Unit Cell Chemistry Simple Cubic, Body Centered Cubic, Face Centered Cubic Crystal Lattice Structu
TLDRThe video script provides an educational overview of three fundamental crystal structures: simple cubic, body-centered cubic (BCC), and face-centered cubic (FCC). It explains that each structure represents a unit cell with varying numbers of atoms: one for simple cubic, two for BCC, and four for FCC. The script delves into the coordination numbers, which are six, eight, and twelve respectively, indicating the number of atoms each atom is attached to. It also discusses the volume efficiency of these structures, with the FCC being the most space-efficient at 74%. The relationships between the edge length of the cube (x) and the atomic radius (r) are given as 2r for simple cubic, \( \frac{4}{\sqrt{3}}r \) for BCC, and \( \sqrt{8}r \) for FCC. The script further explains how to calculate the volume ratio of atoms to the entire cube, confirming the percentages of space utilization for each structure: 52% for simple cubic, 68% for BCC, and 74% for FCC. This comprehensive introduction is valuable for understanding the basics of unit cells in crystallography.
Takeaways
- 📐 **Simple Cubic Structure**: Each unit cell contains one atom, with a coordination number of six, meaning each atom is adjacent to six others.
- 🔴 **Body Centered Cubic (BCC) Structure**: Each unit cell has two atoms, with a coordination number of eight, and the atoms occupy 68% of the cube's volume.
- 🔶 **Face Centered Cubic (FCC) Structure**: Each unit cell contains four atoms, with a coordination number of twelve, and is the most space-efficient with 74% volume utilization.
- 🔢 **Volume Calculation**: In a simple cubic structure, atoms use 52% of the cube's volume, while in BCC it's 68%, and in FCC it's 74%.
- ⚖️ **Density and Atomic Radius**: The edge length of the unit cell can be calculated from the atomic radius for each structure, which is useful for determining the density of an element.
- 🔗 **Edge Length Relationship**: For simple cubic (SC), edge length (x) is 2r; for BCC, x is 4√3r; and for FCC, x is √8r, where r is the atomic radius.
- 🧮 **Volume of Atoms in Unit Cell**: The volume of a single atom is calculated as (4/3)πr³, which is then used to determine the volume fraction of atoms within the unit cell.
- 📏 **Unit Cell Calculations**: Understanding the relationship between the edge length and atomic radius is crucial for calculating the volume and density of the material.
- 🔲 **Efficiency of Packing**: FCC is known as cubic closest packing due to its high efficiency in space utilization, making it the most compact arrangement.
- 📊 **Coordination Numbers**: The coordination number represents how many atoms are directly connected to a single atom, which is six for SC, eight for BCC, and twelve for FCC.
- 🧠 **Understanding Unit Cells**: Grasping the concepts of unit cells, their structures, and how they relate to atomic packing is fundamental to materials science and chemistry.
Q & A
What are the three types of unit cells discussed in the video?
-The three types of unit cells discussed are the simple cubic structure, the body-centered cubic structure, and the face-centered cubic structure.
How many atoms are there in a unit cell for each of the three structures?
-In the simple cubic structure, there is one atom per unit cell. In the body-centered cubic structure, there are two atoms per unit cell. In the face-centered cubic structure, there are four atoms per unit cell.
What is the coordination number for the simple cubic structure?
-The coordination number for the simple cubic structure is six, which means each atom is attached to six other atoms.
What is the relationship between the edge length (x) and the atomic radius (r) for the simple cubic structure?
-For the simple cubic structure, the edge length (x) is equal to twice the atomic radius (2r).
Which cubic structure is known as the cubic closest packing and why?
-The face-centered cubic structure is known as the cubic closest packing because it has the highest volume efficiency at 74%, meaning it maximizes the use of the volume of the cube with the least amount of empty space.
How is the volume of atoms within a unit cell calculated for the simple cubic structure?
-The volume of atoms within a unit cell for the simple cubic structure is calculated by taking the volume of a single spherical atom, which is (4/3)πr^3, and dividing it by the volume of the cube, which is (2r)^3.
What is the coordination number for the body-centered cubic structure?
-The coordination number for the body-centered cubic structure is eight, as each atom is attached to eight other atoms.
How is the edge length (x) related to the atomic radius (r) in the body-centered cubic structure?
-In the body-centered cubic structure, the edge length (x) is equal to (4/√3) times the atomic radius (r).
What percentage of the volume of the cube is occupied by atoms in the body-centered cubic structure?
-In the body-centered cubic structure, 68% of the volume of the cube is occupied by atoms.
How many atoms are considered to be at the edge of a cube in the simple cubic structure?
-In the simple cubic structure, there are eight atoms at the edge of the cube, with each atom contributing one-eighth of its volume to the unit cell.
What is the formula for calculating the volume of a cube?
-The volume of a cube is calculated by raising the edge length (x) to the third power, which is expressed as x^3.
How can you find the edge length of a cubic structure if you are given the atomic radius and the structure type?
-You can find the edge length by using the specific relationship between the edge length and the atomic radius for the given structure type. For example, in a simple cubic structure, x = 2r, in a body-centered cubic structure, x = (4/√3)r, and in a face-centered cubic structure, x = √(8r).
Outlines
📐 Introduction to Unit Cells
This paragraph introduces the concept of unit cells in crystallography, focusing on three types: simple cubic, body-centered cubic, and face-centered cubic structures. Each unit cell is represented by a cube, with one, two, and four atoms respectively. The coordination number, which is the number of atoms attached to a single atom, is six for the simple cubic, eight for the body-centered cubic, and twelve for the face-centered cubic structure. The paragraph also discusses the volume efficiency of each structure, with the face-centered cubic being the most space-efficient, known as cubic closest packing.
🔬 Simple Cubic Structure Details
The focus is on the simple cubic structure, explaining that there's one atom per unit cell, which is calculated by considering the fraction of atoms at the edges. The coordination number is six due to the surrounding atoms in the x, y, and z directions. The edge length of the cube (x) is twice the atomic radius (2r). The volume of the unit cell is x cubed, and the volume occupied by atoms is 52% of the total cube volume, calculated by dividing the volume of the atom (4/3πr cubed) by the cube volume (x cubed, replaced with (2r) cubed).
🔨 Body-Centered Cubic Structure Details
The body-centered cubic structure is explored, noting that there are two atoms per unit cell, with each corner atom contributing an eighth of an atom and the central atom being whole. The coordination number is eight, as the central atom is connected to eight others. The edge length of the cube (x) is derived to be 4 times the atomic radius (r) divided by the square root of 3, using geometric relationships within the structure. The volume ratio of the atoms to the cube is confirmed to be 68%, leaving 32% as empty space.
📊 Volume Calculation for Body-Centered Cubic
This section provides a mathematical derivation for the volume occupied by atoms in the body-centered cubic structure. The volume of two atoms within a unit cell is calculated using the volume of a sphere formula (4/3πr cubed) and then divided by the cube volume (x cubed, with x replaced by 4r/√3). After simplification, the ratio is shown to be 0.68, confirming that 68% of the cube's volume is occupied by atoms, with the remaining 32% being empty space.
Mindmap
Keywords
💡Unit Cell
💡Simple Cubic Structure
💡Body Centered Cubic (BCC) Structure
💡Face Centered Cubic (FCC) Structure
💡Coordination Number
💡Volume Efficiency
💡Edge Length (x)
💡Atomic Radius (r)
💡Density
💡Cubic Closest Packing (CCP)
💡Volume of a Cube
Highlights
Introduction to unit cells with three key structures: simple cubic, body-centered cubic, and face-centered cubic.
Simple cubic structure contains one atom per unit cell.
Body-centered cubic structure has two atoms per unit cell.
Face-centered cubic structure contains four atoms per unit cell.
Coordination numbers are six for simple cubic, eight for body-centered cubic, and twelve for face-centered cubic structures.
Volume efficiency varies: 52% for simple cubic, 68% for body-centered cubic, and 74% for face-centered cubic structures.
Face-centered cubic structure is known as cubic closest packing due to its efficient use of space.
Edge length (x) in simple cubic structure is equal to 2 times the atomic radius (r).
Edge length in body-centered cubic structure is 4 times the square root of 3 times the atomic radius.
Edge length in face-centered cubic structure is equal to the square root of 8 times the atomic radius.
Calculation of the volume of atoms within a unit cell relative to the cube volume for simple cubic structure.
In simple cubic structure, each atom is surrounded by six other atoms, resulting in a coordination number of six.
Volume of a cube is calculated as the cube of its edge length (x^3), where x is 2r in a simple cubic structure.
Derivation of the volume ratio for body-centered cubic structure, confirming 68% volume efficiency.
Explanation of how to relate the edge length to the atomic radius in body-centered cubic structure.
Each corner atom in body-centered cubic structure is considered as one-eighth of an atom, totaling two atoms per unit cell.
The body-centered cubic structure has a coordination number of eight, with each central atom connected to eight corner atoms.
Transcripts
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