Problem 127 - Chemistry: An Atoms-Focused Approach

Question

Substitutional alloys may form when the difference in atomic radii between the alloying elements is less than \(15 \%\) a. Predict which of the following alloys has the greatest mismatch in atomic radii: AuZn, AgZn, or CuZn. b. Find the atomic radii of \(\mathrm{Cu}, \mathrm{Ag}, \mathrm{Au},\) and \(\mathrm{Zn}\) in Appendix 3, and calculate the percent difference in their atomic radii. Are all three alloys expected to form substitutional alloys? c. If gold is alloyed with silver in a 1: 1 ratio, do the atoms still touch along the face diagonal of a face-centered cubic unit cell?

Step-by-Step Solution

Answer

Answer: AuZn and AgZn alloys have the greatest mismatch in atomic radii with a 7.3% difference. Gold and silver atoms do not touch along the face diagonal of a face-centered cubic unit cell when alloyed in a 1:1 ratio.

Step 1: Find the atomic radii of Cu, Ag, Au, and Zn

Using Appendix 3, we will find the atomic radii for Cu, Ag, Au, and Zn. The values are as follows: - Cu: 128 pm - Ag: 144 pm - Au: 144 pm - Zn: 134 pm

Step 2: Calculate the percent difference in atomic radii

We will use the formula for percent difference: Percent difference = \(\frac{|r_1 - r_2|}{(r_1 + r_2) / 2} \times 100\%\) (a) AuZn: Percent difference = \(\frac{|144 - 134|}{(144 + 134) / 2} \times 100\% = 7.3 \%\) (b) AgZn: Percent difference = \(\frac{|144 - 134|}{(144 + 134) / 2} \times 100\% = 7.3 \%\) (c) CuZn: Percent difference = \(\frac{|128 - 134|}{(128 + 134) / 2} \times 100\% = 4.7 \%\) As per the given condition, substitutional alloys form when the difference in atomic radii is less than 15%. Since all percent differences calculated above are less than 15%, all three alloys are expected to form substitutional alloys.

Step 3: Determine the greatest mismatch in atomic radii

Based on the percent difference in atomic radii calculated in Step 2: - AuZn and AgZn have the greatest mismatch with 7.3% difference - CuZn has a smaller mismatch with 4.7% difference

Step 4: Verify if gold and silver atoms still touch along the face diagonal of a face-centered cubic unit cell when alloyed in a 1:1 ratio

The face diagonal length (\(d\)) in a face-centered cubic unit cell can be expressed as: \(d = 2r_1 + 2r_2\) When gold and silver are alloyed in a 1:1 ratio, all the corners and face centers of the face-centered cubic unit cell would be alternatively occupied by gold and silver atoms. For gold and silver atoms to still touch along the face diagonal, we would need: \(d = 2 \times (r_{Au} + r_{Ag}) = 2 \times (144 + 144) = 576\) pm However, the actual face diagonal length of a face-centered cubic unit cell based on the atomic radii of gold is: \(d = 2\sqrt{2} r_{Au} = 2\sqrt{2} \times 144 \approx 407\) pm Since the required face diagonal length (576 pm) to maintain contact between gold and silver atoms is significantly larger than the actual face diagonal length (407 pm), the atoms do not touch along the face diagonal when alloyed in a 1:1 ratio.

Key Concepts

Atomic Radii

Understanding atomic radii is critical when studying substitutional alloys. The atomic radius of an element is the distance from the center of its nucleus to the boundary of the surrounding cloud of electrons. This radius can influence how elements interact and bond with each other. In the context of forming substitutional alloys, the similarity in atomic radii between elements is important to ensure that the resulting alloy has a stable structure. For an alloy to form successfully, the difference in atomic radii between alloying components should be less than 15%, which allows the atoms to interchange positions within the alloy while retaining the overall lattice structure.
In our example, the atomic radii of Cu, Ag, Au, and Zn have been used to calculate their percentage differences to evaluate whether substitutional alloys will form effectively.

• Cu: 128 pm
• Ag: 144 pm
• Au: 144 pm
• Zn: 134 pm

Determining these radii accurately helps in predicting the formation of alloys and understanding their structural properties.

Percent Difference

The percent difference between atomic radii is a vital calculation to determine whether elements are likely to form substitutional alloys. The formula for percent difference is given by: \[ \text{Percent difference} = \frac{|r_1 - r_2|}{(r_1 + r_2) / 2} \times 100\% \] where \(r_1\) and \(r_2\) are the atomic radii of the two elements being compared.
Using this formula, calculations for AuZn, AgZn, and CuZn revealed percent differences of 7.3%, 7.3%, and 4.7%, respectively. These values indicate that all three alloy combinations have a percent difference below the 15% threshold, thus supporting the potential for forming stable substitutional alloys.

• AuZn and AgZn: 7.3% difference
• CuZn: 4.7% difference

This calculation ensures that the atomic sizes are close enough to substitute within each other's lattice structure without creating significant distortions.

Face-Centered Cubic Unit Cell

The face-centered cubic (FCC) unit cell is an important geometric configuration in crystallography where atoms are located at each of the corners and the center of each face of the cube. This structure is common in metals and some alloys due to its density and stability. In an FCC unit cell, the atoms along the edge are not in direct contact; instead, atoms at the face centers touch those at the cube's corners.
When examining alloy compositions like a gold-silver alloy, knowing whether the atoms maintain contact along the face diagonals can indicate structural integrity.
The theoretical face diagonal length in a pure FCC unit cell with radius \(r\) is: \[ d = 2\sqrt{2}r \] In a 1:1 Au-Ag alloy, if the atoms did touch along the diagonal, the distance would theoretically be \(2 \times (144 + 144) = 576\) pm, while the calculated diagonal length of gold alone is approximately 407 pm ( \(2\sqrt{2} \times 144\)). This discrepancy implies that atoms in a 1:1 Au-Ag alloy do not maintain contact, causing breaks along the face diagonal. Therefore, even small variations in atomic arrangements can lead to significant changes in structural properties, a crucial consideration in material sciences.