Question

A solid XY has \(\mathrm{NaCl}\) structure. If radius of \(\mathrm{X}^{+}\)is 100 \(\mathrm{pm}\), the radius of \(\mathrm{Y}\) - ion is (a) \(136.6\) to \(241.6 \mathrm{pm}\) (b) \(341.6\) to \(134.6 \mathrm{pm}\) (c) \(128 \mathrm{pm}\) (d) \(136.8 \mathrm{pm}\)

Short answer

The radius of \( \mathrm{Y}^- \) lies between 136.6 and 241.6 pm. (Option a)

Step-by-step solution

Understand the NaCl Structure

In the NaCl crystal structure, the cations \( \mathrm{X}^{+} \) and anions \( \mathrm{Y}^{-} \) are arranged in a face-centered cubic lattice. The cation typically fits into the octahedral voids created by the anions.

Relationship of Ionic Radii in NaCl Structure

For an NaCl-type structure, the sum of the radii of the cation and anion equals the edge length of the cubic unit cell divided by \( \sqrt{2} \). This is because the ions touch along the face diagonal.

Apply Ideal Radius Ratio Condition

The ideal radius ratio \( r^+ / r^- \) for a cation in an octahedral void is between 0.414 to 0.732 to form a stable structure without distortion. We set \( r^{+} = 100 \) pm.

Calculate Possible Radius of \( Y^- \)

Using the condition from Step 3, calculate the radius of \( \mathrm{Y}^{-} \): \[0.414 = \frac{100}{r^-} \quad \Rightarrow \quad r^- = \frac{100}{0.414} \approx 241.6 \, \text{pm}\]\[0.732 = \frac{100}{r^-} \quad \Rightarrow \quad r^- = \frac{100}{0.732} \approx 136.6 \, \text{pm}\]Thus, the radius of \( \mathrm{Y}^- \) should be in the range of 136.6 pm to 241.6 pm.

Key concepts

NaCl Structure

The NaCl structure is a common crystalline arrangement found in many ionic compounds. Each sodium ion, commonly represented as \( \mathrm{Na}^+ \), pairs with a chloride ion, noted as \( \mathrm{Cl}^- \), creating a lattice that maximizes attraction and minimizes repulsion among ions. This arrangement is described as face-centered cubic (FCC), where each ion is surrounded by six ions of opposite charge, forming octahedral cation sites.

In simpler terms, imagine the structure as a three-dimensional grid. At each corner and center of each face of the cube, a chloride ion sits, while the sodium ions occupy the gaps in between, known as voids. Calculating the size of these ions becomes essential, as it determines how well they fit into the structure. A successful NaCl structure requires perfectly fitting ions so that the crystal remains stable.

In a NaCl crystal, the size of ions is crucial because it affects how well they can nestle into voids left by their neighboring ions. Understanding this geometry is the first step in calculating ionic radii and stabilizing crystal structures.

Cation-Anion Interaction

In ionic crystals such as \({\text{NaCl}}\), the interaction between cations and anions is fundamental. Cations are positively charged ions—like \( \mathrm{X}^+ \) in our discussion—and anions, like \( \mathrm{Y}^- \), hold a negative charge.

These interactions are primarily governed by electrostatic forces: oppositely charged ions attract each other strongly, forming the backbone of the ionic compound's structure. This balance of electrostatic forces is crucial because it stabilizes the structure. If the ionic sizes were inappropriate, like if cations were too large or anions too small, the crystal could become unstable.

In the NaCl structure, the ionic radii align such that each cation fits neatly into the void left by surrounding anions without causing too much distortion to the lattice's shape. It's comparable to a jigsaw puzzle where each piece must fit without gaps for the image to appear complete and stable. Hence, understanding cation-anion interactions helps us predict and explain the physical properties of the salt and its stability.

Radius Ratio Condition

The concept of the radius ratio condition plays a critical role in determining the structural stability of ionic compounds. This condition defines the allowable size relationship between the cation and anion in a stable crystal structure. Specifically, for a cation in an octahedral void—as in the NaCl structure—the ideal radius ratio \( \frac{r^+}{r^-} \) varies from 0.414 to 0.732.

If the radius ratio deviates significantly outside of this range, the structure may experience distortions or even become unstable due to insufficient or excessive packing of the ions. For instance, if the cation is too small compared to the anion, it may not sufficiently fill the voids, leading to instability. Conversely, a too-large cation may push the anions apart, disrupting the lattice.

• This condition ensures that ions are optimally packed, maintaining both strength and stability within the crystal.
• Accurate measurement of ionic radii is crucial, as it informs chemists whether the structural arrangement is likely to be durable.

By adhering to the radius ratio condition, scientists can predict and control the formation of stable crystalline structures in a variety of compounds, not just NaCl.