Understanding and Calculating the Theoretical Density of FCC Copper

To understand the theoretical density of Face-Centered Cubic (FCC) copper, we need to go through a few steps. This article will walk you through the process, using the atomic radius and atomic weight as inputs. We'll also explore why the calculated density differs slightly from the experimental value.

Step 1: Calculate the Volume of the Unit Cell

In an FCC structure, there are four atoms per unit cell. The relationship between the atomic radius r and the edge length a of the FCC unit cell is given by:

a 2√2 r

Given the atomic radius r of copper as 0.128 nm, which is 0.128 × 10-9 m, we can calculate the edge length a:

a 2√2 × 0.128 × 10-9 m ≈ 0.362 × 10-9 m

Next, we calculate the volume V of the unit cell:

V a3 (0.362 × 10-9 m)3 ≈ 4.75 × 10-29 m3

Step 2: Calculate the Mass of the Atoms in the Unit Cell

The atomic weight of copper is 63.55 g/mol. To find the mass of one atom of copper and then four atoms, follow these steps:

Convert the atomic weight to kilograms: Atomic weight 63.55 g/mol 0.06355 kg/mol The mass of one atom of copper is: Mass of one atom 0.06355 kg/mol ÷ 6.022 × 1023 atoms/mol ≈ 1.055 × 10-25 kg The mass of 4 atoms in the unit cell is: Mass of 4 atoms 4 × 1.055 × 10-25 kg ≈ 4.22 × 10-25 kg

Step 3: Calculate the Theoretical Density

The density ρ can be calculated using the mass and volume of the unit cell:

ρ Mass / Volume 4.22 × 10-25 kg / 4.75 × 10-29 m3 ≈ 8890 kg/m3

Conclusion: Theoretical Density vs. Experimental Density

The theoretical density of FCC copper is approximately 8890 kg/m3, which is equivalent to 8.89 g/cm3. However, experimental values for the density of copper are around 8.96 g/cm3. The close ratio of 8.96/12.01 0.746 to the expected 0.75 further validates the theoretical model, considering the FCC structure leaves approximately one-fourth of the atomic space empty.

To refine the theoretical value, we can ignore the empty atomic volume for simplicity. The total volume would be:

V [4/3 × π × (0.128 nm)323 atoms/mol 5.29 × 1021 nm3 5.29 cm3

The mass of one mole of copper is 63.55 g. Thus, the theoretical density is:

d 63.55 g / 5.29 cm3 ≈ 12.01 g/cm3

This theoretical value of 12.01 g/cm3 is very close to the experimental value of 8.96 g/cm3, indicating that the reported theoretical density of FCC copper would be:

Dt 63.55 g / (4/3) × [4/3 × π × (0.128 nm)323 atoms/mol × 10-21 9.01 g/cm3

There is a 1/4 empty space volume over the FCC packing of copper, making this theoretical density very close to the actual experimental value.

Keywords

FCC Copper Theoretical Density Atomic Radius

References

[1] Laidler, K. J., Meiser, J. H. (1982). The Chemical Callable. Benjamin/Cummings Publishing.

[2] Atkins, P. W., de Paula, J. (2006). Physical Chemistry. Oxford University Press.