Problem 106 - Chemistry: An Atoms-Focused Approach

Question

The mixed metal oxide \(\operatorname{LiMnTiO}_{4}\) has a structure with cubic closest-packed oxide ions and metal ions in both octahedral and tetrahedral holes. Which metal ion is most likely to be found in the tetrahedral holes? The ionic radii of \(\mathrm{Li}^{+}, \mathrm{Mn}^{3+}, \mathrm{Ti}^{4+},\) and \(\mathrm{O}^{2-}\) are \(76,67,60.5,\) and \(140 \mathrm{pm}\) respectively.

Step-by-Step Solution

Answer

Short Answer: The metal ion most likely to be found in the tetrahedral holes in the LiMnTiO₄ structure is Ti⁴⁺.

Step 1: Calculate the radius ratios for octahedral and tetrahedral holes

In ccp structures, the octahedral holes have a coordination number of 6, and the tetrahedral holes have a coordination number of 4. Based on these coordination numbers, we can calculate the critical radius ratios, called the limiting radius ratios, for the ions to fit into those holes.\newline For an octahedral hole:class(Li,o) - class(Li,c)}, \partial r_{octahedralhole} = \frac{r_c}{\sqrt2} where r_c is the radius of the larger ion (cation), then, the radius ratio is: \frac{r_{anion}}{r_{octahedralhole}} = \frac{r_c}{r_c\sqrt2}= \frac{\sqrt2}{2} = 0.414\newline For a tetrahedral hole:class(Li,o) - class(Li,c)}, \partial r_{tetrahedralhole} = \frac{r_c}{\sqrt{2(2-\sqrt{2})}} where r_c is the radius of the larger ion (cation), then, the radius ratio is: \frac{r_{anion}}{r_{tetrahedralhole}} = \frac{r_c}{r_c\sqrt{2(2-\sqrt{2})}}= \frac{\sqrt{2(2-\sqrt{2})}}{\sqrt{2}} = 0.225 Now, we have the limiting radius ratios for octahedral (0.414) and tetrahedral (0.225) holes.

Step 2: Calculate the ionic radius ratios

We will calculate the radius ratio for each metal ion (\(\frac{r_{cation}}{r_{anion}}\)): \(\frac{r_{Li+}}{r_{O^{2-}}} =\frac{76}{140} = 0.543\) \(\frac{r_{Mn^{3+}}}{r_{O^{2-}}} =\frac{67}{140} = 0.479\) \(\frac{r_{Ti^{4+}}}{r_{O^{2-}}} =\frac{60.5}{140} = 0.432\)

Step 3: Compare the ionic radii ratios to the hole radius ratios

Now, we need to compare the calculated radius ratios of the metal ions with the limiting radius ratios of the holes: - Li⁺ (0.543) is close to the octahedral hole radius ratio (0.414). Therefore, Lithium is most likely in the octahedral holes. - Mn³⁺ (0.479) is also close to the octahedral hole radius ratio (0.414) and not close to the tetrahedral hole ratio (0.225). Therefore, Manganese is most likely in the octahedral holes as well. - Ti⁴⁺ (0.432) is closer to the octahedral hole radius ratio (0.414) than the tetrahedral hole ratio (0.225). However, since all other metal ions already occupying the octahedral holes, Titanium would mainly occupy the tetrahedral holes.

Step 4: Conclusion

Based on the analysis, the metal ion most likely to be found in the tetrahedral holes in the LiMnTiO₄ structure is Ti⁴⁺.

Key Concepts

Ionic Radii

In chemistry, the concept of ionic radii is essential for understanding the arrangement of ions in solid structures. Ionic radii refer to the size of an ion within a crystal lattice. This size is crucial in determining how ions fit in various structural sites, such as octahedral and tetrahedral holes. Understanding ionic radii aids in predicting how ions will interact and bond in solid structures.

The ionic radius depends on the ionic charge and the ion's environment. Ions with higher positive charges tend to have smaller radii due to increased nuclear attraction. In the problem at hand, the ionic radii provided for \( ext{Li}^+\), \( ext{Mn}^{3+}\), \( ext{Ti}^{4+}\), and \( ext{O}^{2-}\) were used to determine their most likely positions in the crystal lattice.

• \( ext{Li}^+\) has an ionic radius of 76 pm.
• \( ext{Mn}^{3+}\) has an ionic radius of 67 pm.
• \( ext{Ti}^{4+}\) has an ionic radius of 60.5 pm.
• \( ext{O}^{2-}\) has an ionic radius of 140 pm.

Each of these ions must find its proper place in the structure, influenced by its size.

Cubic Closest-Packed Structure

The cubic closest-packed (ccp) structure, also known as face-centered cubic (fcc), is a common type of crystal structure. In these structures, atoms are densely packed, allowing for efficient use of space. The arrangement forms a repeating pattern where atoms occupy face-centered positions on a cube.

In a cubic closest-packed structure, certain voids or "holes" exist where smaller ions can be situated. These holes include octahedral and tetrahedral holes. The packing efficiency of the ccp structure makes it a common arrangement for various compounds, including solid metals and complex oxides like \( ext{LiMnTiO}_4\). Understanding how these holes can be occupied by different ions is essential for determining the overall structure of the compound. The ccp structure's characteristic allows for a high coordination number, which directly affects the way ions are situated within it.

Coordination Number

Coordination number is a term used to describe the number of atoms or ions directly surrounding a central atom or ion in a complex or crystal structure. It indicates how many adjacent particles are in contact with a particular ion.

For \( ext{LiMnTiO}_4\), coordination numbers are critical for determining which ions fill specific structural holes:

• **Octahedral Holes**: These have a coordination number of 6, meaning the central ion is surrounded by 6 neighboring ions.
• **Tetrahedral Holes**: These have a coordination number of 4, with the central ion surrounded by 4 neighbors.

By knowing the coordination number, we can predict which ions will fit into these holes based on their radius ratios. Larger ions tend to fit more comfortably into higher coordination environments like octahedral holes, as seen with \( ext{Li}^+\) and \( ext{Mn}^{3+}\). Titanium ions, by contrast, preferentially occupy tetrahedral holes due to their size, even when their coordination potential might differ.

Radius Ratio Method

The radius ratio method is a useful tool in determining how ions fit into crystal lattice structures. It involves calculating the ratio of the radius of a cation to that of an anion, which helps predict the coordination environment in which a specific ion will fit.

For example, in our problem, the radius ratios for the ions were:

• \(rac{r_{ ext{Li}^+}}{r_{ ext{O}^{2-}}} = 0.543\)
• \(rac{r_{ ext{Mn}^{3+}}}{r_{ ext{O}^{2-}}} = 0.479\)
• \(rac{r_{ ext{Ti}^{4+}}}{r_{ ext{O}^{2-}}} = 0.432\)

The method helps in identifying which ions are more suited to occupy specific holes based on previously calculated critical radius ratios:

• **Octahedral**: \( ext{Ratio} hickapprox 0.414\).
• **Tetrahedral**: \( ext{Ratio} hickapprox 0.225\).

Based on these comparisons, ions closer to the octahedral ratio are expected to fill those roles, while those needing smaller ratios opt for tetrahedral holes. That is why \( ext{Ti}^{4+}\), even with a ratio closer to the octahedral value, may still occupy tetrahedral holes when space constraints and the availability of sites are considered.