Chapter 9: Problem 51
Which void is smallest in close packing? (a) tetrahedral (b) octahedral (c) cubic (d) square
Short Answer
Expert verified
The smallest void in close packing is the tetrahedral void.
Step by step solution
01
Understanding Close Packing
Close packing in crystals refers to the way in which atoms pack together to occupy the least amount of space. In close packing structures, there are voids or interstices where additional atoms can fit. Tetrahedral and octahedral voids are commonly found in these arrangements.
02
Comparing the Voids in Close Packing
Tetrahedral and octahedral voids are the most common types of voids in close packing. A tetrahedral void is formed by four atoms located at the corners of a tetrahedron, while an octahedral void is formed by six atoms located at the corners of an octahedron. Cubic and square voids are not typically considered in close packing as they are not as efficient in space filling.
03
Identifying the Smallest Void
In close packed structures, tetrahedral voids are smaller than octahedral voids. Each tetrahedral void is surrounded by 4 atoms whereas each octahedral void is surrounded by 6 atoms. Since cubic and square voids are not commonly found in these packings, they can be disregarded for this comparison.
04
Conclusion
Based on the arrangement of atoms in close packing and the number of atoms surrounding each type of void, the tetrahedral void is the smallest.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Tetrahedral Void
The concept of a tetrahedral void is vital for understanding close packing in crystals. A tetrahedral void is the space that is encased by four atoms in a tetrahedral shape. Imagine a pyramid with a triangular base - the corners of this pyramid are where the atoms are situated.
These voids are small pockets within a crystal structure that can host a single atom. When atoms settle into a closely packed arrangement, such as face-centered cubic (FCC) or hexagonal close-packed (HCP), these tetrahedral voids form naturally. The size of a tetrahedral void relates to the size of the atoms surrounding it, and it is determined by drawing lines between the centers of the surrounding atoms.
Mathematically, if we consider the radius of atoms forming the void to be 'r', then the radius of the atom that can fit into the tetrahedral void without disturbance is approximately 0.225r. This size relation allows us to understand why in close packing structures, such as those found in metallic crystals, tetrahedral voids are typically the smallest voids present.
These voids are small pockets within a crystal structure that can host a single atom. When atoms settle into a closely packed arrangement, such as face-centered cubic (FCC) or hexagonal close-packed (HCP), these tetrahedral voids form naturally. The size of a tetrahedral void relates to the size of the atoms surrounding it, and it is determined by drawing lines between the centers of the surrounding atoms.
Mathematically, if we consider the radius of atoms forming the void to be 'r', then the radius of the atom that can fit into the tetrahedral void without disturbance is approximately 0.225r. This size relation allows us to understand why in close packing structures, such as those found in metallic crystals, tetrahedral voids are typically the smallest voids present.
Octahedral Void
On the other side of the crystal packing spectrum lies the octahedral void. This void is created when six atoms are positioned at the corners of an octahedron - imagine two pyramids base to base.
These voids are larger than tetrahedral voids and also occur naturally in close packed structures. In both FCC and HCP lattice arrangements, for every particle, there are an equal number of octahedral voids, reflecting how octahedral voids also play an integral role in the close-packing process.
When calculating the size of an atom that can fit in an octahedral void, we find that it can be larger than the one in a tetrahedral void. Specifically, an atom accommodated in an octahedral void can have a radius up to approximately 0.414r, where 'r' is the radius of the atoms forming the void. Because of their size and configuration, octahedral voids are crucial for the incorporation of different atom sizes in alloy formations, finding utility in materials science and engineering.
These voids are larger than tetrahedral voids and also occur naturally in close packed structures. In both FCC and HCP lattice arrangements, for every particle, there are an equal number of octahedral voids, reflecting how octahedral voids also play an integral role in the close-packing process.
When calculating the size of an atom that can fit in an octahedral void, we find that it can be larger than the one in a tetrahedral void. Specifically, an atom accommodated in an octahedral void can have a radius up to approximately 0.414r, where 'r' is the radius of the atoms forming the void. Because of their size and configuration, octahedral voids are crucial for the incorporation of different atom sizes in alloy formations, finding utility in materials science and engineering.
Crystal Structure
Crystal structure is the ordered arrangement of atoms in a three-dimensional pattern. This ordering is defined on the scale of a crystal's unit cell, which is the smallest repeating unit of the crystal and acts as the building block for the extended structure.
In the discussion of crystal packing, most common structures are face-centered cubic (FCC), body-centered cubic (BCC), and hexagonal close-packed (HCP). Each type of crystal structure has a distinctive arrangement of atoms and consequently various types of voids will be formed. For example, the FCC structure, with atoms at each corner and the centers of all the faces of the cube, leads to a high density of tetrahedral and octahedral voids.
The arrangement of atoms in crystal structures not only determines the kind of voids that are created but also the materials' properties. This includes mechanical strength, conductivity, and density, which are crucial for different applications across industries, including electronics, construction, and aerospace.
In the discussion of crystal packing, most common structures are face-centered cubic (FCC), body-centered cubic (BCC), and hexagonal close-packed (HCP). Each type of crystal structure has a distinctive arrangement of atoms and consequently various types of voids will be formed. For example, the FCC structure, with atoms at each corner and the centers of all the faces of the cube, leads to a high density of tetrahedral and octahedral voids.
The arrangement of atoms in crystal structures not only determines the kind of voids that are created but also the materials' properties. This includes mechanical strength, conductivity, and density, which are crucial for different applications across industries, including electronics, construction, and aerospace.
Space Filling in Crystals
Space filling in crystals refers to how atoms pack together to fill space most efficiently. Considering the nature of atoms as spherical particles, the objective in crystal packing is to minimize unused space, which leads to the creation of voids like tetrahedral and octahedral voids.
In a closely packed structure, the objective is to attain the highest possible packing factor - the fraction of space occupied by atoms. The FCC and HCP structures have the highest packing factors of any lattice arrangements, 0.74, signifying that 74% of the space within the structure is filled with atoms, and the rest is void space.
The complexity of space filling also arises from the need to accommodate different sizes of atoms within the same crystal lattice, which can lead to distortions and the formation of defects. Nonetheless, understanding space filling is key to developing new materials with desired properties and to advance fields like nanotechnology and crystallography.
In a closely packed structure, the objective is to attain the highest possible packing factor - the fraction of space occupied by atoms. The FCC and HCP structures have the highest packing factors of any lattice arrangements, 0.74, signifying that 74% of the space within the structure is filled with atoms, and the rest is void space.
The complexity of space filling also arises from the need to accommodate different sizes of atoms within the same crystal lattice, which can lead to distortions and the formation of defects. Nonetheless, understanding space filling is key to developing new materials with desired properties and to advance fields like nanotechnology and crystallography.
