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Chapter 13: Problem 104

The correct order of the lattice energies for the following ionic compounds is (a) \(\mathrm{Al}_{2} \mathrm{O}_{3}>\mathrm{CaO}>\mathrm{MgBr}_{2}>\mathrm{NaCl}\) (b) \(\mathrm{MgBr}_{2}>\mathrm{Al}_{2} \mathrm{O}_{3}>\mathrm{CaO}>\mathrm{NaCl}\) (c) \(\mathrm{Al}_{2} \mathrm{O}_{3}>\mathrm{MgBr}_{2}>\mathrm{CaO}>\mathrm{NaCl}\) (d) \(\mathrm{NaCl}>\mathrm{MgBr}_{2}>\mathrm{CaO}>\mathrm{Al}_{2} \mathrm{O}_{3}\)

Short Answer

Expert verified
The correct order is (a) \(\mathrm{Al}_{2} \mathrm{O}_{3} > \mathrm{CaO} > \mathrm{MgBr}_{2} > \mathrm{NaCl}\).

Step by step solution

01

Define Lattice Energy

Lattice energy is the energy required to completely separate one mole of a solid ionic compound into its gaseous ions. It is influenced by the charges on the ions and the distance between them, given by: \[ U = rac{k imes |Z^+ imes Z^-|}{r} \] where \( U \) is the lattice energy, \( Z^+ \) and \( Z^- \) are the charges on the cation and anion, \( r \) is the distance between the ions, and \( k \) is a proportionality constant.
02

Compare Ionic Charges

The compounds listed are \( \mathrm{Al}_{2} \mathrm{O}_{3}, \) \( \mathrm{CaO}, \) \( \mathrm{MgBr}_{2}, \) and \( \mathrm{NaCl}. \) Their ion pairs are AlO, CaO, MgBr, and NaCl.- \( \mathrm{Al}^{3+} \) and \( \mathrm{O}^{2-} \) contribute the highest charge product: \( +3 \times -2 = -6 \).- \( \mathrm{Ca}^{2+} \) and \( \mathrm{O}^{2-} \): \( +2 \times -2 = -4 \).- \( \mathrm{Mg}^{2+} \) and \( \mathrm{Br}^{-} \) contribute \( +2 \times -1 = -2 \).- \( \mathrm{Na}^{+} \) and \( \mathrm{Cl}^{-} \) have \( +1 \times -1 = -1 \).Higher charge products usually indicate higher lattice energy.
03

Consider Ionic Sizes

Lattice energy is inversely proportional to the distance between ions, \( r \). Smaller ions result in a smaller \( r \), which increases lattice energy. Here are the sizes ranked approximately from smallest to largest:- \( \mathrm{O}^{2-} \) and \( \mathrm{Al}^{3+} \) are smaller than other ions.- \( \mathrm{Ca}^{2+} \) and \( \mathrm{O}^{2-} \) create a smaller \( r \) than \( \mathrm{MgBr}_{2} \).- \( \mathrm{NaCl} \) has relatively large ions, increasing \( r \).
04

Determine Order of Lattice Energies

Based on the charge and size analysis, lattice energies can be ordered as follows:1. \( \mathrm{Al}_{2} \mathrm{O}_{3} \) has the highest lattice energy due to the highest charge product and small ion size.2. \( \mathrm{CaO} \), with moderate charge and small ionic distances, follows.3. \( \mathrm{MgBr}_{2} \) is next, with a smaller charge product than \( \mathrm{CaO} \).4. \( \mathrm{NaCl} \) has the lowest lattice energy because of its comparatively low charge product and larger ionic size.
05

Identify the Correct Option

The correct order of lattice energies is \( \mathrm{Al}_{2} \mathrm{O}_{3} > \mathrm{CaO} > \mathrm{MgBr}_{2} > \mathrm{NaCl} \).This matches option (a) among the given choices.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ionic Compounds
Ionic compounds are formed from the electrostatic attraction between cations and anions. They typically consist of a metal and a non-metal. Understanding their lattice energy is crucial. Lattice energy refers to the energy required to separate the ions in a solid into gaseous ions. This energy is a key factor in determining the stability and strength of the ionic compound. Ionic compounds with high lattice energy are generally more stable and have higher melting and boiling points.

Important factors influencing the lattice energy:
  • Charge of ions: Higher charges typically lead to higher lattice energies.
  • Size of ions: Smaller ions contribute to greater lattice energy because they can get closer to each other.
Understanding these factors can help predict how ionic compounds behave under different conditions.
Charge Product
The charge product of the ions in an ionic compound directly affects its lattice energy. This charge product is calculated as the product of the charges on the cation and anion. The greater this product, the higher the lattice energy likely is. In simple terms, more charged ions attract each other more strongly. For instance, comparing the charge products of various compounds:
  • \[ \mathrm{Al}_{2} \mathrm{O}_{3}, \] with \[ +3 imes -2 = -6,\]
  • \[ \mathrm{CaO}, \] with \[ +2 \times -2 = -4,\]
  • \[ \mathrm{MgBr}_{2}, \] with \[ +2 \times -1 = -2,\]
  • \[ \mathrm{NaCl},\] with \[ +1 \times -1 = -1.\]
The sequence reflects the predicted order of lattice energies based purely on charge products. Typically, a higher total charge means a stronger force holding the compound together, suggesting a higher lattice energy.
Ionic Size
The size of ions in a compound is critical when evaluating lattice energy. The smaller the ions, the closer they can be to each other in the crystal lattice. This closeness increases the electrostatic forces that hold the compound together. In simpler terms, it's like the closer two magnets are, the stronger their attraction.

Comparing ion sizes:
  • \[ \mathrm{Al}^{3+} \text{ and } \mathrm{O}^{2-} \] are smaller than the other ions involved.
  • \[ \mathrm{Ca}^{2+} \text{ } \] and \[ \mathrm{O}^{2-} \] form a tighter bond than larger ions.
  • \[ \mathrm{Mg}^{2+}\] and \[ \mathrm{Br}^{-}\] are larger, leading to a greater distance.
  • \[ \mathrm{Na}^{+} \] and \[ \mathrm{Cl}^{-}\] in \[ \mathrm{NaCl} \] have larger radii, increasing the distance.
This ranking by size correlates with the distance between ions, crucial for understanding the variation in lattice energies.
Order of Lattice Energies
To determine the order of lattice energies for a series of ionic compounds, both the charge product and ionic size must be considered. These factors together dictate how strongly ions stick together in the crystal lattice. Reviewing the given compounds:
  • \[ \mathrm{Al}_{2} \mathrm{O}_{3} \] stands out due to its high charge product and small ionic size.
  • \[ \mathrm{CaO} \] follows, supported by a moderate charge and small ion size.
  • \[ \mathrm{MgBr}_{2} \] comes next with a lesser charge impact compared to \[ \mathrm{CaO} \].
  • \[ \mathrm{NaCl} \], with the least charge and larger ions, has the lowest lattice energy.
This sequence helps in predicting properties such as melting and boiling points and informs practical applications where the stability of an ionic compound is a deciding factor.

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