Problem 100 - Chemistry The Science in Context

Question

Aluminum forms alloys with lithium \((LiAl),\) gold \(\left(\mathrm{Au} \mathrm{Al}_{2}\right)\) and titanium \(\left(\mathrm{Al}_{3} \mathrm{Ti}\right) .\) On the basis of their crystal lattices, each of these alloys is considered a substitutional alloy. a. Do these alloys fit the general size requirements for substitutional alloys? The atomic radii for Li, Al, Au, and Ti are \(152,143,144,\) and \(147 \mathrm{pm},\) respectively. b. If the unit cell of LiAl is bec, what is the density of LiAl?

Step-by-Step Solution

Answer

If so, calculate the density of LiAl if its unit cell is bcc. Answer: Yes, all the given alloys fit the general size requirements for substitutional alloys as their percentage differences in atomic radii are less than 15%. The density of LiAl with a bcc unit cell is 0.708 g/cm³.

Step 1: a. Size requirements for substitutional alloys

To determine if these alloys fit the general size requirements for substitutional alloys, we need to compare the atomic radii of the elements involved. The difference in atomic radii should not exceed 15%. We will calculate the percentage difference for each alloy to check this. Percentage difference = \(\frac{|\text{Atomic radius 1 - Atomic radius 2}|}{\text{Average of the two atomic radii}} \times 100\) For LiAl: Percentage difference = \(\frac{|152-143|}{\frac{152+143}{2}} \times 100\) = 6.25% For AuAl2: Percentage difference = \(\frac{|144-143|}{\frac{144+143}{2}} \times 100\) = 0.69% For Al3Ti: Percentage difference = \(\frac{|147-143|}{\frac{147+143}{2}} \times 100\) = 2.78% All percentages are less than 15%, so these alloys fit the general size requirements for substitutional alloys.

Step 2: b. Density of LiAl if the unit cell is bcc

To calculate the density of LiAl, we will need the lattice parameter (a), which we can find using the atomic radius and the unit cell type (bcc). Then, we can use the molar mass of LiAl and Avogadro's number to find the density. 1. Find the lattice parameter (a): For a bcc unit cell, \(a = \dfrac{4r_\text{Al}}{\sqrt{2}}\), and r\(_\text{Al}\) is the atomic radius of Aluminum. Substitute the given value of atomic radius: \(a = \dfrac{4(143)}{\sqrt{2}} = 305.7\mathrm{pm}\) 2. Calculate the volume of the unit cell: \(V_\text{unit cell} = a^3 = (305.7\mathrm{pm})^3 = 2.866\times 10^{-23}\mathrm{cm}^3\) 3. Determine the number of atoms per unit cell for LiAl: For a bcc unit cell, there are 2 Al atoms and 2 Li atoms per unit cell. 4. Calculate the molar mass of LiAl: Molar mass of LiAl = Atomic mass of Li + Atomic mass of Al = 6.939 + 26.982 = 33.921 g/mol 5. Compute the mass of one unit cell of LiAl: \(V_\text{unit cell} = N_\text{Avogadro} \times \frac{M_\text{LiAl}}{n_\text{atoms}} \times \rho_\text{LiAl}\) \(\Rightarrow \rho_\text{LiAl} = \frac{V_\text{unit cell} \times n_\text{atoms}}{N_\text{Avogadro} \times M_\text{LiAl}} = \frac{2.866\times 10^{-23}\mathrm{cm}^3 \times 4}{(6.022\times 10^{23}\mathrm{atoms/mol) \times (0.033921 kg/mol)}}\) \(\rho_\text{LiAl} = 0.708\,\mathrm{g/cm^3}\) The density of LiAl with a bcc unit cell is 0.708 g/cm³.

Key Concepts

Atomic Radii

The atomic radius is an important measure in understanding atoms and how they interact to form alloys. The atomic radius is essentially the size of an atom, which varies from one element to another. It's measured in picometers (pm). When elements form alloys, such as substitutional alloys, their atomic radii play a crucial role in determining whether the alloy will have stable configurations. For substitutional alloys, the difference in atomic radii between the elements should generally not exceed 15%. This ensures that atoms can replace one another within the alloy's structure without causing significant distortions.

• Lithium (Li): 152 pm
• Aluminum (Al): 143 pm
• Gold (Au): 144 pm
• Titanium (Ti): 147 pm

The exercise demonstrates this by showing the percentage differences for the LiAl, AuAl extsubscript{2}, and Al extsubscript{3}Ti alloys, all of which meet the size requirements, indicating their potential stability as substitutional alloys.

Crystal Lattice

A crystal lattice is an organized, repeating 3D arrangement of atoms, ions, or molecules in a crystalline material. It's like the scaffold or framework of a crystal. In the context of alloys and metals, the crystal lattice determines how atoms are positioned relative to one another. Substitutional alloys depend on this lattice structure, as atoms of different elements substitute at lattice points occupied by the main metal atoms. For instance, a body-centered cubic (bcc) lattice is a common structure in which each corner of the cube is an atom, and there is a single atom at the center. This is seen in the LiAl alloy. Understanding the lattice structure is crucial because it influences the material's properties, such as strength, ductility, and thermal conductivity. Changes in the lattice caused by substituting different atoms can lead to various material properties, underlying why precise control of atomic radii and lattice parameters is necessary when designing new materials.

Density Calculation

Calculating the density of an alloy involves considering its crystal structure, atomic masses, and how atoms are arranged within the unit cell. Density, defined as mass per unit volume, is vital for understanding the material's usability in applications.For a body-centered cubic (bcc) structure, calculations begin with determining the lattice parameter, followed by the volume of the unit cell. In our exercise with the LiAl alloy:

• Lattice parameter, calculated using the atomic radius of aluminum: \( a = \frac{4r_\text{Al}}{\sqrt{2}} \).
• The volume of the unit cell: \( V_\text{unit cell} = a^3 \).

Next, count the atoms in the unit cell, which determines how much mass is in that volume. The molar mass for LiAl combines the atomic masses of lithium and aluminum. Finally, using Avogadro's number, convert these masses into density. In the provided example, the calculated density of LiAl is 0.708 g/cm\(^3\). Understanding density helps design materials for specific needs, whether it's reducing weight in aerospace applications or determining load-bearing capabilities in construction.

Lattice Parameter

The lattice parameter is the physical dimension of the unit cell in a crystal lattice. This parameter is vital because it directly influences the volume of the unit cell, which impacts calculations for various physical properties, including density and structural stability.For body-centered cubic (bcc) structures, like those found in certain substitutional alloys, the lattice parameter can be derived from an element’s atomic radius. The exact formula used depends on the structure. In the LiAl alloy, the formula \( a = \frac{4r_\text{Al}}{\sqrt{2}} \) is applicable. Manipulating this parameter allows material scientists to predict how densely atoms are packed in a metal and what properties the final product might possess. Small changes in lattice parameters can lead to significant changes in material properties by affecting how tightly atoms pack in a given volume.Understanding lattice parameters is essential for both theorizing the properties of new materials and tailoring existing materials to meet specific practical requirements.