Silver nanoparticles are embedded in clothing fabric to kill...

Silver nanoparticles are embedded in clothing fabric to kill odor-causing bacteria. Silver crystallizes in an fcc unit cell. Aow many unit cells are present in a cubic silver particle with an edge length of $25 \mathrm{nm} ?$ b. How many silver atoms are there in this particle? c. If there are $1360 \mu \mathrm{g}$ of silver per sock, how many of these particles does this mass correspond to?

Silver nanoparticles are embedded in clothing fabric to kill odor-causing bacteria. Silver crystallizes in an fcc unit cell.
Aow many unit cells are present in a cubic silver particle with an edge length of $25 \mathrm{nm} ?$
b. How many silver atoms are there in this particle?
c. If there are $1360 \mu \mathrm{g}$ of silver per sock, how many of these particles does this mass correspond to?

Key Concepts

Atomic Counting in Crystals

This concept involves using the known number of atoms per unit cell and the total number of unit cells in a particle to calculate the total number of atoms. It is essential in materials science to understand the atomic composition of nanoparticles based on their crystallographic structure.

Mass-to-Particle Conversion

This concept deals with converting a given mass of a substance into the number of particles or structures it contains, using known parameters such as the mass of a single particle, conversion between micrograms and grams, and possibly Avogadro’s number. It is central to quantitatively linking the mass of a material to its microscopic building blocks.

Unit Cell

A unit cell is the smallest repeating unit in a crystal lattice that, when repeated in all directions, creates the entire crystalline structure. Understanding the dimensions of the unit cell allows one to calculate the number of such cells in a larger crystal or particle, a key step in relating microscopic properties to macroscopic ones.

Face-Centered Cubic (FCC) Structure

This refers to a type of crystal lattice where atoms are arranged at each of the corners and the centers of all the faces of the cube, resulting in a highly symmetric and densely packed structure. In an FCC unit cell, there are four whole atoms per cell, a fact fundamental to computations involving atomic counts in crystalline materials.

Nanoparticle Volume Calculation

When dealing with nanoparticles, especially those with simple geometric shapes like cubes, the total volume can be calculated from the edge length. Dividing the particle's total volume by the volume of one unit cell determines the number of unit cells present in the nanoparticle, bridging the gap between nanoscale structure and bulk material properties.

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