Problem 105 - Chemistry: An Atoms-Focused Approach

Question

The ceramic material barium titanate \(\left(\mathrm{BaTiO}_{3}\right)\) is used in devices that measure pressure. The radii of \(\mathrm{Ba}^{2+}, \mathrm{T} \mathrm{i}^{4+},\) and \(\mathrm{O}^{2-}\) are \(135,60.5,\) and \(140 \mathrm{pm},\) respectively. If the \(\mathrm{O}^{2-}\) ions are in a closest-packed structure, which hole(s) can accommodate the metal cations?

Step-by-Step Solution

Answer

Answer: In BaTiO₃, Ba²⁺ ions can fit in cubic holes, while Ti⁴⁺ ions can fit in octahedral holes within the closest-packed structure of O²⁻ ions.

Step 1: Calculate the radius ratios

To find out which hole(s) can accommodate the metal cations, we need to calculate the radius ratios for both Ba²⁺ and Ti⁴⁺ ions. The radius ratio is defined as the ratio of the radius of the cation to the radius of the anion. For Ba²⁺: Radius ratio = \(\frac{r_{Ba^{2+}}}{r_{O^{2-}}}\) For Ti⁴⁺: Radius ratio = \(\frac{r_{Ti^{4+}}}{r_{O^{2-}}}\)

Step 2: Input the given values

Substitute the given values of the radii of Ba²⁺, Ti⁴⁺, and O²⁻ ions: For Ba²⁺: Radius ratio = \(\frac{135}{140}\) For Ti⁴⁺: Radius ratio = \(\frac{60.5}{140}\)

Step 3: Calculate the radius ratios

Compute the radius ratios for both metal cations: For Ba²⁺: Radius ratio = \(\frac{135}{140} = 0.964\) For Ti⁴⁺: Radius ratio = \(\frac{60.5}{140} = 0.432\)

Step 4: Determine which hole(s) can accommodate the metal cations

Now, we can compare the calculated radius ratios to the guidelines for the hole structures: 1. Tetrahedral holes: 0.225 < radius ratio < 0.414 2. Octahedral holes: 0.414 < radius ratio < 0.732 3. Cubic holes: 0.732 < radius ratio < 1 For Ba²⁺: The radius ratio is 0.964, which falls within the range for cubic holes. Therefore, the Ba²⁺ ions can be accommodated in cubic holes. For Ti⁴⁺: The radius ratio is 0.432, which falls within the range for octahedral holes. Therefore, the Ti⁴⁺ ions can be accommodated in octahedral holes. So, in Barium Titanate (BaTiO₃), Ba²⁺ ions can fit in cubic holes, while Ti⁴⁺ ions can fit in octahedral holes within the closest-packed structure of O²⁻ ions.

Key Concepts

Barium Titanate

Barium titanate, with the chemical formula \(\text{BaTiO}_3\), is a key material in the realm of advanced ceramics. It stands out due to its extensive use in electronic devices, especially as a dielectric material in capacitors and as a pressure sensor in transducers. The interesting part about barium titanate is its crystal structure that enables its unique properties, such as ferroelectricity. This means that its electrical polarization can be reversed by an external electric field, making it incredibly useful in various applications.

This ceramic is a perovskite structure, which is a type of crystal structure that has the general formula \(\text{ABX}_3\). Here, 'A' and 'B' are two different cations, and 'X' is an anion that connects them. In barium titanate, barium (Ba) and titanium (Ti) are the cations, while oxygen (O) acts as the anion.

The arrangement of these ions within the crystal lattice plays a crucial role in the material's properties. Understanding which hole or void each ion occupies in the lattice helps in comprehending how the material will behave under various conditions.

Ionic Radii

When discussing barium titanate, it's important to consider the ionic radii of its components: \(\text{Ba}^{2+}\), \(\text{Ti}^{4+}\), and \(\text{O}^{2-}\). Ionic radius is essentially the effective distance from the center of an ion to the outermost electron, and it's a critical factor in determining how ions fit together in a crystal structure.

In barium titanate, the ionic radii are given as:

• \(\text{Ba}^{2+}: 135\text{ pm}\)
• \(\text{Ti}^{4+}: 60.5\text{ pm}\)
• \(\text{O}^{2-}: 140\text{ pm}\)

These measurements indicate the relative sizes of the ions, which helps in predicting how they will pack together. Larger ions like \(\text{Ba}^{2+}\) will generally occupy larger voids or holes in the crystal lattice than smaller ions like \(\text{Ti}^{4+}\).

This understanding of ionic radii aids in the analysis of how these ions can be organized in a stable crystal structure, influencing the properties of the material.

Radius Ratio

The concept of the radius ratio is vital in identifying the type of void each ion fits into within a crystal. It is calculated as the ratio of the cation's radius to the anion's radius. With barium titanate, we focus on the radius ratios for \(\text{Ba}^{2+}\) and \(\text{Ti}^{4+}\) concerning the \(\text{O}^{2-}\) ion.

Here's how the radius ratios are calculated:

• For \(\text{Ba}^{2+}: \frac{135}{140} = 0.964\)
• For \(\text{Ti}^{4+}: \frac{60.5}{140} = 0.432\)

The resulting numbers tell us about the fitting of these cations into the lattice:

* A radius ratio of \(0.964\) indicates that \(\text{Ba}^{2+}\) fits into cubic holes.
* A radius ratio of \(0.432\) suggests \(\text{Ti}^{4+}\) can fit into octahedral holes.

This analysis of radius ratios helps predict the stability and type of crystal structure these ions will collectively form when together in a material.

Cation Accommodation

In crystal structures, accommodating cations within an anion framework involves considering how these cations fit into available space. The efficiency and stability of a crystal depend largely on how well the cations are accommodated by the anions they associate with.

In the case of barium titanate, which involves a closest-packed array of \(\text{O}^{2-}\) ions, the cations \(\text{Ba}^{2+}\) and \(\text{Ti}^{4+}\) must find appropriate "holes"—or voids—in this framework. These voids can be of different types based on their shape and size:

• Tetrahedral holes: Generally smaller and can accommodate smaller cations.
• Octahedral holes: Intermediate in size, suitable for medium-sized cations.
• Cubic holes: Largest, suitable for larger cations like \(\text{Ba}^{2+}\).

Therefore, \(\text{Ba}^{2+}\) ions, given their radius ratio of \(0.964\), are best suited for cubic holes. Meanwhile, \(\text{Ti}^{4+}\) ions, with a radius ratio of \(0.432\), can be accommodated within octahedral holes.

This type of structural analysis is crucial for determining how these cations contribute to the overall properties and functionality of the material, as their positioning affects the way in which they interact and bond with other ions.