Final answer:
The percentage of atoms on the surface of a cubic nanoparticle with an edge length of 8.0 nm and atomic diameter of 250 pm is approximately 8.51%.
Step-by-step explanation:
To calculate the percentage of atoms on the surface of a cubic nanoparticle, first, we need to understand that the surface atoms are those that do not contribute to the interior volume of the particle. The cubic nanoparticle has a total of eight corners, each shared by eight nanoparticles, meaning each corner contributes 1/8 of an atom to the surface. Additionally, each edge of the nanoparticle has atoms that, aside from the corners, are shared by four nanoparticles and therefore contribute half an atom each to that specific nanoparticle. Also, each face atom is shared by two nanoparticles.
Given that the particle edge length is 8.0 nm (8000 pm), and the atomic diameter is 250 pm, each side of the cube would accommodate 32 atoms (8000 pm / 250 pm = 32 atoms per edge). Here is how you can calculate the number on the surface:
Atoms solely at corners: 8 corners × (1/8 contribution per corner) = 1 atom.
Atoms on edges excluding corners: 12 edges × (32 - 2) atoms/edge × 1/4 contribution = 90 atoms.
Atoms on faces excluding edges: 6 faces × (30 × 30) atoms/face × 1/2 contribution = 2700 atoms.
So, the total number of atoms on the surface is 1 (corners) + 90 (edges) + 2700 (faces) = 2791 atoms.
To find the total number of atoms in the nanoparticle, consider that the nanoparticle is filled with a simple cubic structure of atoms:
Total atoms in the cube: 32 atoms/edge × 32 atoms/edge × 32 atoms/edge = 32768 atoms.
The percentage of surface atoms can then be calculated using the formula:
Percentage of surface atoms = (Number of surface atoms / Total number of atoms) × 100
Plugging in the numbers: (2791 / 32768) × 100 ≈ 8.51%
Hence, approximately 8.51% of the atoms are on the surface of the nanoparticle.