Chapter 10: Problem 70
The compounds \(\mathrm{Na}_{2} \mathrm{O}, \mathrm{CdS}\), and \(\mathrm{Zr} \mathrm{I}_{4}\) all can be described as cubic closest packed anions with the cations in tetrahedral holes. What fraction of the tetrahedral holes is occupied for each case?
Short Answer
Expert verified
In cubic closest packed anions with cations in tetrahedral holes, the fraction of occupied tetrahedral holes for Na2O is 1, for CdS is 0.5, and for ZrI4 is 0.125.
Step by step solution
01
Find the ratio of cations to anions for each compound
For each compound, we need to find the ratio of the cations to the anions.
1. Na2O: 2 Na+ cations and 1 O2- anion
2. CdS: 1 Cd2+ cation and 1 S2- anion
3. ZrI4: 1 Zr4+ cation and 4 I- anions
02
Calculate the number of tetrahedral holes in a ccp structure
In a cubic closest packed structure, there are twice as many tetrahedral holes as there are anions. We will use this information to calculate the number of tetrahedral holes for each compound.
1. Na2O: Number of tetrahedral holes = 2 × 1 (anions) = 2
2. CdS: Number of tetrahedral holes = 2 × 1 (anions) = 2
3. ZrI4: Number of tetrahedral holes = 2 × 4 (anions) = 8
03
Calculate the fraction of occupied tetrahedral holes
Now, we can find the fraction of occupied tetrahedral holes for each compound by dividing the number of cations (found in step 1) by the number of tetrahedral holes (found in step 2).
1. Na2O: Fraction of occupied tetrahedral holes = 2 (cations) / 2 (tetrahedral holes) = 1
2. CdS: Fraction of occupied tetrahedral holes = 1 (cations) / 2 (tetrahedral holes) = 0.5
3. ZrI4: Fraction of occupied tetrahedral holes = 1 (cations) / 8 (tetrahedral holes) = 0.125
In conclusion, the fraction of occupied tetrahedral holes for each compound is:
1. Na2O: 1
2. CdS: 0.5
3. ZrI4: 0.125
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
cubic closest packed
The term "cubic closest packed" (or ccp) refers to a highly efficient arrangement of atoms in a crystal lattice. In this structure, atoms are packed together as tightly as possible, like oranges stacked in a crate, and they fill up the available space with maximum efficiency. In a cubic closest packed system, each atom is surrounded by twelve equidistant neighbors. This arrangement leads to an optimal packing efficiency.
The ccp structure is one of the most common arrangements in crystal structures, particularly for metals and ionic compounds. It is also known as "face-centered cubic" due to the layered arrangement that repeats every three atomic layers. While it shares the same packing efficiency as the hexagonal closest packed (hcp) structure, it differs in its stacking sequence.
In a cubic closest packed structure, one important feature is the presence of two types of holes or voids amongst the packed atoms: octahedral holes and tetrahedral holes. These holes are not empty spaces but are regions where smaller atoms or ions can sit, allowing for a lower energy configuration when combined with other atoms. In the context of an ionic compound, these holes are critical to maintaining charge balance and structural stability.
The ccp structure is one of the most common arrangements in crystal structures, particularly for metals and ionic compounds. It is also known as "face-centered cubic" due to the layered arrangement that repeats every three atomic layers. While it shares the same packing efficiency as the hexagonal closest packed (hcp) structure, it differs in its stacking sequence.
In a cubic closest packed structure, one important feature is the presence of two types of holes or voids amongst the packed atoms: octahedral holes and tetrahedral holes. These holes are not empty spaces but are regions where smaller atoms or ions can sit, allowing for a lower energy configuration when combined with other atoms. In the context of an ionic compound, these holes are critical to maintaining charge balance and structural stability.
anion
An anion is a negatively charged ion, meaning it has more electrons than protons. In the context of ionic compounds, anions are typically formed when an atom gains one or more electrons. The type of element that generally forms anions is nonmetals. Common examples include oxygen, sulfur, and the halogens.
Anions are important in determining the structure and stability of ionic compounds. They often form part of the lattice framework in a crystal structure. For example, in a cubic closest packed structure, the anions often form the base of the lattice while the cations fill the interstitial spaces such as the octahedral and tetrahedral holes.
In the chemical compounds like \( ext{Na}_2 ext{O}\), \( ext{CdS}\), and \( ext{ZrI}_4\), the anions (O\(^{2-}\), S\(^{2-}\), and I\(^{-}\) respectively) provide the larger "skeleton" of the lattice structure, and their arrangement is crucial for defining the compound's overall geometry and chemical properties.
Anions are important in determining the structure and stability of ionic compounds. They often form part of the lattice framework in a crystal structure. For example, in a cubic closest packed structure, the anions often form the base of the lattice while the cations fill the interstitial spaces such as the octahedral and tetrahedral holes.
In the chemical compounds like \( ext{Na}_2 ext{O}\), \( ext{CdS}\), and \( ext{ZrI}_4\), the anions (O\(^{2-}\), S\(^{2-}\), and I\(^{-}\) respectively) provide the larger "skeleton" of the lattice structure, and their arrangement is crucial for defining the compound's overall geometry and chemical properties.
cation
Cations are positively charged ions; they have more protons than electrons. Cations are typically formed when a metal loses one or more electrons. Metals such as sodium (Na), zinc (Zn), and calcium (Ca) are common examples that become cations when they lose electrons.
In an ionic compound, cations are smaller than their anion counterparts. They play a crucial role in the overall charge balance of the compound. In crystal structures with cubic closest packed anions, cations often fit into smaller interstitial spaces, such as tetrahedral or octahedral holes. These holes are formed due to the way anions are packed together, leaving small gaps that cations can occupy.
In instances like \( ext{Na}_2 ext{O}\), \( ext{CdS}\), and \( ext{ZrI}_4\), cations (Na\(^{+}\), Cd\(^{2+}\), and Zr\(^{4+}\) respectively) reside in these interstitial spaces to balance charge, enable strong ionic bonding, and maintain the stability of the crystalline structure. The presence and distribution of cations directly influence the material's properties such as density, ionic conductivity, and melting point.
In an ionic compound, cations are smaller than their anion counterparts. They play a crucial role in the overall charge balance of the compound. In crystal structures with cubic closest packed anions, cations often fit into smaller interstitial spaces, such as tetrahedral or octahedral holes. These holes are formed due to the way anions are packed together, leaving small gaps that cations can occupy.
In instances like \( ext{Na}_2 ext{O}\), \( ext{CdS}\), and \( ext{ZrI}_4\), cations (Na\(^{+}\), Cd\(^{2+}\), and Zr\(^{4+}\) respectively) reside in these interstitial spaces to balance charge, enable strong ionic bonding, and maintain the stability of the crystalline structure. The presence and distribution of cations directly influence the material's properties such as density, ionic conductivity, and melting point.
fraction of occupancy
The fraction of occupancy refers to the proportion of available interstitial spaces (like tetrahedral holes) that are occupied by cations in a crystal lattice. This concept is vital in structural chemistry as it helps us understand how tightly the crystal is packed and how ions are distributed within the lattice.
To calculate the fraction of occupancy, it's essential to know the ratio of cations to anions in the compound. The next step involves calculating the number of tetrahedral holes available in a cubic closest packed structure, which is typically twice the number of anions. By dividing the number of cations by the number of available tetrahedral holes, we can determine the fraction of occupied holes.
For instance, in \( ext{Na}_2 ext{O}\), the fraction of occupied tetrahedral holes is 1, which implies full occupancy. In contrast, \( ext{CdS}\) has a fraction of 0.5, suggesting that only half of the available tetrahedral spaces are filled by cations. Finally, in \( ext{ZrI}_4\), with a fraction of 0.125, only a small portion of the holes are occupied, indicating a lower packing density and different structural properties compared to the other compounds.
To calculate the fraction of occupancy, it's essential to know the ratio of cations to anions in the compound. The next step involves calculating the number of tetrahedral holes available in a cubic closest packed structure, which is typically twice the number of anions. By dividing the number of cations by the number of available tetrahedral holes, we can determine the fraction of occupied holes.
For instance, in \( ext{Na}_2 ext{O}\), the fraction of occupied tetrahedral holes is 1, which implies full occupancy. In contrast, \( ext{CdS}\) has a fraction of 0.5, suggesting that only half of the available tetrahedral spaces are filled by cations. Finally, in \( ext{ZrI}_4\), with a fraction of 0.125, only a small portion of the holes are occupied, indicating a lower packing density and different structural properties compared to the other compounds.
