Problem 100 - Chemistry: The Central Science

Question

Pure iron crystallizes in a body-centered cubic structure, but small amounts of impurities can stabilize a face-centered cubic structure. Which form of iron has a higher density?

Step-by-Step Solution

Answer

The body-centered cubic structure (BCC) of iron has a higher density compared to the face-centered cubic structure (FCC).

Step 1: Calculate the Volume of the Unit Cell for Each Structure

We will calculate the volume for both body-centered cubic (BCC) and face-centered cubic (FCC) unit cells. For BCC, the edge length (a) can be related to the atomic radius (r) by the Pythagorean theorem: a = 4r / √3 For FCC, we have the analogous relationship: a = 2√2r Now, we find the volume of the unit cell for each structure: - Volume of BCC Unit Cell: V_BCC = (4r / √3)^3 - Volume of FCC Unit Cell: V_FCC = (2√2r)^3

Step 2: Determine the Number of Atoms per Unit Cell

Each structure has a different number of atoms in the unit cell: - For BCC, there is 1 atom at the center and 1/8 of an atom at each of the 8 corners, so the number of atoms in total is: n_BCC = 1 + 8*(1/8) = 2 atoms - For FCC, there is 1/8 of an atom at each of the 8 corners and 1/2 of an atom at each of the 12 face centers, so the number of atoms in total is: n_FCC = 8*(1/8) + 12*(1/2) = 4 atoms

Step 3: Calculate the Density for Each Structure

The density (ρ) can be calculated by dividing the mass of the atoms in each unit cell by the volume of the unit cell. Since we want to compare the densities, we can use a constant, M_Iron, to represent the mass of one iron atom. Density of BCC: ρ_BCC = 2*M_Iron / V_BCC Density of FCC: ρ_FCC = 4*M_Iron / V_FCC

Step 4: Compare the Densities

Now we can compare ρ_BCC and ρ_FCC to determine which structure has higher density: If ρ_BCC > ρ_FCC, then the BCC structure has a higher density. If ρ_BCC < ρ_FCC, then the FCC structure has a higher density. If ρ_BCC = ρ_FCC, then both structures have the same density. Since the radius (r) is the same for both structures, we can compare the densities directly: ρ_BCC = 2*M_Iron / (4r / √3)^3 ρ_FCC = 4*M_Iron / (2√2r)^3 ρ_BCC / ρ_FCC = [(2*M_Iron) / (64r^3 / 3√3)] / [(4*M_Iron) / (16√2r^3)] ρ_BCC / ρ_FCC = (3√3) / (2√2) ρ_BCC / ρ_FCC ≈ 1.63 Since ρ_BCC / ρ_FCC > 1, the body-centered cubic structure (BCC) of iron has a higher density compared to the face-centered cubic structure (FCC).