Question 10.8: Using Unit-Cell Dimensions to Calculate the Radius of an Ato......
Using Unit-Cell Dimensions to Calculate the Radius of an Atom
Silver metal crystallizes in a cubic closest-packed arrangement with the edge of the unit cell having a length d = 407 pm. What is the radius in picometers of a silver atom?
STRATEGY AND SOLUTION
Cubic closest-packing uses a face-centered cubic unit cell. Looking at any one face of the cube head-on shows that the face atoms touch the corner atoms along the diagonal of the face but that corner atoms do not touch one another along the edges. Each diagonal is therefore equal to four atomic radii, 4r:
Because the diagonal and two edges of the cube form a right triangle, we can use the Pythagorean theorem to set the sum of the squares of the two edges equal to the square of the diagonal, d^2 + d^2 = (4r)^2 and then solve for r, the radius of one atom:
\begin{aligned}& \qquad d^2+d^2=(4 r)^2 \\& 2 d^2=16 r^2 \text { and } r^2=\frac{d^2}{8} \\& \text { thus } \quad r=\sqrt{\frac{d^2}{8}}=\sqrt{\frac{(407 \mathrm{pm})^2}{8}}=144 \ \mathrm{pm}\end{aligned}