Determine the radius of the largest atom that can be accommodated in the interstices of BCC iron (Fe) without stress. (Hint: the center of the largest site is at ½, ¼, 0; draw a unit cell - it helps.)
Radius (in centimeters):
3.61*10^-9
To determine the radius of the largest atom that can be accommodated in the interstices of BCC iron (Fe) without stress, we need to understand the crystal structure of BCC iron and the concept of interstitial sites.
BCC stands for Body-Centered Cubic, which is a crystal structure with a unit cell consisting of eight atoms at the corners of a cube and one atom in the center of the cube. Within this structure, there are interstitial sites, which are empty spaces between atoms that can be occupied by smaller atoms. In this case, we are trying to determine the largest atom that can fit into these interstitial sites without causing stress.
The hint provided mentions that the center of the largest site is at ½, ¼, 0. To visualize this, we can draw a unit cell of BCC iron. The unit cell has eight corner atoms and one centered atom. The largest site mentioned in the hint is located at the center of one of the faces in the middle of the unit cell.
Now, let's determine the radius of the largest atom that can fit into this site. We know that the coordinates of the center of the site are ½, ¼, 0. In BCC iron, the atoms are arranged in a body-centered manner, meaning that the atoms on one face of the unit cell are connected to the atom in the center. In this case, the atom in the center is connected to the atoms at the corners of the unit cell.
To calculate the radius, we need to measure the distance between the center of the site and one of the corner atoms. Based on the crystal structure, the diagonal of the face of the unit cell is equal to four atomic radii. Therefore, the distance between the center of the site and one of the corners is equal to the diagonal of the face divided by two.
To calculate the diagonal of the face, we can use the Pythagorean theorem. The diagonal is the square root of the sum of the squares of the three sides. In this case, two sides are equal to the atomic radius. So, the diagonal of the face is equal to the square root of two times the atomic radius.
Once we have the diagonal of the face, we divide it by two to find the distance between the center of the site and one of the corners. This distance gives us the radius of the largest atom that can fit into the interstitial site without causing stress.
So, to summarize the steps:
1. Draw a unit cell of BCC iron.
2. Identify the location of the largest interstitial site mentioned in the hint.
3. Use the coordinates of the site to find the distance between the center of the site and one of the corner atoms.
4. Calculate the diagonal of the face of the unit cell using the Pythagorean theorem.
5. Divide the diagonal by two to find the radius of the largest atom that can fit into the interstitial site without causing stress.
Performing these calculations will give you the radius in centimeters.