Find the packing efficiency of silver metal?
Find the packing efficiency of silver metal?
Solution and Explanation
The packing efficiency of a metallic crystal is a measure of how efficiently the constituent atoms are packed together in the crystal lattice. It is defined as the ratio of the volume occupied by the atoms in the unit cell to the total volume of the unit cell.
For face-centered cubic (FCC) metals such as silver (Ag), the packing efficiency is given by:
packing efficiency = (number of atoms per unit cell) x (volume of one atom) / (volume of the unit cell)
In an FCC lattice, each corner atom is shared between eight unit cells, while each face-centered atom is shared between two unit cells. Therefore, the number of atoms per unit cell in an FCC lattice is 4.
The volume of one atom of silver can be calculated using the atomic radius, which is known to be 144 picometers (pm) for silver. The volume of one atom can be calculated using the formula for the volume of a sphere:
volume of one atom = (4/3) x π x (atomic radius)3
Substituting the value of the atomic radius of silver, we get:
volume of one atom = (4/3) x π x (144 pm)3 = 2.52 x 10-23 cm3
The volume of the unit cell in an FCC lattice can be calculated using the formula:
volume of the unit cell = (a3) / 4,
where 'a' is the length of one side of the unit cell.
The length of one side of the unit cell can be calculated using the relationship between the atomic radius and the length of the diagonal of the unit cell:
a = (2 x atomic radius) / √2
Substituting the value of the atomic radius of silver, we get:
a = (2 x 144 pm) / √2 = 408.3 pm
Converting this value to centimeters, we get:
a = 4.083 x 10-8 cm
Substituting this value in the formula for the volume of the unit cell, we get:
volume of the unit cell = (4.083 x 10-8 cm)3 / 4 = 6.21 x 10-24 cm3
Now, substituting the values for the number of atoms per unit cell, the volume of one atom, and the volume of the unit cell, we get:
packing efficiency = (4 x 2.52 x 10-23 cm3) / (6.21 x 10-24 cm3) = 0.74
Therefore, the packing efficiency of silver metal is approximately 0.74 or 74%.
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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.