Chapter 10: Problem 23
The fraction of volume occupied by atoms in a primitive cubic unit cell is nearly: (a) \(0.48\) (b) \(0.52\) (c) \(0.55\) (d) \(0.68\)
Short Answer
Expert verified
The fraction of volume occupied by atoms in a primitive cubic unit cell is nearly 0.52.
Step by step solution
01
Understand the Geometry of the Primitive Cubic Unit Cell
A primitive cubic unit cell has one atom at each of its eight corners. Each corner atom is shared among eight adjacent unit cells, so only 1/8th of each corner atom actually 'belongs' to one unit cell. Since there are eight corners, the unit cell contains a total of one atom's volume.
02
Calculate the Volume of One Atom
Assuming the atom is a sphere, the volume of one atom is given by the formula for the volume of a sphere: \( V_{atom} = \frac{4}{3}\pi r^3 \). Here, \( r \) is the radius of the atom.
03
Relate the Atom's Radius to the Cube's Edge Length
In a primitive cubic unit cell, the edge length of the cube, \( a \), is equal to twice the atom's radius because an atom is situated in each corner touching along the edge. Thus, \( a = 2r \).
04
Calculate the Volume of the Unit Cell
The volume of the cubic unit cell is the cube's edge length cubed, given by the formula \( V_{cell} = a^3 \). Substituting the relation from Step 3, we get \( V_{cell} = (2r)^3 = 8r^3 \).
05
Find the Fraction of Volume Occupied by Atoms
Now, divide the volume of one atom by the volume of the unit cell to obtain the fraction of the volume occupied by atoms: \( \frac{V_{atom}}{V_{cell}} = \frac{\frac{4}{3}\pi r^3}{8r^3} = \frac{\pi}{6} \).
06
Calculate the Numerical Value of the Fraction
Calculating the numerical value of \( \frac{\pi}{6} \) yields a value near to 0.52.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Crystal Lattice Structure
The crystal lattice structure of a material is the framework around which the entire solid is built. It is a repetitive three-dimensional arrangement of atoms, ions, or molecules and serves as the defining characteristic for different types of crystalline solids. One fundamental pattern is that of the primitive cubic unit cell.
In a primitive cubic structure, the unit cell, which is the smallest repeating unit of the lattice, is shaped like a cube. The structure can be visualized as spheres representing atoms, positioned at each corner of the cube. Because each of these corner atoms is shared with adjacent unit cells, only a fraction (1/8th) of each corner atom's volume is actually within the given unit cell.
This sharing means that despite there being eight corners, the unit cell contains the volume of just one entire atom. Understanding this geometric arrangement is key to determining the efficient use of space within the unit cell, which can affect the material's properties such as density and packing.
In a primitive cubic structure, the unit cell, which is the smallest repeating unit of the lattice, is shaped like a cube. The structure can be visualized as spheres representing atoms, positioned at each corner of the cube. Because each of these corner atoms is shared with adjacent unit cells, only a fraction (1/8th) of each corner atom's volume is actually within the given unit cell.
This sharing means that despite there being eight corners, the unit cell contains the volume of just one entire atom. Understanding this geometric arrangement is key to determining the efficient use of space within the unit cell, which can affect the material's properties such as density and packing.
Volume of Atoms in Unit Cell
To connect the microscale structure of a crystal to its macroscopic physical properties, one must understand the volume of atoms in a unit cell. In the case of a primitive cubic lattice, since one whole atom's volume is contained within the boundaries of a unit cell, this calculation is vital for quantifying the space utilization within the cell.
Given that atoms are typically approximated as spheres, the formula for the volume of one atom is \( V_{atom} = \frac{4}{3}\pi r^3 \), with \( r \) being the atomic radius. The edge length of the cube (\( a \)) is directly related to the radius, \( a = 2r \), since two radii span the length of the edge, touching at just one point along this length.
The volume of the unit cell can then be calculated using the edge length and is represented as \( V_{cell} = a^3 \). Therefore, by substituting in the relationship \( a = 2r \), we find that \( V_{cell} = 8r^3 \), or eight times the volume of the atom. Using these relationships, we can determine how much of the cell is 'filled' by the atomic volume, which is of significant interest for understanding the material's density and structural efficiency.
Given that atoms are typically approximated as spheres, the formula for the volume of one atom is \( V_{atom} = \frac{4}{3}\pi r^3 \), with \( r \) being the atomic radius. The edge length of the cube (\( a \)) is directly related to the radius, \( a = 2r \), since two radii span the length of the edge, touching at just one point along this length.
The volume of the unit cell can then be calculated using the edge length and is represented as \( V_{cell} = a^3 \). Therefore, by substituting in the relationship \( a = 2r \), we find that \( V_{cell} = 8r^3 \), or eight times the volume of the atom. Using these relationships, we can determine how much of the cell is 'filled' by the atomic volume, which is of significant interest for understanding the material's density and structural efficiency.
Solid State Chemistry
Solid state chemistry is an area of study focusing on the synthesis, structure, and properties of solid phase materials. Much of solid state chemistry deals with crystallography, which explores the arrangement of atoms within a crystal lattice and how these arrangements impact the material's overall characteristics. From conducting electrical currents to mechanical strength, properties of solids are often predicated on their lattice structure.
In the context of our primitive cubic unit cell example, solid state chemistry helps in understanding not just the geometric arrangement of the atoms but also its implications. Knowing the fraction of the volume occupied by atoms within a unit cell, calculated as \( \frac{\pi}{6} \) or approximately 0.52, offers insight into the atomic packing efficiency of the structure. This numeric value reflects how tightly packed the atoms are within the material and impacts properties like thermal expansion, conductivity, and britellness.
In applications where materials are subjected to extreme conditions, knowing the density and how atoms interact within their lattice can inform engineers how the material will behave, making the study of solid state chemistry crucial for material science and various industrial applications.
In the context of our primitive cubic unit cell example, solid state chemistry helps in understanding not just the geometric arrangement of the atoms but also its implications. Knowing the fraction of the volume occupied by atoms within a unit cell, calculated as \( \frac{\pi}{6} \) or approximately 0.52, offers insight into the atomic packing efficiency of the structure. This numeric value reflects how tightly packed the atoms are within the material and impacts properties like thermal expansion, conductivity, and britellness.
In applications where materials are subjected to extreme conditions, knowing the density and how atoms interact within their lattice can inform engineers how the material will behave, making the study of solid state chemistry crucial for material science and various industrial applications.
