy using Coulomb's Law is:\[U = \frac{k \cdot Q_1 \cdot Q_2}{r}\]Where:

• \(U\) is the lattice energy,
• \(k\) is a proportionality constant,
• \(Q_1\) and \(Q_2\) are the ionic charges, and
• \(r\) is the distance between ion centers.

Understanding this relationship helps predict and explain differences in lattice energies for various ionic compounds.

Metal Oxides

Metal oxides are compounds formed by a metal ion and an oxide ion (O\(^{2-}\)). These compounds are ionic in nature, composed of charged particles held together in a lattice by electrostatic forces.
In these structures, metal ions exhibit different charges, such as 1+, 2+, or 3+, impacting the compound's lattice energy. Metal oxides generally have high melting points due to the strong attraction between their ions.
The strength of this ionic bond, and hence the lattice energy, depends significantly on the charge of the metal ion:

• Higher charged metal ions lead to strong ionic bonds.
• This results in greater lattice energy, affecting the stability and properties of the metal oxide.

Understanding these concepts can shed light on why a metal oxide might have a particular lattice energy, as seen in the provided exercise.

Ionic Solids

Ionic solids are solid forms of ionic compounds, which include a wide array of materials such as salts and metal oxides. These solids are structured in a lattice, where positive and negative ions alternate positions.
This arrangement maximizes attractive forces while minimizing repulsive forces. These solids have distinct characteristics:

• They typically have high melting and boiling points.
• Their electrical conductivity is high when melted or dissolved in water, but low in solid form.
• Their rigidity and brittleness are pronounced due to the strong ionic bonds.

The lattice energy of ionic solids, such as a metal oxide, depends largely on the size and charge of the ions. A smaller and higher-charged ion contributes to a higher lattice energy due to stronger interactions and shorter distances between ions. Therefore, discussing ionic solids helps us appreciate how ionic charge and lattice energy relate to the properties and stability of these materials.