Problem 113 - Chemistry The Central Science In Si Units

Question

Platinum nanoparticles of diameter \(-2 \mathrm{nm}\) are important catalysts in carbon monoxide oxidation to carbon dioxide. Platinum crystallizes in a face- centered cubic arrangement with an edge length of \(392.4 \mathrm{pm} .\) (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 \(1 \mathrm{pm}=1 \times 10^{-12} \mathrm{~m}\) and \(1 \mathrm{nm}=1 \times 10^{-9} \mathrm{~m} .\) (b) Esti- mate how many platinum atoms are on the surface of a \(2.0-\mathrm{nm}\) Pt sphere, using the surface area of a sphere \(\left(4 \pi r^{2}\right)\) and assuming that the "footprint" of one \(\mathrm{Pt}\) atom can be estimated from its atomic diameter of \(280 \mathrm{pm}\) (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 \(5.0-\mathrm{nm}\) platinum nanoparticle. (e) Which size of nanoparticle would you expect to be more catalytically active and why?

Step-by-Step Solution

Answer

The 2.0-nm nanoparticle has a higher percentage of surface atoms and is expected to be more catalytically active.

Step 1: Convert Units

First, convert the diameter of the nanoparticle to \((2.0 \, \text{nm})\) and \((5.0 \, \text{nm})\) to meters. Since 1 nm = \(1 \, \times \, 10^{-9} \, \text{m}\), the radius \(r\) for the \((2.0 \, \text{nm})\) nanoparticle is \(1.0 \, \text{nm} = 1.0 \, \times 10^{-9} \, \text{m}\), and for the \((5.0 \, \text{nm})\) it is \(2.5 \, \text{nm} = 2.5 \, \times 10^{-9} \, \text{m}\).

Step 2: Volume of the Nanoparticle

Calculate the volume of the sphere using \((V = \frac{4}{3} \pi r^3)\). For \((r = 1.0 \, \text{nm})\), \(V = \frac{4}{3} \pi (1.0 \, \times 10^{-9} \, \text{m})^3\). Similarly, calculate for \((r = 2.5 \, \text{nm})\).

Step 3: Convert Crystal Lattice Parameters

Convert the edge length of the face-centered cubic (FCC) crystal unit, \(392.4 \, \text{pm}\), to meters: \(392.4 \, \text{pm} = 392.4 \, \times 10^{-12} \, \text{m}\).

Step 4: Calculate the Volume of the Unit Cell

The volume of the FCC unit cell is \(a^3\), where \(a = 392.4 \, \text{pm}\). Convert it to meters and calculate: \( (392.4 \, \times 10^{-12} \, \text{m})^3\).

Step 5: Calculate Number of Atoms per Unit Cell

In a face-centered cubic unit cell, there are 4 platinum atoms per unit cell due to the atom arrangement within the lattice structure.

Step 6: Determine Volume per Atom

Divide the volume of the unit cell by the number of atoms per unit cell to find the volume occupied by each atom.

Step 7: Estimate Number of Atoms in the Nanoparticle

Divide the volume of the nanoparticle by the volume occupied by a single atom calculated in Step 6 for both nanoparticle sizes.

Step 8: Surface Area and Surface Atoms

Using the surface area formula \((4 \pi r^2)\), calculate the surface area of the nanoparticle. Determine the number of atoms on the nanoparticle surface by dividing the surface area by the area occupied by one platinum atom (using atomic diameter).

Step 9: Calculate Surface Atoms Percentage

For both nanoparticle sizes, calculate the percentage of atoms on the surface using \(\frac{\text{Number of surface atoms}}{\text{Total number of atoms}} \times 100\).

Step 10: Assess Catalytic Activity

Compare the results of the surface-to-volume atom percentages. Nanoparticles with higher percentage of surface atoms are expected to be more catalytically active due to more available active sites on the surface.

Key Concepts

Face-centered cubic

Platinum, like many metals, crystallizes in a specific arrangement of atoms called a face-centered cubic (FCC) structure. This cubic structure is unique because of the way platinum atoms are arranged within it.
Each cube in this arrangement has an atom at each corner and one at the center of each face of the cube.
This results in a total of four platinum atoms per unit cell because each corner atom is shared among eight cubes and each face atom is shared among two.
This efficient packing contributes to platinum's high density and stability. Understanding this crystal structure is key to determining how many atoms can fit into a nanoparticle of a particular size.
It's also critical in calculating how such structures affect the properties and reactivity of platinum nanoparticles.

Catalysis

Catalysis is a process where the speed of a chemical reaction is increased by a substance called the catalyst, which isn't consumed in the reaction. Platinum nanoparticles are renowned for their exceptional catalytic properties.
They are highly effective in facilitating reactions like the oxidation of carbon monoxide to carbon dioxide. This reaction is an important process for reducing harmful emissions in catalytic converters of cars.
The effectiveness of platinum as a catalyst comes from its ability to provide a surface where reactants can be adsorbed and react more easily. Nanoparticles have a significant advantage in catalysis because their small size offers a high surface area for the same amount of material, making more active sites available for reactions.
This is why optimizing the size of the nanoparticles is crucial for improving catalytic efficiency.

Surface area to volume ratio

The surface area to volume ratio is a measure that describes how much surface area the exterior of an object has relative to its volume. This concept is especially important in the context of nanoparticles.
As the size of the nanoparticles decreases, the surface area to volume ratio increases substantially.
A higher ratio means that a larger proportion of atoms are present on the surface of the nanoparticle. For instance, a 2 nm platinum nanoparticle will have a much higher surface area to volume ratio compared to a 5 nm nanoparticle, meaning more platinum atoms are exposed on the surface.
In the context of catalysis, this elevated ratio allows for more sites for chemical reactions, boosting the nanoparticle's catalytic activity. Comparing different sizes, the 2 nm particles are expected to perform better in catalytic processes due to a greater proportion of surface atoms.

Nanoparticle size effect

The size of nanoparticles significantly affects their physical and chemical properties, a phenomenon known as the size effect. When considering platinum nanoparticles, as the size decreases to the nanoscale, distinct changes happen.
Properties like melting point, electronic structure, and catalytic capability shift dramatically compared to their bulk counterparts.
Smaller nanoparticles have a greater fraction of their atoms located on their surface.
This can result in enhanced reactivity and different electronic properties. For example, the catalytic activity of metal nanoparticles increases as their size decreases because more atoms are available to interact with reactants.
However, there is a balance to find since particles too small can become unstable. An understanding of this size effect is crucial for designing nanoparticles with desired properties for specific applications.
For platinum, selecting the appropriate nanoparticle size can make a significant difference in achieving optimal performance in industrial processes like catalysis.