Please help. My teacher never went over crystal lattices and the book doesn't cover it either :(
Copper crystallizes in a face-centered cubic lattice. If the edge of the unit cell is 351pm, what is the radius of the copper atom?
4r = a*21/2
a is the edge length, r is radius.
r= a/2✅2(2 root 2)where be the edge length
No problem! I can help you understand how to approach this problem.
To determine the radius of the copper atom in a face-centered cubic (FCC) lattice, we need to consider the relationship between the edge length of the unit cell and the atomic radius.
In an FCC lattice, there are atoms at each corner of the unit cell and additional atoms at the center of each face. This arrangement results in a total of four atoms per unit cell.
The diagonal of the face-centered cubic unit cell can be calculated using the Pythagorean theorem. Let's call the edge length of the unit cell "a" and the diagonal of the unit cell "d". The diagonal can be calculated as follows:
d = √(a^2 + a^2 + a^2) = √3a
Since the edge length of the unit cell is given as 351 pm, we can substitute this value into the equation to find the diagonal:
d = √3(351 pm) ≈ 607.32 pm
Next, we need to determine the distance from the center of a face to the corner of the unit cell. This distance is equal to half the length of the diagonal, which can be expressed as:
d_face = d/2 = √3a/2 = 0.866a
Given that the edge length of the unit cell is 351 pm, we can substitute this into the equation to find the distance from the center of a face to the corner:
d_face = 0.866(351 pm) ≈ 304.02 pm
Now, in an FCC lattice, the face diagonal is twice the distance from the center of a face to the corner. So, we have:
Face diagonal = 2d_face ≈ 2(304.02 pm) = 608.04 pm
Finally, to find the radius of the copper atom, we need to subtract half the face diagonal from the edge length:
Radius = (Edge length - 0.5 * Face diagonal) / 2
Substituting the known values, we have:
Radius = (351 pm - 0.5 * 608.04 pm) / 2
Calculating this expression gives us:
Radius ≈ 547.96 pm
Therefore, the radius of the copper atom in the FCC lattice is approximately 547.96 pm.