Atom X occupies the fcc lattice sites as well as alternate tetrahedral voids of the same lattice. The packing efficiency (in %) of the resultant solid is closest to
- 25
- 35
- 55
- 75
The Correct Option is B
Approach Solution - 1
Approach Solution -2
Let's, Calculate the number of atoms per unit cell in the FCC lattice:
In an FCC unit cell, there are 8 corner atoms, each shared by 8 unit cells and 6 face-centered atoms, each shared by 2 unit cells.
Total number of atoms in FCC unit cell: \(8\times\frac{1}{8}+6\times\frac{1}{2}\) = 1+3 = 4
Calculate the number of tetrahedral voids per unit cell in the FCC lattice:
In an FCC unit cell, there are 8 tetrahedral voids. If only alternate tetrahedral voids are occupied, then 4 tetrahedral voids are occupied per unit cell.
Total number of occupied tetrahedral voids = 4
Calculate the total number of atoms in the modified structure:
Total atoms in FCC lattice = 4 and Total atoms in tetrahedral voids = 4
Total number of atoms per unit cell=4+4=8
The relationship between the edge length a and the atomic radius r, since tetrahedral voids forms at \(\frac{1}{4}th\) of body diagonal
\(\frac{a\sqrt{3}}{4}=2r\)
\(a=\frac{8r}{\sqrt{3}}\)
Packing efficiency (PE) is the ratio of the volume occupied by atoms to the volume of the unit cell, multiplied by 100%:
Packing efficiency = \(\frac{8\times\frac{4}{3}\pi r^3}{a^3}\times100\)
\(\frac{8\times\frac{4}{3}\pi r^3}{(\frac{8r}{\sqrt{3}})^3}\times100\simeq35\%\)
So, The correct option is (B): 35
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Concepts Used:
Solid State
Solids are substances that are featured by a definite shape, volume, and high density. In the solid-state, the composed particles are arranged in several manners. Solid-state, in simple terms, means "no moving parts." Thus solid-state electronic devices are the ones inclusive of solid components that don’t change their position. Solid is a state of matter where the composed particles are arranged close to each other. The composed particles can be either atoms, molecules, or ions.
Types of Solids:
Based on the nature of the order that is present in the arrangement of their constituent particles solids can be divided into two types;
- Amorphous solids behave the same as super cool liquids due to the arrangement of constituent particles in short-range order. They are isotropic and have a broad melting point (range is about greater than 5°C).
- Crystalline solids have a fixed shape and the constituent particles are arranged in a long-range order.