Problem 46 - Chemistry: An Atoms-Focused Approach

Question

A number of crystalline transition metals (including titanium, zirconium, and hafnium) can store hydrogen as metal hydrides for use as fuel in a hydrogen- powered vehicle. Are the H atoms (radius \(37 \mathrm{pm}\) ) more likely to be in the tetrahedral or octahedral holes of these three metals whose atomic radii are \(147,160,\) and \(159 \mathrm{pm},\) respectively?

Step-by-Step Solution

Answer

Answer: The hydrogen atoms are more likely to be found in the tetrahedral holes of titanium, zirconium, and hafnium.

Step 1: Understanding Tetrahedral and Octahedral Holes

In a crystal lattice, there are empty spaces between the atoms where other smaller atoms can fit. These empty spaces are called holes. The tetrahedral hole occurs when an atom is surrounded by four other atoms in a tetrahedral arrangement. The octahedral hole occurs when an atom is surrounded by six other atoms in a octahedral arrangement.

Step 2: Determining Radius Ratios

The radius ratio is defined as the ratio of the radius of the smaller atom to the larger atom. The radius ratios for tetrahedral and octahedral holes should lie within a certain range for the smaller atom to fit comfortably within the hole. - Tetrahedral hole: The ideal radius ratio range is \(0.225\) to \(0.414\). - Octahedral hole: The ideal radius ratio range is \(0.414\) to \(0.732\).

Step 3: Given Atomic Radii

We are given the radius of H atoms, which is \(37 \mathrm{pm}\). The atomic radii of the three transition metals are as follows: 1. Titanium (Ti): \(147 \mathrm{pm}\) 2. Zirconium (Zr): \(160 \mathrm{pm}\) 3. Hafnium (Hf): \(159 \mathrm{pm}\)

Step 4: Calculating Radius Ratios for the Transition Metals

Now, we will calculate the radius ratios for each transition metal and hydrogen atom: 1. Titanium (Ti): - Radius ratio \(= \frac{37}{147} \approx 0.2517\) 2. Zirconium (Zr): - Radius ratio \(= \frac{37}{160} \approx 0.2313\) 3. Hafnium (Hf): - Radius ratio \(= \frac{37}{159} \approx 0.2327\)

Step 5: Comparing Radius Ratios and Determining Lattice Position

Comparing the calculated radius ratios with the ideal radius ratio ranges for tetrahedral and octahedral holes, we observe that all the radius ratios fall within the range for tetrahedral holes (\(0.225\) to \(0.414\)). Therefore, the hydrogen atoms are more likely to be found in the tetrahedral holes of titanium, zirconium, and hafnium.

Key Concepts

Metal Hydrides

Transition metals like titanium, zirconium, and hafnium have the ability to absorb hydrogen, forming compounds known as metal hydrides. These compounds are crucial in hydrogen storage for applications, such as fueling hydrogen-powered vehicles.

When these metals form hydrides, they trap hydrogen atoms within the crystal lattice structure. This storage is facilitated by the holes or voids created by the arrangement of the metal atoms. The ability of these metals to store hydrogen effectively makes them an essential component in the advancement of clean energy technology.

The process of forming metal hydrides involves a balance of temperature, pressure, and hydrogen concentration. This reversible process allows hydrogen to be adsorbed and released as needed, making it a practical option for energy storage and consumption.

Tetrahedral Holes

In a crystal lattice, the tetrahedral hole is one of the two most common types of voids where smaller atoms like hydrogen can reside.

These holes are formed when four larger atoms come together, creating a space in the shape of a tetrahedron.
Tetrahedral holes are relatively smaller compared to octahedral holes, which makes them suitable for smaller atomic radii.

The radius ratio needed for an atom to fit into a tetrahedral hole is between 0.225 and 0.414.

• Atoms within this range can fit snugly, allowing for stable compound formation.
• The crystal structure determines the availability and the localization of these holes.

For titanium, zirconium, and hafnium, the radius ratios of hydrogen fall perfectly into the tetrahedral range, indicating that these metals naturally store hydrogen in these types of holes.

Octahedral Holes

Octahedral holes arise in a crystal lattice at the intersection of six surrounding larger atoms. They form when these atoms arrange themselves in an octahedral geometry.

Compared to tetrahedral holes, octahedral holes are larger and can accommodate slightly bigger atoms.

To effectively fit an atom into an octahedral hole, the radius ratio should be between 0.414 and 0.732.

• This broader range allows for more flexibility in terms of atom size.
• Octahedral holes are crucial in ionic compounds where larger ions are placed between smaller counterions in the lattice.

However, with a hydrogen radius of 37 pm, none of the calculated radius ratios with titanium, zirconium, or hafnium fall into the required range for octahedral holes.
Therefore, hydrogen is more likely to occupy tetrahedral holes in these metals.