Chapter 11: Problem 51
A face-centered cubic cell contains \(8 \mathrm{X}\) atoms at the corners of the cell and \(6 \mathrm{Y}\) atoms at the faces. What is the empirical formula of the solid?
Short Answer
Expert verified
The empirical formula is \( \text{XY}_3 \).
Step by step solution
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Face-Centered Cubic
A face-centered cubic (FCC) structure is a common way that atoms arrange themselves in a crystal lattice. Imagine a cube where atoms sit at each corner, as well as in the middle of each face. This arrangement is both efficient and symmetrical, which is why it's found in many metals. The unique thing about FCC is how atoms are shared between neighboring unit cells, affecting how we count the atoms in a given structure.
For example, in this structure, each corner atom is shared among eight unit cells. Thus, any individual unit cell gets only one-eighth share of each corner atom. Additionally, the atoms on the faces of the cube are shared between two adjacent cells. This sharing must be accounted for when calculating the total number of atoms that belong to a single unit cell.
For example, in this structure, each corner atom is shared among eight unit cells. Thus, any individual unit cell gets only one-eighth share of each corner atom. Additionally, the atoms on the faces of the cube are shared between two adjacent cells. This sharing must be accounted for when calculating the total number of atoms that belong to a single unit cell.
Empirical Formula
The empirical formula is a simple way to express the ratio of different types of atoms in a chemical compound. It shows the smallest whole-number ratio of atoms of each element in the compound. This is important in understanding the chemical identity of the compound.
When working with crystal structures like the FCC, determining the empirical formula involves accounting for how atoms are shared. Using the example of the FCC unit cell where we had 8 corner atoms and 6 face atoms, calculations showed that the overall number of atoms in a single cell is all about these shared contributions. By counting carefully, the empirical formula gets reduced to \( \text{XY}_3 \), indicating one \( \text{X} \) atom for every three \( \text{Y} \) atoms.
When working with crystal structures like the FCC, determining the empirical formula involves accounting for how atoms are shared. Using the example of the FCC unit cell where we had 8 corner atoms and 6 face atoms, calculations showed that the overall number of atoms in a single cell is all about these shared contributions. By counting carefully, the empirical formula gets reduced to \( \text{XY}_3 \), indicating one \( \text{X} \) atom for every three \( \text{Y} \) atoms.
Unit Cell
A unit cell is the smallest repeating unit in a crystalline solid that, when stacked together in three dimensions, recreates the entire structure. It's like the basic building block of a crystal lattice. Visualize it like a three-dimensional box, where atoms or molecules form a particular geometric pattern which repeats throughout the material.
In a face-centered cubic unit cell, you find atoms at each of the corners and at the center of each of the faces. Understanding the concept of a unit cell helps in visualizing how materials are structured at an atomic level, and aids in calculating the empirical formula of crystals. This unit cell serves as a microcosm of the whole crystal, representing both the symmetry and the spatial efficiency of the arrangement.
In a face-centered cubic unit cell, you find atoms at each of the corners and at the center of each of the faces. Understanding the concept of a unit cell helps in visualizing how materials are structured at an atomic level, and aids in calculating the empirical formula of crystals. This unit cell serves as a microcosm of the whole crystal, representing both the symmetry and the spatial efficiency of the arrangement.
Atoms in a Crystal
Atoms in a crystal are arranged in specific patterns that maximize space efficiency and minimize energy. This structured pattern is what makes crystals strong and durable. The arrangement of atoms within the crystal, such as the face-centered cubic arrangement, dictates the properties of the material, including its density, orientation, and even color.
In many crystal structures, like those found in metals, atoms are closely packed to maximize interactions among them. This arrangement is vital for understanding both the chemical and physical properties of the material. The careful arrangement allows us to predict how the material will behave under different conditions, using information from the basic atomic structure to intricate physical properties.
In many crystal structures, like those found in metals, atoms are closely packed to maximize interactions among them. This arrangement is vital for understanding both the chemical and physical properties of the material. The careful arrangement allows us to predict how the material will behave under different conditions, using information from the basic atomic structure to intricate physical properties.
