Calaulating Atomic Radius from Crystal Structure
At room temperature the most stable crystal structure of ion has a body-centered cubic unit cell with an edge length of 287 pm (1 pm =10^-12 m). What is the radius in picometers of the iron atoms?
COLLECT AND ORGANIZE The crystal structuer is based on a bcc unit cell (Figure 11.10b). This means that the iron atoms do not touch along cell edges or face diagonals but do touch along the body diagonals. We know the edge length of the cell: 287 pm.
ANALYZE Atoms touch along the body diagonals of a bcc unit cell, so the radius (r) of the atoms is related to the cell's edge length (l) by Equation 11.2:
r= 0.4330l
SOLVE We insert the edge-length value into this equation to solve for r:
r= 0.4330*287 pm = 124 pm
THINK ABOUT IT The average atomic radius of iron atoms is 126 pm, so the result of this calculation is reasonable. It should be because the values for atomic radii are derived from analyses of crystal structures in the first place.
Practice Exercise At 1070 degree the most stable structure of iron has an fcc unit cell with an edge length of 361 pm. What is the atomic radius of iron at this temperature?