Sterling silver, an alloy of silver and copper, contains 7.5 percent copper by mass and has
a density of 10.3 g/cm3. How many copper atoms are in a sterling silver sphere whose radius is
2.0 in?
To determine the number of copper atoms in a sterling silver sphere, we need to follow a series of steps:
Step 1: Calculate the mass of the sterling silver sphere.
Since we know the density of sterling silver (10.3 g/cm3) and the radius of the sphere (2.0 in), we need to convert the radius to centimeters.
1 inch = 2.54 cm
Hence, the radius of the sterling silver sphere in cm is 2.0 in x 2.54 cm/in = 5.08 cm.
The volume of a sphere can be calculated using the formula:
Volume = (4/3) x π x r³
Substituting the values, we get
Volume = (4/3) x 3.1416 x (5.08 cm)³ ≈ 548.19 cm³
Since density = mass/volume, we can rearrange the equation to solve for mass:
Mass = density x volume
Mass = 10.3 g/cm3 x 548.19 cm³ ≈ 5643.00 g
Step 2: Calculate the mass of copper in the sterling silver sphere.
Since the sterling silver contains 7.5% copper by mass, we can calculate the mass of copper in the sphere using the following equation:
Mass of copper = 7.5% x mass of sterling silver sphere
Mass of copper = (7.5/100) x 5643.00 g ≈ 423.23 g
Step 3: Calculate the number of moles of copper.
To convert grams of copper to moles, we need to divide the mass of copper by the molar mass of copper.
The molar mass of copper (Cu) is approximately 63.55 g/mol.
Number of moles of copper = Mass of copper / Molar mass of copper
Number of moles of copper = 423.23 g / 63.55 g/mol ≈ 6.66 mol
Step 4: Calculate the number of copper atoms.
Avogadro's number (6.022 x 10²³) represents the number of atoms in one mole of a substance. We can use this constant to convert the number of moles of copper to the number of copper atoms.
Number of copper atoms = Number of moles of copper x Avogadro's number
Number of copper atoms = 6.66 mol x 6.022 x 10²³ atoms/mol ≈ 4.01 x 10²⁴ atoms
Therefore, there are approximately 4.01 x 10²⁴ copper atoms in the sterling silver sphere with a radius of 2.0 inches.