What is a Unit Cell?
Do you want to learn how to calculate the total density of a unit cell calculator? If yes, then you are in the right place. Here, we will learn to calculate the density of an fcc unit cell. Before you calculate the density of an fcc unit cell, you must know what a unit cell is. A unit cell can be described as the smallest group of atoms that have the overall symmetry of a crystal. It is a structure that can be used to build an entire lattice by repetition in three dimensions.
It is important to remember that crystalline solids also exhibit a regular and repeating pattern of constituent particles. Before you calculate the density of a unit cell, you must also be able to visualize its complete structure of it. The image of a unit cell is given below for your reference.
(Image to be added soon)
Benefits of Visualization
This visualization will go a long way in helping students calculate the density of a unit cell. Students can also use this technique to calculate the density of body-centered cubic unit cells. On the same note, students should also keep in mind that the diagrammatic representation of three-dimensional arrangements of constituent particles of a crystal is known as a crystal lattice. In a crystal lattice, each particle is depicted as a particular point in space.
What is Lattice?
Before moving on to the next topic of how to calculate the density of a body-centered cubic unit cell let us understand what a lattice is. According to experts, lattice can be described as a framework that resembles a three-dimensional periodic array of points. It is this framework on which an entire crystal is built.
Bravais Lattices
Every student who wishes to know the answer to the question of how to calculate the density of empty space within a unit cell has to also learn the theory of this topic. It all started in 1850 when M. A. Bravais showed the world that identical points can also be arranged spatially. This arrangement would produce 14 different types of regular patterns. Students might also be interested to know that these 14 space lattices are known as Bravais lattices.
Calculation of Density of Unit Cell
To further continue the discussion on how to calculate the density of empty space within a unit cell, it is also vital for students to note that a crystal lattice of a solid could also be described in terms of its unit cells.
As mentioned above, a crystal lattice consists of a very large number of unit cells, and every lattice point is occupied by one constituent particle. Hence, the density of unit cell fcc can be viewed as a three-dimensional structure that contains one or more atoms. All of this information can be used to determine the density of unit cell fcc. To understand all of these how to find the density of a one-unit cell clearly.
Example
Let’s take an example. Let’s assume that an individual has a unit cell that has an edge ‘a.’ The volume of that unit cell is ‘v,’ and the density of the unit cell is given as the ratio of mass and volume of the unit cell.
Further, the mass of the unit cell is equal to the product of the number of total atoms in the unit cell and the mass of every atom in the unit cell. Now, one has to learn how to find the density of a one-unit cell to arrive at the final answer. Let us begin the answer by noting down everything that we already know.
Mass of unit cell = number of total atoms in unit cell x mass of every atom = z x m
In this equation for finding out the density of unit cell material science, ‘z’ is the total number of atoms in a unit cell, and ‘m’ is the mass of every atom. Students must also be familiar with the fact that the mass of an atom can be calculated with the help of Avogadro number and molar mass, that is, M / Na
Here, ‘M’ is the molar mass, and ‘Na’ is the Avogadro’s number. Further, the volume of the unit cell or ‘V’ = A3
Hence, the density of the unit cell = Mass of the unit cell / Volume of the unit cell
This further means that density of the unit cell is equal to m/V = z x m / A3 = z x M / A3 x Na
It can be concluded that if one knows the total number of atoms in a unit cell, edge length, and molar mass, then one can calculate the density of the unit cell.
Significance of the Given Technique
This same technique can be used to calculate the density of NaCl unit cells and the density of polonium unit cells. We also learned that the density of a unit cell equals extended lattice, and it is easy to determine the density given the length of the unit cell.
General Expression for Density of Unit Cells Derived for Various Cases
Till now, we have learned how to calculate the density of a NaCl unit cell, the density of a polonium unit cell, the fact that the density of a unit cell equals an extended lattice, and it is relatively easy to determine the density given the length of a unit cell.
What is the Planar Density of the (101) Plane in a Face-Centered Cubic (fcc) Unit Cell
Now, let us go through the answer to the question of what is the planar density of the (101) plane in a face-centered cubic (fcc) unit cell. Students should also be familiar with the general expressions used for the density of unit cells. These expressions are derived for various cases. And to help students to understand this topic better, we have formulated a list of important expressions. The list is given below.
Primitive Unit Cell
In a primitive unit cell, the total number of atoms in a unit cell is equal to one. This means that the density of a primitive unit cell is equal to 1 x M / A3 x Na.
Body-Centered Cubic Unit Cell
When it comes to a body-centered cubic unit cell, then the total number of atoms in a unit cell is two. This means that the value of z is two. Hence, the density of a body-centered cubic unit cell is equal to 2 x M / A3 x Na.
Face-Centered Cubic Unit Cell
The total number of atoms present in a face-centered cubic unit cell is four. Hence, the density of a face-centered cubic unit cell is 4 x M / A3 x Na.
Fun Facts about Unit Cells
Did you know that primitive cells only have lattice points at the corners of their cell? In contrast to primitive cells, unit cells, instead, have lattice points at their edges, face centers, and body centers! This means that an individual can examine the entire crystal by simply investigating a single unit cell.
Explanation of Relation between Lattice Constant and Density
Consider 'a' as the lattice constant of the cubic crystal.
We know that the density of the crystal is represented by 'P'.
V = a³ represents the volume of the unit cell (cubic crystal).
n = Total number of atoms / unit cell
M represents the material's atomic weight.
N is the Avogadro constant in this equation.
Derivation of Formula for Density of Unit Cell
Now, the mass of 1 atom is equal to M/N.
For n number of molecules, the mass of the unit cell will be equal to nM/N. ….(i)
Also, density = P = mass per unit volume. So, P = m/V.
Now, P = m/a³ ….. (ii)
and here, m= mass
So, from (i) and (ii), P = nM / N a³.
Relation between Lattice Constant and Density
Density = Number of atoms per unit cell Atomic weight / Avogadro number x (Lattice constant)^3.
Diamond Cubic Structure
In a diamond cubic structure, eight corner atoms are present along with six face-centered atoms and four more atoms.
The number of atoms contributed to the unit cell (by the corner atom) is ⅛ × 8 = 1.
Similarly, The number of atoms contributed to the unit cell (by the face-centered atom) is ½ × 6 = 3.
Here, a diamond cubic structure is a repeating pattern consisting of 8 atoms. A diamond cubic structure is considered to be FCC.
The materials that are only made out of carbon are an example of their type of cubic structure. It can be better understood by the concepts of Crystallography.
FAQs on The Density of a Unit Cell
1. What is meant by the density of a unit cell in solid-state chemistry?
The density of a unit cell is defined as the ratio of the mass of the atoms within the unit cell to the total volume of that unit cell. Since the unit cell is the smallest repeating structural unit of a crystalline solid, its density is representative of the density of the entire crystal. It is a crucial parameter that helps in understanding the packing efficiency and nature of a solid.
2. What is the general formula used to calculate the density of a unit cell?
The density (ρ) of a unit cell can be calculated using the formula: ρ = (Z × M) / (a³ × Nₐ). To use this formula, you need to know the following values:
- Z: The total number of atoms per unit cell.
- M: The molar mass (or atomic mass) of the substance in g/mol.
- a: The edge length of the unit cell, usually in cm.
- Nₐ: Avogadro's constant, which is approximately 6.022 × 10²³ mol⁻¹.
3. How do you calculate the density for a body-centred cubic (BCC) unit cell?
To calculate the density of a body-centred cubic (BCC) unit cell, you follow these steps: First, identify the number of atoms (Z) in a BCC structure, which is 2 atoms per unit cell. Then, substitute this value into the general density formula. The calculation would be: ρ = (2 × M) / (a³ × Nₐ), where 'M' is the molar mass and 'a' is the edge length of the cubic cell.
4. What are the key variables in the unit cell density formula, and what does each represent?
The density formula ρ = (Z × M) / (a³ × Nₐ) contains four key variables:
- Z (Number of atoms per unit cell): This value depends on the crystal structure (e.g., Z=1 for simple cubic, Z=2 for BCC, Z=4 for FCC).
- M (Molar Mass): This is the mass of one mole of the substance, found on the periodic table.
- a (Edge Length): This is the length of one side of the unit cell. Its cube (a³) gives the volume of the unit cell.
- Nₐ (Avogadro's Constant): This constant connects the atomic scale (molar mass) to the macroscopic scale.
5. Why is the density of a single unit cell considered equal to the density of the entire macroscopic crystal?
A crystalline solid is composed of a vast number of identical unit cells repeating in a three-dimensional pattern, forming the crystal lattice. The unit cell is the fundamental building block that contains all the structural and chemical information of the crystal. Because this arrangement is uniform and repetitive, the mass-to-volume ratio within one unit cell is the same as the mass-to-volume ratio for the entire crystal. Therefore, calculating the density of one unit cell is sufficient to determine the density of the whole substance.
6. How does the calculation for the density of a face-centred cubic (FCC) unit cell differ from that of a simple cubic (SC) unit cell?
The primary difference in the density calculation between an FCC and an SC unit cell lies in the value of Z (the number of atoms per unit cell). An FCC unit cell contains 4 atoms (8 corners × 1/8 + 6 faces × 1/2), while a simple cubic unit cell contains only 1 atom (8 corners × 1/8). Consequently, for the same element and similar edge length, the density of an FCC structure would be four times greater than that of a simple cubic structure, reflecting its more efficient packing of atoms.
7. If you know the density, crystal structure, and edge length of an unknown element, how can you identify it?
If the density (ρ), crystal structure (which gives you Z), and edge length (a) are known, you can identify the element by calculating its molar mass (M). By rearranging the density formula, you get: M = (ρ × a³ × Nₐ) / Z. After calculating 'M', you can compare this value to the molar masses on the periodic table to determine the identity of the unknown element. This is a practical application of the unit cell density concept.
8. How do the number of atoms (Z) and the edge length (a) inversely affect the density of a crystal?
The variables Z and 'a' have opposing effects on density. The density is directly proportional to Z; a higher number of atoms packed into the same volume results in a higher density. In contrast, density is inversely proportional to the volume (a³). As the edge length 'a' increases, the volume of the unit cell increases cubically, causing the atoms to be spread over a larger space. This leads to a significant decrease in density, assuming Z and M remain constant.
