Question

The cubic unit cell of aluminium has an edge length of \(400 \mathrm{pm}\). Its density is \(2.8 \mathrm{~g} \mathrm{~cm}^{-3}\). The number of atoms present per unit cell will be . (atomic mass of \(\mathrm{Al}=27)\)

Short answer

The number of aluminium atoms per unit cell is 4.

Step-by-step solution

Calculate Volume of Unit Cell

The volume of the unit cell is calculated using the edge length provided. The edge length of the cubic unit cell is given as \(400 \text{ pm}\). First, convert the edge length from picometers to centimeters: \(400 \text{ pm} = 400 \times 10^{-12} \text{ m} = 4 \times 10^{-8} \text{ cm}\). Then, the volume \( V \) of the cubic unit cell is \( V = (4 \times 10^{-8} \text{ cm})^3 = 6.4 \times 10^{-23} \text{ cm}^3\).

Determine Mass of Unit Cell

The mass of the unit cell can be determined using the density formula \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). Rearranging gives \( \text{Mass} = \text{Density} \times \text{Volume} \). Using the given density of aluminium \(2.8 \text{ g/cm}^3\) and the volume calculated in Step 1, we have \( \text{Mass} = 2.8 \text{ g/cm}^3 \times 6.4 \times 10^{-23} \text{ cm}^3 = 1.792 \times 10^{-22} \text{ g}\).

Calculate Number of Moles of Atoms in Unit Cell

The number of moles in the unit cell can be calculated using the formula \( n = \frac{\text{Mass}}{\text{Molar Mass}} \). The molar mass of aluminium is 27 g/mol. \( n = \frac{1.792 \times 10^{-22} \text{ g}}{27 \text{ g/mol}} \approx 6.63 \times 10^{-24} \text{ mol}\).

Calculate Number of Atoms in Unit Cell

Use Avogadro's number to find the number of atoms. Avogadro's number \( N_A \) is \( 6.022 \times 10^{23} \text{ atoms/mol} \). Therefore, the number of atoms \( N \) is \( N = 6.63 \times 10^{-24} \text{ mol} \times 6.022 \times 10^{23} \text{ atoms/mol} \approx 4\).

Conclusion: How many atoms per unit cell?

The number of atoms present per unit cell of aluminium is 4. This aligns with the face-centered cubic (FCC) structure characteristic of aluminium.

Key concepts

Density Calculation

Density is a fundamental concept in material science and chemistry. It gives us an idea of how closely packed the atoms or molecules are within a given substance. For a solid material like aluminium, density is calculated using the formula:
\[\text{Density} = \frac{\text{Mass}}{\text{Volume}}\]
In the case of a cubic unit cell, we determine the volume using the length of the cube's edge. After converting the edge length from picometers to centimeters, we calculate the volume by cubing the edge length.
Given the density, we can then rearrange the formula to solve for mass, multiplying it by the volume to find the mass contained within that unit cell.

• The volume of the cell gives an insight into the spatial arrangement of atoms.
• Density helps in relating the structural properties of a crystal to its physical characteristics.

Understanding density is crucial for predicting the behavior and properties of materials.

Avogadro's Number

Avogadro's Number, denoted as \(N_A\), is a fundamental constant in chemistry. It is the bridge between the microscale world of atoms and the macroscale world of grams and liters. The value of Avogadro's Number is \(6.022 \times 10^{23}\) atoms/mol. It tells us how many atoms are in one mole of any substance.
This number is immensely useful because it allows chemists to convert between atomic scale measurements and bulk material quantities.

• Provides a count of entities (atoms, molecules) in a mole.
• Makes it easier to relate macroscopic and microscopic properties of substances.

In our exercise, using Avogadro's Number helps us determine the number of aluminium atoms in a unit cell by multiplying it by the moles calculated earlier. It's the reason we can interpret properties of substances on an atomic level.

Face-Centered Cubic Structure

The face-centered cubic (FCC) structure is one of the most common crystal arrangements. In this structure, atoms are located at each corner and the center of each face of the cube. It's important because it explains many properties:

• Each corner atom is shared among eight cubes, while face-centered atoms are shared with only two.
• This means that in a single FCC unit cell, the net number of atoms is four.

Due to this arrangement, materials with FCC structures tend to have high density. Plus, they typically exhibit excellent ductility, meaning they can deform under tensile stress. This understanding of spatial arrangement aids in predicting the behavior of a material under certain conditions, which is crucial in various applications from engineering to materials science.

Molar Mass Calculation

Molar mass is the mass of one mole of a substance (in grams per mole). It allows conversion between grams and moles, making it a critical element for stoichiometric calculations in chemistry.
For aluminium, the molar mass is 27 g/mol, a figure derived from its atomic mass. This value enables us to calculate how many moles are in the given mass of the aluminium unit cell. By using the equation:
\[ n = \frac{\text{Mass}}{\text{Molar Mass}} \] we can find the number of moles present.

• Understanding the molar mass is fundamental in linking mass of a substance to the amount (in moles).
• It is crucial for calculating the number of atoms or molecules in a given mass of a substance.

With this knowledge, students can relate the physical mass they measure to the number of atoms present, blending macroscopic observations with atomic scale insights. Thus, it's a foundational concept that cements a comprehensive understanding of chemical reactions and material properties.