Question

The fraction of total volume occupied by the atoms present in a simple cube is

• A. $\frac{\pi}{6}$
• B. $\frac{\pi}{3\sqrt{2}}$
• C. $\frac{\pi}{4\sqrt{2}}$
• D. $\frac{\pi}{4}$

Answer 1

The Correct Option is A

The correct answer is A) \(\frac{\pi}{6}\)

For simple cube, \(\, \, \, \, \, \, \, \, \, Radius(r)=\frac{a}{2}\, \, \, \, \, \, \, \, \, \, \, \, \,\)[a = edge length) 

The volume of the atom =\(\frac{4}{3}\pi \Bigg(\frac{a}{2}\Bigg)^3\) Packing fraction\(=\frac{Z\times\frac{1}{4}\pi r^3}{a^3}\) 

The volume of the sphere (atoms) in a unit cell. For simple cubic, Z = 1 atom Packing fraction \(=\frac{\frac{4}{3}\pi\Bigg(\frac{a}{2}\Bigg)^3}{a^3}=\frac{\pi}{6}\)

Read more from chapter: cube 

Answer 2

The correct answer is A) \(\frac{\pi}{6}\)

Real Life Applications

• A real-life example of a packing fraction of a cube is ice and metals. Ice is made of molecules of water that are crystallized and thus come close together increasing the packing fraction.
• A diamond is a solid that is crystallized in a simple cubic structure. But even though the density of diamonds is high they are not that tightly packed. Metals are dense because of their packing fraction. They have an FCC and BCC crystal structure. But there are some metals with simple cubic packing that are Silicon, copper, and tungsten.
• Packing fractions can be used during material science to make a new material. It can be used in engineering, to design a material with a certain property and also to calculate the electrical conductivity

Question can also be asked as

1. What is the packing fraction of the cubic lattice?
2. What percentage of unit cell volume is occupied by the atoms?
3. What is the fraction of unit cell volume that is occupied by the atoms?

Answer 3

The correct answer is A) \(\frac{\pi}{6}\)

Cube is a solid object bounded by 6 square faces. Of them, 3 meets at each vertex, 12 edges, and 8 vertices. It is a three-dimensional figure with ice cubes, dice are some examples of the same. 

Properties of cube

• It is a square prism and all the sides of a cube are square.
• The angle between the two faces is 90°
• Opposite faces and the edges are parallel
• There are three sides and three edges to which each vertex is connected.

 Volume of cube 

The space occupied by the cube is its volume. The formulas are as follows: 

Based on the side length: a*a*a 

Based on diagonal : (√3×d3)/9

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Related Concepts
Cube	Three-dimensional figures	Surface area and volume
Cube and cube root	Perimeter	Geometry formula