Animated Solution for Chemistry - States of Matter: The cubic unit cell structure of a compound containing cation M and anion X is shown below. When compared to the anion, the cation has smaller ionic radius. Choose the correct statement(s).

Imagine you are shrinking down to the atomic level and stepping inside this crystalline world. What do you see? We have a perfect cubic room—our unit cell. At every single corner of this room, there is a white sphere, which represents our anion, $X$. Right in the dead center of the room, floating perfectly in the middle, is a black sphere, our cation, $M$.

This specific arrangement is a classic in solid-state chemistry. It is known as the Cesium Chloride (CsCl) structure. To understand the macroscopic properties of this crystal, we first need to figure out its empirical formula. We do this by calculating the effective number of atoms, denoted by $Z$, inside a single unit cell.

Let's start with the anions at the corners. A corner of a cube isn't just owned by one room; it is the meeting point of eight identical rooms. Therefore, only $\frac{1}{8}$ of each corner atom actually belongs to our unit cell. Since there are $8$ corners, the effective number of $X$ atoms is:

Now, look at the cation $M$. It is sitting comfortably at the body center, completely enclosed within our unit cell. It isn't sharing its space with anyone else. Thus, its contribution is a full $1$.

Since we have exactly one $M$ and one $X$ effectively inside the unit cell, their ratio is $1:1$. This makes the empirical formula $MX$. Option (A) is absolutely correct!

Coordination and Geometry

Now, let's talk about neighbors. In the atomic world, your nearest neighbors define your coordination geometry.

Focus on the central cation $M$. If you look around from the center, you will see exactly $8$ anions $X$ sitting at the corners, all at the exact same distance. This means the coordination number of $M$ is $8$, forming a cubic geometry.

But what about the anion $X$ at the corner? It might seem lonely, but remember, the crystal lattice extends infinitely. That corner $X$ is surrounded by $8$ adjacent unit cells, and each of those cells has a cation $M$ at its center. So, the corner $X$ is also surrounded by $8$ cations! Both $M$ and $X$ share the exact same cubic coordination geometry. Therefore, Option (B) is incorrect.

Bond Lengths and Voids

Next, we need to measure the distance between $M$ and $X$, which is the bond length. In this structure, the cation and anion are in direct contact along the body diagonal of the cube.

If the edge length of our cube is $a$, the mathematics of 3D geometry tells us that the total length of the body diagonal is $\sqrt{3}a$. Since the cation is exactly in the middle, the distance from the center to any corner is exactly half of the body diagonal:

The question asks for the ratio of this bond length to the edge length $a$. Let's set up the ratio:

We know that $\sqrt{3}\approx 1.732$. Dividing this by $2$ gives us exactly $0.866$. The math aligns perfectly! Option (C) is correct.

Finally, let's evaluate the ionic radii. The cation $M$ is sitting in the empty space created by the $8$ corner anions. This space is called a cubic void. For a cation to perfectly fit into a cubic void without rattling around or pushing the anions too far apart, the ideal radius ratio $\frac{r_M}{r_X}$ must be:

Let's calculate this value:

The question suggests a ratio of $0.414$. However, $0.414$ is the magic number for an octahedral void (coordination number 6), not a cubic void. Therefore, Option (D) is a trap and is incorrect.

Our thrilling journey through the unit cell concludes with Options (A) and (C) standing victorious!