Problem 22 - Chemistry: The Central Science

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) Using a periodic table, arrange the following substances according to their expected lattice energies, listing them from lowest lattice energy to the highest: \(\mathrm{ScN}, \mathrm{KBr}, \mathrm{MgO}, \mathrm{NaF}\). Compare your list with the data in Table \(8.2\).

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 of an ionic solid decreases as the sizes of the ions increase. (b) The order of the substances from lowest lattice energy to highest is KBr, NaF, MgO, ScN. This order should be consistent with the data provided in Table 8.2.

Step 1: Understanding Lattice Energy

Lattice energy is the energy required to separate one mole of an ionic solid into its gaseous ions. It depends on the charges of the ions and the sizes (ions' radii) of the ions involved in the crystal. One formula that you can use to relate lattice energy with charges and sizes of ions is the Born-Lande equation: \[E = -\dfrac{N A q_{+} q_{-} e^{2}}{4 \pi \varepsilon_{0} r_{0}} (1 - \dfrac{1}{n})\] where \(E\) is the lattice energy, \(N\) is Avogadro's number, \(A\) is the Madelung constant, \(q_{+}\) is the charge of the cation, \(q_{-}\) is the charge of the anion, \(e\) is the elementary charge, \(\varepsilon_{0}\) is the vacuum permittivity, \(r_{0}\) is the equilibrium distance between the ions, and \(n\) is the Born exponent.

Step 2: Part (a): How Charges and Sizes Affect Lattice Energy

(i) As the charges of the ions increase, the attractive forces between the ions increase, making it more difficult to separate them. So, the lattice energy of an ionic solid increases as the charges on the ions increase. (ii) As the sizes (ions' radii) of the ions increase, their distances from each other increase, resulting in weaker electrostatic forces between them, making it easier to separate them. So, the lattice energy of an ionic solid decreases as the sizes of the ions increase.

Step 3: Part (b): Arranging Substances According to Lattice Energies

To arrange the substances, we need to consider their charges and ions' radii. We know the charges of the ions from the periodic table: - ScN: Sc has a charge of +3, N has a charge of -3. - KBr: K has a charge of +1, Br has a charge of -1. - MgO: Mg has a charge of +2, O has a charge of -2. - NaF: Na has a charge of +1, F has a charge of -1. Looking at the charges, ScN should have the highest lattice energy, followed by MgO, and then KBr and NaF with similar lattice energies. Moving left to right across a period in the periodic table, ions generally get smaller. So, Na+ is larger than K+, and F- is larger than Br-, which makes NaF have a higher lattice energy than KBr. So, the order from lowest lattice energy to highest is KBr, NaF, MgO, ScN.

Step 4: Part (b): Comparing with Table 8.2 Data

Compare the given order with the data in Table 8.2. The actual values may be slightly different, but the order should remain the same.

Key Concepts

Ionic Solid

Ionic solids are types of crystalline structures where ions are the basic building blocks. These structures are formed when electrostatic forces hold oppositely charged ions together. The strength of these ionic bonds contributes to various properties of ionic solids, such as high melting points, brittleness, and electrical conductivity in molten states or when dissolved in water.
The lattice energy of an ionic solid is crucial as it quantifies the stability of the structure by measuring the energy required to separate one mole of the ionic solid into ions.

• Ionic solids are composed of ions arranged in a repeating pattern.
• Their stability is significantly influenced by the strength of these electrostatic forces, which are in turn determined by the charges and sizes of the ions.

Understanding ionic solids paves the way for further exploration into how lattice energy impacts their properties.

Born-Lande Equation

The Born-Lande equation is fundamental in calculating the lattice energy of ionic solids. It provides a mathematical relationship that helps understand how different factors contribute to the energy required to separate ions. The equation is:
\[ E = -\dfrac{N A q_{+} q_{-} e^{2}}{4 \pi \varepsilon_{0} r_{0}} \left(1 - \dfrac{1}{n}\right) \]

• In this equation, \(E\) represents the lattice energy.
• \(N\) is Avogadro's number, reflecting the number of ions in a mole.
• \(A\) is the Madelung constant, which is a geometric factor that depends on the crystal structure.
• \(q_{+}\) and \(q_{-}\) are the charges of the cation and anion, respectively.
• \(r_{0}\) is the equilibrium distance between ions, and \(n\) is the Born exponent, related to repulsion between ions.

This equation illustrates the direct relationship between the magnitude of ion charges and lattice energy, and the inverse relationship between ionic size and lattice energy.

Electrostatic Forces

Electrostatic forces are the forces of attraction and repulsion between charged particles. In the context of ionic solids, these forces are responsible for holding the ions in place. The strength of these forces directly influences the lattice energy.
When the charges on the ions increase, the electrostatic force becomes stronger, thereby increasing the lattice energy required to separate the ions. Conversely, when ions are larger, the distance between them increases, weakening the electrostatic forces, resulting in a lower lattice energy.

• Electrostatic forces are crucial for maintaining the integrity of ionic solids.
• Stronger electrostatic attractions increase the stability and hardness of the solid.
• They are calculated using Coulomb’s law as part of the Born-Lande equation.

Understanding these forces allows us to predict how ionic materials will behave under different conditions.

Ion Charges

Ion charges play a vital role in determining the lattice energy of an ionic solid. The charges of the ions directly affect the electrostatic forces between them. Higher charges mean stronger attractions between the ions, leading to higher lattice energy.
For example, in compounds like \(\mathrm{ScN}\) and \(\mathrm{MgO}\), the ions have higher charges compared to \(\mathrm{KBr}\) and \(\mathrm{NaF}\). This results in higher lattice energies for \(\mathrm{ScN}\) and \(\mathrm{MgO}\).

• Higher ion charges lead to increased electrostatic attractions.
• Compounds with higher charged ions are usually more stable.
• In calculations, ion charges are squared, amplifying their impact on lattice energy.

Recognizing the importance of ion charges assists in predicting which ionic solids will be more stable and have higher melting points.

Ion Sizes

The size of ions significantly affects the lattice energy of an ionic solid. As ion sizes increase, the distance between the ions in a crystal lattice also increases. This leads to decreased electrostatic attraction, thereby reducing the lattice energy needed to separate the ions.
Smaller ions can pack together more closely, resulting in stronger attractions and higher lattice energy. Considerations around ion size help understand why certain ionic compounds have higher stability.

• Larger ions generally mean lower lattice energy due to weaker attractions.
• Smaller ions form more compact structures, increasing stability.
• Ion size impacts melting points and other physical properties of ionic solids.

By understanding how ion sizes affect lattice energy, students can better predict and compare the properties of different ionic compounds.