Final answer:
To find the atomic radius of iron in a body-centered cubic structure, calculate the cube's diagonal using the Pythagorean theorem and then divide it by 4, since the diagonal is 4 times the radius.
Step-by-step explanation:
The question is asking to determine the atomic radius of iron when it crystallizes in a body-centered cubic structure with a given edge length. In a body-centered cubic structure, the length of the diagonal that passes through the center of the cube is equal to four times the atomic radius (4r), since the diagonal crosses two radii in the center and half radius at each of the corners of the cube. The diagonal can be calculated using the Pythagorean theorem as the square root of three times the edge length (√3 * edge length).
To find the atomic radius of iron, we use the given edge length of 287 pm:
- Calculate the diagonal using the Pythagorean theorem: diagonal = √3 * edge length = √3 * 287 pm.
- Since the diagonal equals 4r (where r is the atomic radius), we set the diagonal equal to 4r and solve for r: r = diagonal / 4.
- Substitute the calculated diagonal in the formula and solve for r to find the atomic radius of iron.