To calculate the stoichiometry and the number of atoms in a gold nanoparticle, we need to make a few assumptions:

1. Assume gold atoms in the nanoparticle form a face-centered cubic (fcc) lattice structure, which is common for bulk gold.
2. Assume the diameter of the gold nanoparticle is small enough so that it consists of only a single layer of atoms.

The fcc lattice structure has 4 atoms per unit cell. The number of unit cells in a nanoparticle can be approximated as:

Number of unit cells = (Number of atoms on the surface) / (Number of atoms per unit cell)

To find the number of atoms on the surface, we can calculate the total number of atoms in a spherical shell with an outer diameter of 5 nm and an inner diameter of 0 nm (since it has a single layer of atoms).

Number of atoms on the surface = 4π * (radius)^2 * density of atoms

Assuming the density of gold is 19.3 g/cm^3 and the atomic mass of gold is 196.97 g/mol, we can calculate the number of atoms in a gold nanoparticle.

1. Convert the diameter of the nanoparticle to radius:
Radius = diameter/2 = 5 nm / 2 = 2.5 nm = 2.5 * 10^-7 cm

2. Calculate the number of atoms on the surface:
Number of atoms on the surface = 4π * (radius)^2 * density of atoms
Number of atoms on the surface = 4π * (2.5 * 10^-7 cm)^2 * 19.3 g/cm^3 / (196.97 g/mol) * (6.022 * 10^23 atoms/mol)

3. Calculate the number of unit cells:
Number of unit cells = (Number of atoms on the surface) / (Number of atoms per unit cell)
Number of unit cells = Number of atoms on the surface / 4

4. Calculate the number of atoms in the nanoparticle:
Number of atoms = Number of unit cells * number of atoms per unit cell

Let's calculate the values:

Radius = 2.5 * 10^-7 cm
Number of atoms on the surface = 4π * (2.5 * 10^-7 cm)^2 * 19.3 g/cm^3 / (196.97 g/mol) * (6.022 * 10^23 atoms/mol)
Number of unit cells = Number of atoms on the surface / 4
Number of atoms = Number of unit cells * number of atoms per unit cell

Please note that the actual number of atoms in a gold nanoparticle could be different due to surface roughness, defects, and other factors.

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