Problem 21 - A Complete Resource Book in Chemistry for JEE Main

Question

In simple cubic crystal the radius ratio should be (a) \(\geq 0.732\) (b) \(\leq 0.732\) (c) \(\leq 0.424\) (d) \(\leq 0.225\)

Step-by-Step Solution

Answer

(c) \(\leq 0.424\) is most relevant for simple cubic structures.

Step 1: Understanding the Radius Ratio

The radius ratio in crystal structures refers to the ratio of the radius of the smaller atom to the radius of the larger atom. This determines the type of void or interstitial site that can accommodate the smaller ion or atom.

Step 2: Review Simple Cubic Structure

A simple cubic structure has atoms arranged at the corners of a cube. Each edge of this cube is equal to twice the radius of the atom because in simple cubic structures, atoms touch each other along the edge of the cube.

Step 3: Determine Radius Ratio for Simple Cubic

For a simple cubic structure, the radius ratio can be calculated using the condition for maximum packing efficiency. The limiting radius ratio for a simple cubic is determined by the geometry of the cube, where an atom at the center of a cube touches the corner atoms. The common coordination number of simple cubic is 6, and typically a cubic coordination is achieved with radius ratios around 0.414. Thus, typical radius ratios for simple cubic should fall less than 0.732.

Step 4: Evaluate Options

Now we analyze the given options. Options (b) \(\leq 0.732\), (c) \(\leq 0.424\), and (d) \(\leq 0.225\) suggest that the radius ratio could be less than these values. However, for a stable simple cubic structure, radius ratios much greater than typical \(\approx 0.414\) are not suited for simple cubic. Thus, options (c) and possibly (d) are most suited for a simple cubic configuration.

Key Concepts

Simple Cubic Structure

In crystalline materials, a simple cubic structure is one of the simplest forms of crystal lattices. Imagine a box made entirely of identical balls; these balls are atoms, and the box is now a crystal lattice.
In this structure, each atom is positioned at the corner of a cube. All atoms in a simple cubic lattice touch each other along the cube's edges. Here's what to remember:

• The cube's edge length equals twice the radius of the atom because they are touching.
• This is an efficient way to pack atoms, but not the most efficient compared to other cubic structures like body-centered or face-centered cubic lattices.

When studying simple cubic structures, remember that these have very low packing efficiency and are not as common in nature because atoms prefer more stable, tightly packed arrangements.

Crystal Packing Efficiency

When dealing with crystals, packing efficiency measures how tightly atoms are packed in a structure. It helps to understand how much space inside a crystal structure is actually occupied by atoms versus empty voids.

Packing efficiency is often expressed as a percentage. For simple cubic structures, the packing efficiency is rather low. Here's why:

• Atoms in a simple cubic arrangement occupy roughly only 52.4% of the total volume of the cube.
• This means there is a lot of empty space, leading to less stability and less dense materials compared to other structures.

Keep in mind that while simple cubic structures are useful for learning about crystal lattices, they are not typically the most stable or efficient structures in the real world.

Coordination Number in Crystals

The coordination number refers to the number of atoms directly surrounding a central atom in a crystal structure.
In simple cubic structures, the coordination number is an essential concept:

• For simple cubic structures, each atom is in direct contact with 6 other atoms. Hence, the coordination number is 6.
• This lower coordination number contributes to lower stability and packing efficiency.

Higher coordination numbers correlate with more closely packed structures and improved stability. While the simple cubic structure offers a clear understanding of basic lattice arrangements, its low coordination number is a trade-off between simplicity and efficiency.