For this problem, we're being asked to determine the edge length of a body-centered cubic unit cell

Below is an image that shows the smallest unit cell that is repeatedly stacked for a BCC unit cell

The **cell edge** can be represented as** "l"** where the direction from a corner of cube to the farthest corner is the** body diagonal "bd"**

**"fd" **or **face diagonal** a line drawn from one vertex to the opposite corner of the same face. If the edge is "l", we can write an equation as:

$\overline{){{\mathbf{fd}}}^{{\mathbf{2}}}{\mathbf{}}{\mathbf{=}}{\mathbf{}}{{\mathbf{l}}}^{{\mathbf{2}}}{\mathbf{}}{\mathbf{+}}{\mathbf{}}{{\mathbf{l}}}^{{\mathbf{2}}}}\phantom{\rule{0ex}{0ex}}{{\mathbf{fd}}}^{{\mathbf{2}}}{\mathbf{}}{\mathbf{=}}{\mathbf{}}{\mathbf{2}}{{\mathbf{l}}}^{{\mathbf{2}}}\phantom{\rule{0ex}{0ex}}\overline{){{\mathbf{bd}}}^{{\mathbf{2}}}{\mathbf{}}{\mathbf{=}}{\mathbf{}}{{\mathbf{fd}}}^{{\mathbf{2}}}{\mathbf{}}{\mathbf{+}}{\mathbf{}}{{\mathbf{l}}}^{{\mathbf{2}}}}\phantom{\rule{0ex}{0ex}}{\mathbf{bd}}^{\mathbf{2}}\mathbf{}\mathbf{=}\mathbf{}{\mathbf{fd}}^{\mathbf{2}}\mathbf{}\mathbf{+}\mathbf{}{\mathbf{l}}^{\mathbf{2}}\phantom{\rule{0ex}{0ex}}{\mathbf{bd}}^{\mathbf{2}}\mathbf{}\mathbf{=}{\mathbf{2}}{{\mathbf{l}}}^{{\mathbf{2}}}\mathbf{}\mathbf{+}\mathbf{}{\mathbf{l}}^{\mathbf{2}}\phantom{\rule{0ex}{0ex}}{\mathbf{bd}}^{\mathbf{2}}\mathbf{}\mathbf{=}{\mathbf{3}}{{\mathbf{l}}}^{{\mathbf{2}}}\mathbf{}$

Atoms along the body diagonal touch each other which makes the **length of bd as 4 times the radius "r"**

When spheres of radius r are packed into a body-centered cubic unit cell, they occupy 68.0% of the available volume. Use this to calculate the value of l, the length of the edge of the cell, in terms of r.

A. l = 2.79r

B. l = 1.44r

C. l = 2.31r

D. l = (4*2^{1/2})r

E. l = r

Frequently Asked Questions

What scientific concept do you need to know in order to solve this problem?

Our tutors have indicated that to solve this problem you will need to apply the Unit Cell concept. You can view video lessons to learn Unit Cell. Or if you need more Unit Cell practice, you can also practice Unit Cell practice problems.

What is the difficulty of this problem?

Our tutors rated the difficulty of*When spheres of radius r are packed into a body-centered cub...*as medium difficulty.