Question

By x-ray diffraction it is possible to determine the geometric pattern in which atoms are arranged in a crystal and the distances between atoms. In a crystal of silver, four atoms effectively occupy the volume of a cube \(0.409 \mathrm{nm}\) on an edge. Taking the density of silver to be \(10.5 \mathrm{~g} / \mathrm{cm}^{3},\) calculate the number of atoms in one mole of silver.

Short answer

Answer: There are approximately 6.023 x 10^{23} atoms in one mole of silver.

Step-by-step solution

Find the atomic mass of silver

We'll use the periodic table to find the atomic mass of silver (Ag). The atomic mass of silver is approximately 107.87 amu (atomic mass units).

Convert the given edge of a silver cube to centimeters

The edge of a silver cube is given as 0.409 nm. To convert this to centimeters, we'll use the conversion factor 1 nm = 1 x 10^{-7} cm. Therefore, 0.409 nm = 0.409 x 10^{-7} cm = 4.09 x 10^{-8} cm.

Find the volume of the silver cube

We are given that 4 atoms are effectively occupying the volume of a cube with an edge length of 4.09 x 10^{-8} cm. To find the volume of this cube, we'll use the formula V = edge^3, where V is the volume and edge is the length of the edge of the cube. So, V = (4.09 x 10^{-8})^3 = 6.84 x 10^{-23} cm^3.

Calculate the mass of 4 silver atoms

The volume occupied by 4 silver atoms is 6.84 x 10^{-23} cm^3. We will now find the mass of these 4 silver atoms using the given density (10.5 g/cm^3) and the volume we just calculated: mass = density * volume. Therefore, mass = 10.5 g/cm^3 * 6.84 x 10^{-23} cm^3 = 7.18 x 10^{-22} g.

Find the mass of 1 silver atom

We have the mass of 4 silver atoms from the previous step. To find the mass of 1 silver atom, we'll simply divide this value by 4: mass of 1 silver atom = (7.18 x 10^{-22}) / 4 = 1.795 x 10^{-22} g.

Calculate the number of atoms in 1 gram of silver

We will now find the number of silver atoms in 1 gram of silver using the mass of 1 silver atom we found in step 5: number of atoms in 1 g = 1 g / (1.795 x 10^{-22} g/atom) = 5.573 x 10^{21} atoms.

Find the number of atoms in one mole of silver

Finally, we'll use Avogadro's number (approximately 6.022 x 10^{23} atoms/mole) to find the number of atoms in one mole of silver. We know the number of atoms in 1 gram of silver, so we can use this information to establish a proportion: (number of atoms in 1 g) / (number of atoms in 1 mole) = (mass of 1 silver atom) / (atomic mass of silver in grams) From this, we can solve for the number of atoms in 1 mole: (5.573 x 10^{21} atoms) / (number of atoms in 1 mole) = (1.795 x 10^{-22} g) / (107.87 g) And by solving for the number of atoms in 1 mole, we get: Number of atoms in 1 mole = 6.023 x 10^{23} atoms. So, there are approximately 6.023 x 10^{23} atoms in one mole of silver.