Problem 41 - Chemistry: An Atoms-Focused Approach

Question

Vanadium and carbon form vanadium carbide, an interstitial alloy. Given the atomic radii of \(\mathrm{V}(135 \mathrm{pm})\) and \(\mathrm{C}(77 \mathrm{pm}),\) which holes in a cubic closest-packed array of vanadium atoms do you think the carbon atoms are more likely to occupy-octahedral or tetrahedral?

Step-by-Step Solution

Answer

Answer: Carbon atoms are more likely to occupy tetrahedral holes.

Step 1: Calculate the octahedral hole size

The size of the octahedral hole in a cubic closest-packed structure can be calculated as: Octahedral hole size = \(\sqrt{2}\) x radius of a vanadium atom Using the given atomic radius of vanadium (135 pm), the calculation becomes: Octahedral hole size = \(\sqrt{2}\) x 135 pm ≈ 190.92 pm

Step 2: Calculate the tetrahedral hole size

The size of the tetrahedral hole in a cubic closest-packed structure can be calculated as: Tetrahedral hole size = \(\sqrt{3}/2\) x radius of a vanadium atom Using the given atomic radius of vanadium (135 pm), the calculation becomes: Tetrahedral hole size = \(\sqrt{3}/2\) x 135 pm ≈ 116.88 pm

Step 3: Compare hole sizes with carbon's atomic radii

Now that we have both the octahedral and tetrahedral hole sizes, we will compare them to the atomic radius of carbon (77 pm) given in the exercise: Octahedral hole size = 190.92 pm Tetrahedral hole size = 116.88 pm Carbon's atomic radius = 77 pm

Step 4: Determine the hole in which the carbon atom fits

Comparing the hole sizes with the atomic radius of carbon, we can see that the carbon atom is too small to occupy the octahedral hole as it's atomic radius (77 pm) is smaller than the octahedral hole size (190.92 pm). But, the carbon atom fits well into the tetrahedral hole as its atomic radius (77 pm) is closer to the size of the tetrahedral hole (116.88 pm). Therefore, it is more likely that the carbon atoms will occupy the tetrahedral holes in a cubic closest-packed array of vanadium atoms.

Key Concepts

Octahedral Holes

When atoms are packed in a three-dimensional structure, spaces or "holes" are created between them. In a cubic closest-packed (ccp) structure, octahedral holes are one kind of space found between the atoms. These holes have a symmetrical, eight-sided shape like an octahedron.
To imagine an octahedral hole, picture two stacked triangular pyramids. These can fit in the spaces surrounded by six atoms in the ccp structure.

• Each unit cell in a ccp structure has four octahedral holes.
• These holes are relatively large, making them ideal for larger atoms to fit inside.

To calculate the size of these holes, we multiply the atomic radius by \(\sqrt{2}\). This formula shows the relationship between the vanadium atoms and the space they leave in the structure. In our exercise, this formula helped us find the octahedral hole's size was approximately 190.92 pm, which is too large for a carbon atom with a radius of 77 pm.

Tetrahedral Holes

Tetrahedral holes are another type of space created in a ccp structure. These are smaller than octahedral holes. To picture a tetrahedral hole, think of a four-faced pyramid – a tetrahedron. This space is formed when four atoms come together in a ccp structure.
Unlike octahedral holes, tetrahedral holes are smaller and found in different locations within the unit cell.

• Each unit cell in a ccp structure contains eight tetrahedral holes.
• These smaller holes can better accommodate smaller atoms or ions.

Calculating the size of tetrahedral holes involves multiplying the atomic radius by \(\sqrt{3}/2\). Using this formula, we determined the hole's size as approximately 116.88 pm. Within this exercise, we found that carbon atoms fit more snugly into these holes than into the larger octahedral holes.

Cubic Closest-Packed Structure

The cubic closest-packed (ccp) structure is a common and efficient way that atoms stack together. It is also called face-centered cubic (fcc) because one atom is present on each face of a cube.
This arrangement aims to minimize empty space, allowing atoms to pack tightly together.

• Each atom in a ccp structure is surrounded by 12 others, offering high stability.
• The structure develops a repeating pattern, filling space in a uniform manner.

The ccp structure naturally creates both octahedral and tetrahedral holes. These vacancies allow other kinds of atoms or ions, like carbon in our vanadium carbide example, to fit into the ccp arrangement when forming alloys. Understanding these spaces and how they work is essential when exploring materials like interstitial alloys.