one edge of a cube is measured and found to be 2.162 cm. An atom of copper is 0.2560 nm in diameter. how many atoms of copper are contained in the cube.
I would do this. I don't think it is an exact way but...
0.2560 nm = 2.56E-8 cm diameter so how many can be placed on an edge length. That will be 2.56E-8 cm x # atoms = 2.162 cm and solve for # atoms. Then cube that. I think that will over count because the end atom (and there are two ends) will be counted twice and the corner atoms will be counted three times.
To calculate the number of atoms of copper contained in the cube, we need to find the volume of the cube and then determine how many copper atoms can fit within that volume.
First, let's find the volume of the cube using the given edge length. The volume of a cube is calculated by cubing the length of one of its edges. So, in this case, the volume (V) of the cube can be calculated as:
V = (edge length)^3
V = (2.162 cm)^3
Now, let's perform the calculation:
V = 10.441057688 cm^3 (rounded to 10.441 cm^3)
Keep in mind that the unit of length used for the cube's edge length (cm) needs to be converted to the same unit used for the copper atom's diameter (nm). 1 cm is equal to 10^-7 nm, so we can convert the volume from cm^3 to nm^3.
Now, let's perform the conversion:
V = 10.441 cm^3 * (10^-7 nm/cm)^3
V = 10.441 * 10^-21 nm^3
Now, let's calculate how many copper atoms can fit within this volume.
The volume of a sphere (which approximates the shape of a copper atom) is given by the formula:
V = (4/3) * π * (radius)^3
The radius of a sphere is half of its diameter, so the radius of a copper atom (diameter = 0.2560 nm) can be calculated as:
radius = 0.2560 nm / 2
Now, let's perform the calculation:
radius = 0.1280 nm
The volume of a single copper atom can be calculated as:
V_atom = (4/3) * π * (0.1280 nm)^3
Now, let's perform the calculation:
V_atom = 0.2758914564 nm^3
Finally, we can find the number of copper atoms that can fit within the cube by dividing the volume of the cube by the volume of a single copper atom.
n_atoms = V / V_atom
Now, let's perform the calculation:
n_atoms = 10.441 nm^3 / 0.2758914564 nm^3
n_atoms ≈ 37.8458
Therefore, there are approximately 37.8458 atoms of copper contained in the cube.