Problem 24 - Chemistry: The Central Science in SI Units, Global Edition

Question

(a) Does the lattice energy of an ionic solid increase or decrease (i) as the charges of the ions increase, (ii) as the sizes of the ions increase? (b) Arrange the following substances not listed in Table 8.1 according to their expected lattice energies, listing them from lowest lattice energy to the highest: MgS, KI, GaN, LiBr.

Step-by-Step Solution

Answer

(a) (i) The lattice energy of an ionic solid increases as the charges of the ions increase. (ii) The lattice energy decreases as the sizes of the ions increase. (b) The substances can be arranged according to their expected lattice energies in the following order (lowest to highest): KI, LiBr, MgS, GaN.

Step 1: Understanding Lattice Energy

Lattice energy is the energy required to separate one mole of an ionic solid into its constituent gaseous ions. It depends on the charges of the ions and the distance between them. The lattice energy can be calculated using the Born-Lande equation or the Born-Haber cycle, but for this exercise, we will stick to the basic concept. A key principle to keep in mind is that the larger the charge on the ions and the smaller the distance between the ions, the greater the lattice energy.

Step 2: Determining the effect of ion charges and sizes on lattice energy

Now, let's address part (a) of the exercise. (i) As the charges of the ions increase, the electrostatic force between the ions increases, which results in a higher lattice energy. (ii) As the sizes of the ions increase, the distance between the ions also increases, which results in a lower lattice energy.

Step 3: Arranging the given substances based on their expected lattice energies

For part (b), we should consider the ion charges and sizes for each substance to rank their lattice energies. Let's analyze each substance: 1. MgS: Magnesium ion (Mg^(2+)) has a charge of +2 and sulfur ion (S^(2−)) has a charge of -2. Both Mg^(2+) and S^(2−) are relatively small ions. 2. KI: Potassium ion (K^+) has a charge of +1 and iodide ion (I^−) has a charge of -1. Both K^+ and I^− are larger ions than Mg^(2+) and S^(2−). 3. GaN: Gallium ion (Ga^(3+)) has a charge of +3 and nitride ion (N^(3−)) has a charge of -3. Both Ga^(3+) and N^(3−) are relatively small ions. 4. LiBr: Lithium ion (Li^+) has a charge of +1 and bromide ion (Br^−) has a charge of -1. The Li^+ ion is smaller than K^+, but the Br^− ion is larger than I^−. Considering the information above, we can arrange the substances in order of their expected lattice energies from lowest to highest: Lowest Lattice Energy: KI (Lowest charges and larger ions) < LiBr (Lowest charges and smaller ions) < MgS (Higher charges and larger ions) < GaN (Highest charges and smaller ions): Highest Lattice Energy So, the final order is: KI, LiBr, MgS, GaN.

Key Concepts

Ionic Solids

Ionic solids are a type of crystalline material composed primarily of ions held together by strong electrostatic forces. These solids consist of positively charged cations and negatively charged anions arranged in a repeating pattern. This structure results in distinct properties that make ionic solids unique.

Some notable characteristics of ionic solids include:

• High melting and boiling points due to strong ionic bonds.
• Hard and brittle texture.
• Generally soluble in polar solvents like water.
• Capable of conducting electricity in molten state or when dissolved in water due to the mobility of ions.

In the context of lattice energy, ionic solids demonstrate the fundamental concepts of ion interaction. The attraction between oppositely charged ions forms the basis for lattice energy, crucial in understanding these materials' stability and properties.

Born-Lande Equation

The Born-Lande equation is a mathematical formula used to estimate the lattice energy of ionic solids. It is particularly useful in predicting the stability and properties of these compounds. The equation calculates lattice energy by considering both the size of the ions and their charge.

The formula for the Born-Lande equation is:
\[ U = rac{{N_A imes A imes Z^+ imes Z^- imes e^2}}{{4 ext{πε}_0r}} imes (1-rac{1}{n}) \]
where:

• \( U \) = Lattice energy
• \( N_A \) = Avogadro's number
• \( A \) = Madelung constant (depends on crystal structure)
• \( Z^+ \) and \( Z^- \) = Charges on the cation and anion
• \( e \) = Elementary charge
• \( ε_0 \) = Permittivity of free space
• \( r \) = Distance between ions
• \( n \) = Born exponent (related to repulsion)

This equation highlights the significance of ionic charge and size in determining lattice energy, supporting observations in ionic solids' behavior.

Electrostatic Force

Electrostatic force is the fundamental force of attraction or repulsion between charged particles. In ionic solids, this force plays a key role in maintaining the structure and stability of the material.

When discussing electrostatic forces in ionic compounds, we note that:

• The force is directly proportional to the product of the charges involved. Higher charges lead to stronger forces.
• The force is inversely proportional to the square of the distance between the ions. Closer ions experience stronger attraction.

Thus, the characteristics of the ions, such as their charge and radius, directly contribute to the magnitude of electrostatic forces. This understanding is crucial for explaining why lattice energy increases with higher charges and smaller distances between ions. Such interactions define the robustness and physical properties of ionic solids.

Born-Haber Cycle

The Born-Haber cycle is a theoretical model that helps analyze the formation of ionic compounds by breaking down the formation process into multiple enthalpy changes. It allows us to calculate lattice energy indirectly by considering other thermochemical data.

The cycle typically involves steps such as:

• Formation of gaseous atoms from the solid elements.
• Ionization of atoms to form ions.
• The recombination of ions to form the solid compound.

By applying Hess's law, which states that the total enthalpy change is the same regardless of the path taken, the lattice energy can be derived from the sum of enthalpies involved in these steps. The Born-Haber cycle thus provides insight into the energetic aspects of bond formation in ionic solids, highlighting the importance of lattice energy in the compound's stability.