Platinum nanoparticles of diameter 2 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 Å. (b) Estimate how many platinum atoms are on the surface of a 2.0-nm Pt sphere, using the surface area of a sphere 14pr22 and assuming that the 'footprint' of one Pt atom can be estimated from its atomic diameter of 2.8 A .

Verified step by step guidance

1. Calculate the surface area of the platinum sphere using the formula for the surface area of a sphere: \( A = 4\pi r^2 \). Convert the diameter of the sphere from nanometers to angstroms to match the units of the atomic diameter.
2. Determine the 'footprint' area of one platinum atom. Since the atomic diameter is given as 2.8 Å, the area occupied by one atom can be approximated as a circle with radius 1.4 Å. Use the formula for the area of a circle: \( A = \pi r^2 \).
3. Divide the total surface area of the sphere by the area of one platinum atom to estimate the number of platinum atoms on the surface of the sphere.
4. Consider the arrangement of atoms on the surface. Since platinum crystallizes in a face-centered cubic structure, the surface atoms may not be packed as tightly as in the bulk. Adjust the estimate if necessary, considering the packing efficiency.
5. Review the assumptions made, such as the spherical shape and uniform distribution of atoms, to ensure the estimate is reasonable.

Key Concepts

Surface Area of a Sphere

The surface area of a sphere is calculated using the formula 4πr², where r is the radius. This concept is crucial for determining how much area is available for atoms to occupy on the surface of a nanoparticle. In this case, the radius of the platinum sphere must be converted from nanometers to centimeters or meters to ensure consistency in units when performing calculations.

Atomic Diameter and 'Footprint'

The atomic diameter of an element, such as platinum, represents the effective size of an atom when considering its arrangement in a solid. The 'footprint' of an atom refers to the area it occupies on the surface of a material. For platinum, with an atomic diameter of 2.8 Å, this footprint can be approximated as a circle with a radius equal to half the atomic diameter, which is essential for estimating how many atoms fit on the surface of the nanoparticle.

Nanoparticle Catalysis

Nanoparticles, like platinum nanoparticles, exhibit unique catalytic properties due to their high surface area-to-volume ratio. This increased surface area allows for more active sites for chemical reactions, such as the oxidation of carbon monoxide to carbon dioxide. Understanding the role of nanoparticles in catalysis is vital for applications in environmental chemistry and industrial processes, as they can significantly enhance reaction rates.