To solve this problem, we need to calculate the volume of the silver core and the gold shell separately, then use the density and molar mass of each material to find the number of atoms.

Step 1: Calculate the volume of the silver core and the gold shell
The volume of a sphere is given by the formula V = 4/3 * π * r^3. The radius is half the diameter, so the radius of the silver core is 1 nm and the radius of the gold shell is 1.5 nm.

The volume of the silver core is V_silver = 4/3 * π * (1 nm)^3 = 4.19 nm^3.

The volume of the gold shell is the total volume minus the volume of the silver core. The total volume is V_total = 4/3 * π * (1.5 nm)^3 = 14.14 nm^3. So, the volume of the gold shell is V_gold = V_total - V_silver = 14.14 nm^3 - 4.19 nm^3 = 9.95 nm^3.

Step 2: Calculate the number of atoms
The number of atoms in a given volume is given by the formula n = V * ρ / M, where ρ is the density and M is the molar mass.

For silver, the density is 10.49 g/cm^3 and the molar mass is 107.87 g/mol. For gold, the density is 19.32 g/cm^3 and the molar mass is 197.0 g/mol.

Note that 1 nm^3 = 1e-21 cm^3, so we need to convert the volumes from nm^3 to cm^3.

The number of silver atoms is n_silver = V_silver * ρ_silver / M_silver = 4.19e-21 cm^3 * 10.49 g/cm^3 / 107.87 g/mol = 4.09e-22 mol.

The number of gold atoms is n_gold = V_gold * ρ_gold / M_gold = 9.95e-21 cm^3 * 19.32 g/cm^3 / 197.0 g/mol = 9.97e-22 mol.

Step 3: Convert moles to atoms
The number of atoms in a mole is given by Avogadro's number, which is approximately 6.022e23.

The total number of silver atoms is N_silver = n_silver * Avogadro's number = 4.09e-22 mol * 6.022e23 = 2.46e2 atoms.

The total number of gold atoms is N_gold = n_gold * Avogadro's number = 9.97e-22 mol * 6.022e23 = 6.00e2 atoms.

So, approximately 246 silver atoms and 600 gold atoms are required to prepare the nanomaterials.

Question

To solve this problem, we need to calculate the volume of the silver core and the gold shell separately, then use the density and molar mass of each material to find the number of atoms.

Step 1: Calculate the volume of the silver core and the gold shell
The volume of a sphere is given by the formula V = 4/3 * π * r^3. The radius is half the diameter, so the radius of the silver core is 1 nm and the radius of the gold shell is 1.5 nm.

The volume of the silver core is V_silver = 4/3 * π * (1 nm)^3 = 4.19 nm^3.

The volume of the gold shell is the total volume minus the volume of the silver core. The total volume is V_total = 4/3 * π * (1.5 nm)^3 = 14.14 nm^3. So, the volume of the gold shell is V_gold = V_total - V_silver = 14.14 nm^3 - 4.19 nm^3 = 9.95 nm^3.

Step 2: Calculate the number of atoms
The number of atoms in a given volume is given by the formula n = V * ρ / M, where ρ is the density and M is the molar mass.

For silver, the density is 10.49 g/cm^3 and the molar mass is 107.87 g/mol. For gold, the density is 19.32 g/cm^3 and the molar mass is 197.0 g/mol.

Note that 1 nm^3 = 1e-21 cm^3, so we need to convert the volumes from nm^3 to cm^3.

The number of silver atoms is n_silver = V_silver * ρ_silver / M_silver = 4.19e-21 cm^3 * 10.49 g/cm^3 / 107.87 g/mol = 4.09e-22 mol.

The number of gold atoms is n_gold = V_gold * ρ_gold / M_gold = 9.95e-21 cm^3 * 19.32 g/cm^3 / 197.0 g/mol = 9.97e-22 mol.

Step 3: Convert moles to atoms
The number of atoms in a mole is given by Avogadro's number, which is approximately 6.022e23.

The total number of silver atoms is N_silver = n_silver * Avogadro's number = 4.09e-22 mol * 6.022e23 = 2.46e2 atoms.

The total number of gold atoms is N_gold = n_gold * Avogadro's number = 9.97e-22 mol * 6.022e23 = 6.00e2 atoms.

So, approximately 246 silver atoms and 600 gold atoms are required to prepare the nanomaterials.

Knowee AI · Accepted Answer