Problem 113 - Chemistry: The Central Science

Question

Platinum nanoparticles of diameter \(\sim 2 \mathrm{~nm}\) are important catalysts in carbon monoxide oxidation to carbon dioxide. Platinum crystallizes in a face-centered cubic arrangement with an edge length of \(3.924 \AA\). (a) Estimate how many platinum atoms would fit into a \(2.0-\mathrm{nm}\) sphere; the volume of a sphere is \((4 / 3) \pi r^{3}\). Recall that \(1 \dot{A}=1 \times 10^{-10} \mathrm{~m}\) and \(1 \mathrm{~nm}=1 \times 10^{-9} \mathrm{~m}\). (b) Estimate how many platinum atoms are on the surface of a \(2.0\)-nm Pt sphere, using the surface area of a sphere \(\left(4 \pi r^{2}\right)\) and assuming that the "footprint" of one Pt atom can be estimated from its atomic diameter of \(2.8 \mathrm{~A}\). (c) Using your results from (a) and (b), calculate the percentage of \(\mathrm{Pt}\) atoms that are on the surface of a \(2.0-\mathrm{nm}\) nanoparticle. (d) Repeat these calculations for a \(5.0\)-nm platinum nanoparticle. (e) Which size of nanoparticle would you expect to be more catalytically active and why?

Step-by-Step Solution

Answer

For a 2.0-nm platinum nanoparticle, there are approximately 289 Pt atoms, with 133 atoms on the surface. About 46% of the atoms are on the surface. For a 5.0-nm platinum nanoparticle, there are around 4437 Pt atoms, with 622 atoms on the surface. Approximately 14% of the atoms are on the surface. Since catalysts are more active when more atoms are on the surface, the smaller 2.0-nm nanoparticle would be more catalytically active.

Step 1: (a) Calculate the number of Pt atoms in a 2.0-nm sphere

First, we will find the volume of the platinum nanoparticle using the given diameter. The volume of a sphere is given by \(\frac{4}{3}\pi r^3\), where r is the radius of the sphere. Next, calculate the volume of one platinum atom. To do this, we need to find the volume of the unit cell. Platinum has a face-centered cubic arrangement with an edge length of 3.924 Å. The volume of a unit cell is given by \(a^3\), where a is the edge length of the cell. Since a face-centered cubic unit cell contains 4 atoms, we can divide this volume by 4 to find the volume of one atom. Now, divide the volume of the nanoparticle by the volume of one platinum atom to estimate how many platinum atoms fit into a 2.0-nm sphere.

Step 2: (b) Calculate the number of Pt atoms on the surface of a 2.0-nm Pt sphere

First, find the surface area of the nanoparticle using the formula \(4\pi r^2\), where r is the radius of the sphere. Next, we need to find the "footprint" of one Pt atom. We can do this by assuming the atom's footprint is a square with side length equal to the atomic diameter (2.8 Å). Compute the area of one "footprint" and divide the surface area of the nanoparticle by the area of the footprint to estimate the number of platinum atoms present on the surface of the nanoparticle.

Step 3: (c) Calculate the percentage of Pt atoms that are on the surface of a 2.0-nm nanoparticle

Divide the number of surface Pt atoms (calculated in part b) by the total number of Pt atoms in the nanoparticle (calculated in part a) and multiply the result by 100 to get the percentage of surface atoms in the nanoparticle.

Step 4: (d) Repeat the calculations for a 5.0-nm platinum nanoparticle

Follow the same steps as done in parts (a), (b), and (c) with a nanoparticle of diameter 5.0 nm.

Step 5: (e) Determine which size nanoparticle would be more catalytically active and explain why

Compare the percentage of surface atoms of both 2.0-nm and 5.0-nm platinum nanoparticles. The nanoparticle with a higher percentage of surface atoms is expected to be more catalytically active. This is because surface atoms are more accessible, and thus more likely to be involved in reaction processes.

Key Concepts

Catalysis

Catalysis is a fascinating process where a substance, called a catalyst, speeds up a chemical reaction without being consumed. In this context, platinum nanoparticles act as catalysts for the conversion of carbon monoxide into carbon dioxide. This is an essential reaction, especially in catalytic converters in vehicles to reduce harmful emissions. The effectiveness of a catalyst often depends on its surface area. This is because reactions typically occur at the surface, where reactant molecules can interact directly with the catalyst.

• Platinum atoms at the surface of the nanoparticles provide sites for the CO molecules to bind and then react with oxygen to form CO₂.
• The unique properties of nanoparticles can significantly enhance catalysis efficiency because more surface atoms are available for reactions compared to larger particles.

Smaller nanoparticles tend to have a higher percentage of their atoms at the surface, making them more available for catalytic activity. This explains why a 2.0 nm particle might be more catalytically active than a 5.0 nm particle. The increased surface-to-volume ratio in smaller particles is crucial for boosting catalysis. Catalysts increase the rate of reaction; however, the reaction's tendency remains unchanged.

Surface Area Calculation

Calculating the surface area of platinum nanoparticles is essential for understanding their catalytic behavior. In the given exercise, the surface area of a spherical nanoparticle is determined using the formula for the surface area of a sphere:\[ 4\pi r^2 \]where \( r \) is the radius. For a 2.0 nm platinum nanoparticle:

• First, convert the diameter to radius by halving it, which results in 1.0 nm, or 10 Å.
• Then, using the formula, the surface area becomes \( 4\pi (10)^2 \approx 1257 \) square Ångströms.

Once the surface area is known, estimating the number of atoms on this surface involves calculating the footprint area of a single platinum atom. Assuming a square footprint,

• The side equals the atomic diameter, 2.8 Å, making the footprint area \( 2.8^2 \approx 7.84 \) square Ångströms.

By dividing the total surface area of the nanoparticle by this atomic footprint area, one can determine the approximate number of platinum atoms residing on the surface. This surface number is critical, as these atoms are key contributors to catalytic activity.

Face-Centered Cubic Structure

The face-centered cubic (FCC) structure is one of the most common and stable atomic arrangements in crystals, including platinum. In an FCC structure, atoms are located at each corner and the center of all cube faces, resulting in a highly packed arrangement. Platinum being FCC-structured implies:

• There are four atoms per unit cell.
• One atom at each of the 8 corners of the cube shares its position equally with adjoining cells, while 6 face-centered atoms share half.

The edge length of platinum's FCC unit cell is 3.924 Å, from which we can determine the volume of the unit cell by cubing this length:\[ a^3 \approx (3.924)^3 \]This allows us to calculate the volume occupied by a single platinum atom by dividing the cell volume by four (as an FCC unit cell contains four atoms). Understanding this structure helps in figuring out both the density and mass of atoms within a particle and plays a crucial role in determining how many atoms can fit into a given nanoparticle. Such a crystal arrangement in nanoparticulate form offers more active sites, enhancing the catalytic properties due to the increased surface availability of catalytic atoms.