Animated Solution for Chemistry - States of Matter: Copper crystallises in fcc lattice with a unit cell edge of 361 pm. The radius of copper atom is

Imagine you are looking directly at one face of a Face-Centered Cubic (FCC) unit cell. What do you see? You will find an atom at each of the four corners and one atom sitting perfectly in the center of the face.

Because of this central atom, the corner atoms are pushed slightly apart. They no longer touch along the edges of the cube. Instead, the atoms make contact strictly along the face diagonal.

Let the edge length of the unit cell be $a$ and the radius of each atom be $r$.

If we draw a diagonal across the face of the cube, we form a right-angled triangle with the edges. According to the Pythagorean theorem, the length of this face diagonal is $\sqrt{a^2+a^2}=\sqrt{2}a$.

Now, let's look at the atoms along this diagonal. The diagonal passes through the center of the face atom (contributing a full diameter, $2r$) and connects to the centers of the two corner atoms (each contributing a radius, $r$).

Therefore, the total length of the face diagonal in terms of the atomic radius is $r+2r+r=4r$.

Equating the two geometric perspectives, we get our master equation:

The problem provides the edge length $a=361\text{ pm}$. Let's substitute this into our formula:

We know that $\sqrt{2}\approx 1.414$. Substituting this value:

Performing the division, we find:

Since the options are given as integers, we round off our result to the nearest whole number.

This perfectly matches option (c). Always remember to visualize the lattice structure before jumping into the formulas!