Question

Which of the following expressions is correct in case of a \(\mathrm{CsCl}\) unit cell (edge length, \(a\) )? (a) \(r_{c}+r_{a}=a\) (b) \(r_{c}+r_{a}=a / \sqrt{2}\) (c) \(r_{c}+r_{a}=\sqrt{3} a / 2\) (d) \(r_{c}+r_{a}=a / 2\)

Short answer

(c) \(r_c + r_a = \frac{\sqrt{3}a}{2}\) is the correct expression for the sum of the radii of the ions in a CsCl unit cell.

Step-by-step solution

Understanding the Structure of CsCl

Cesium chloride (\text{CsCl}) crystallizes in a body-centered cubic (BCC) structure. In this structure, the Cs ion is at the center of the cube and the Cl ion is at one of the corners. The radius of the cesium ion is denoted by \(r_c\) and that of the chloride ion by \(r_a\). The closest approach between ions happens along the body diagonal of the cube.

Calculating the Body Diagonal

The body diagonal's length in a cube with edge length \(a\) is given by the formula \(d = a \sqrt{3}\). In the case of CsCl, the body diagonal is also equal to the sum of the diameters of the Cs and Cl ions, which is \(2(r_c + r_a)\).

Relating Ion Radii to the Body Diagonal

Setting the body diagonal equal to the sum of the diameters of the ions, we get \(a \sqrt{3} = 2(r_c + r_a)\). Dividing both sides by 2 gives us the relation between the ion radii and the edge length of the cube: \(r_c + r_a = \frac{a \sqrt{3}}{2}\).

Choosing the Correct Expression

Comparing to the given options, the expression that matches our derived relation is (c) \(r_c + r_a = \frac{\sqrt{3}a}{2}\).

Key concepts

Body-Centered Cubic Structure

When exploring the fascinating realm of solid-state chemistry, the body-centered cubic (BCC) structure plays a pivotal role in understanding many of the intrinsic properties of compounds. One practical example is Cesium Chloride (CsCl), a compound known for its BCC structure. In this arrangement, an atom (or ion) rests at the center of a cube, effectively touching the eight corners where other equivalent atoms (or ions) are located. This structure is significant as it directly influences the properties of the solid, such as density and how the atoms or ions pack within the crystal.

It’s important to visualize the geometry: Imagine a three-dimensional checkerboard, with alternating pieces representing different ions. The central atom, in this case, Cs, has a direct line of sight to the corners of the cube, all equidistant. The BCC structure has a packing efficiency of about 68%, which correlates to how much of the space within the cube is 'actively' filled by the atoms themselves. Understanding this can be quite enlightening when approaching crystal lattice calculations or deducing ionic relationships within a solid.

Ionic Radius

Diving a bit deeper, let's discuss the ionic radius — a term that denotes the size of an ion in a crystal lattice. Ionic radii are significant when addressing the spatial relationships between ions in a crystal. They illustrate how closely ions pack together and essentially determine the distances within a crystal structure. For instance, in the CsCl crystal, we denote the radius of the cesium ion as \(r_c\) and that of the chloride ion as \(r_a\).

It's a bit like sizing up spheres to pack into a box — the sizes of the spheres will determine how they fit. Understanding ionic radii is not just about the static measure from the nucleus to the electron cloud's outer edge; it's about grasping how ions will interact with one another. When ions nestle together in the crystal, their radii will determine how they align and ultimately, the structure of the crystal itself. This concept is essential when investigating properties like ionic bonding strength and reactiveness.

Crystal Lattice Calculations

Now let's tie it all together with crystal lattice calculations. These calculations are the mathematical expressions that describe the meticulous architecture of crystal structures, providing us with a deeper understanding of the material's characteristics. Through such computations, we can relate the ionic radii to the unit cell dimensions, which is exactly what we see with the CsCl problem.

To elucidate this relation, look at the body diagonal length within the cube. Following the Pythagorean theorem, the length of the diagonal is calculated by extending the edge length of the cube \(a\) through three dimensions, resulting in the formula \(d = a \sqrt{3}\). This brings the radii into play, essentially saying that the combined diameter of the Cs and Cl ions must equal the cube's body diagonal. It's through these calculations that solid-state chemists unveil the underlying harmony within materials — allowing them to predict how these materials will behave in a variety of conditions. Truly, mathematics becomes the language through which we decode the secrets of crystallography.