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Ch.14 - Chemical Kinetics
Brown - Chemistry: The Central Science 14th Edition
Brown14th EditionChemistry: The Central ScienceISBN: 9780134414232Not the one you use?Change textbook
All textbooksBrown 14th EditionCh.14 - Chemical KineticsProblem 113c,d
Chapter 14, Problem 113c,d

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. (c) Using your results from (a) and (b), calculate the percentage of Pt atoms that are on the surface of a 2.0-nm nanoparticle. (d) Repeat these calculations for a 5.0-nm platinum nanoparticle.

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1
Determine the number of platinum atoms in a 2.0-nm nanoparticle by calculating the volume of the nanoparticle and using the density of platinum and Avogadro's number.
Calculate the number of atoms in a face-centered cubic (FCC) unit cell of platinum using the given edge length and the fact that there are 4 atoms per FCC unit cell.
Estimate the number of surface atoms by considering the geometry of the nanoparticle and the arrangement of atoms in the FCC structure.
Calculate the total number of atoms in the nanoparticle using the volume and density of platinum, and compare it to the number of surface atoms.
Determine the percentage of surface atoms by dividing the number of surface atoms by the total number of atoms and multiplying by 100.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Surface Area to Volume Ratio

The surface area to volume ratio is a critical concept in nanotechnology, particularly for nanoparticles. As the size of a particle decreases, its surface area increases relative to its volume, leading to a higher proportion of atoms being located on the surface. This is significant for catalytic processes, as surface atoms are often more reactive and play a crucial role in chemical reactions.
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Face-Centered Cubic (FCC) Structure

The face-centered cubic (FCC) structure is a type of crystal lattice arrangement where atoms are located at each corner and the centers of all the cube faces. In platinum, this arrangement allows for a specific calculation of atomic density and packing efficiency, which is essential for determining the number of atoms in a given volume and subsequently calculating the surface atoms in nanoparticles.
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Atomic Density and Nanoparticle Calculation

Atomic density refers to the number of atoms per unit volume in a material. For nanoparticles, calculating the atomic density is crucial for determining how many atoms are present in a given size. This information is used to find the total number of atoms in the nanoparticle and to assess how many of those are on the surface, which is vital for understanding their catalytic properties.
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Related Practice
Textbook Question

One of the many remarkable enzymes in the human body is carbonic anhydrase, which catalyzes the interconversion of carbon dioxide and water with bicarbonate ion and protons. If it were not for this enzyme, the body could not rid itself rapidly enough of the CO2 accumulated by cell metabolism. The enzyme catalyzes the dehydration (release to air) of up to 107 CO2 molecules per second. Which components of this description correspond to the terms enzyme, substrate, and turnover number?

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Textbook Question

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 (4πr2) and assuming that the 'footprint' of one Pt atom can be estimated from its atomic diameter of 2.8 A .

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Textbook Question

In a hydrocarbon solution, the gold compound (CH3)3AuPH3 decomposes into ethane (C2H6) and a different gold compound, (CH3)AuPH3. The following mechanism has been proposed for the decomposition of (CH3)3AuPH3:

Step 1: (CH3)3AuPH3 k1⇌k-1 (CH3)3Au + PH3 (fast)

Step 2: (CH3)3Au k2→ C2H6 + (CH3)Au (slow)

Step 3: (CH3)Au + PH3 k3→ (CH3)AuPH3 (fast)

(e) What is the rate law predicted by this mechanism?

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Textbook Question

In a hydrocarbon solution, the gold compound (CH3)3AuPH3 decomposes into ethane (C2H6) and a different gold compound, (CH3)AuPH3. The following mechanism has been proposed for the decomposition of (CH3)3AuPH3:

Step 1: (CH3)3AuPH3 k1⇌k-1 (CH3)3Au + PH3 (fast)

Step 2: (CH3)3Au k2→ C2H6 + (CH3)Au (slow)

Step 3: (CH3)Au + PH3 k3→ (CH3)AuPH3 (fast)

(f) What would be the effect on the reaction rate of adding PH3 to the solution of (CH3)3AuPH3?

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Textbook Question

Enzymes are often described as following the two-step mechanism:

E + S ⇌ ES (fast)

ES → E + P (slow)

where E = enzyme, S = substrate, ES = enzyme9substrate complex, and P = product.

(a) If an enzyme follows this mechanism, what rate law is expected for the reaction?