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Chapter 3: Problem 100

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

Expert verified
Answer: There are approximately 6.009 × 10^{23} silver atoms in one mole of silver.

Step by step solution

01

Find the volume of the cube occupied by four silver atoms

We are given that four silver atoms occupy the volume of a cube with edge length 0.409 nm. To find the volume of this cube, we need to cube the edge length. Volume of a single cube = (edge length)^3 = (0.409 nm)^3 = 6.838 × 10^{-2} nm^3.
02

Convert the volume to cm^3

To be consistent with the units of the density, we need to convert the volume from nm^3 to cm^3. The conversion factor is 1 nm = 10^{-7} cm. 6.838 × 10^{-2} nm^3 × (10^{-7} cm/nm)^3 = 6.838 × 10^{-23} cm^3.
03

Find the mass of four silver atoms using density

The density of silver is given as 10.5 g/cm^3. We can use the formula of density, which is: density = mass/volume mass = density × volume mass of four silver atoms = (10.5 g/cm^3) × (6.838 × 10^{-23} cm^3) = 7.1799 × 10^{-22} g.
04

Find the mass of one mole of silver atoms

We need to find the mass of one mole of silver atoms, which is the molar mass. Silver is represented by the symbol Ag, and its atomic mass is approximately 107.87 amu (atomic mass units). When we convert atomic mass units to grams, we get the molar mass. Molar mass of Ag = 107.87 g/mol.
05

Calculate the number of silver atom sets in one mole of silver

Since four silver atoms have a mass of 7.1799 × 10^{-22} g, we can find out how many sets of four atoms are present in one mole of silver: Number of sets of four silver atoms = (molar mass of Ag) / (mass of four silver atoms) = (107.87 g/mol) / (7.1799 × 10^{-22} g) = 1.50233 × 10^{23} sets.
06

Calculate the number of silver atoms in one mole

Since one set contains four silver atoms, we can now find the total number of silver atoms in one mole by multiplying the number of sets by four: Number of silver atoms in one mole of silver = 4 × (1.50233 × 10^{23} sets) = 6.009 × 10^{23} silver atoms. The number of silver atoms in one mole of silver is approximately 6.009 × 10^{23}.

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Most popular questions from this chapter

Dimethylhydrazine, the fuel used in the Apollo lunar descent module, has a molar mass of \(60.10 \mathrm{~g} / \mathrm{mol}\). It is made up of carbon, hydrogen, and nitrogen atoms. The combustion of \(2.859 \mathrm{~g}\) of the fuel in excess oxygen yields \(4.190 \mathrm{~g}\) of carbon dioxide and \(3.428 \mathrm{~g}\) of water. What are the simplest and molecular formulas for dimethylhydrazine?

A crude oil burned in electrical generating plants contains about \(1.2 \%\) sulfur by mass. When the oil burns, the sulfur forms sulfur dioxide gas: $$ \mathrm{S}(s)+\mathrm{O}_{2}(g) \longrightarrow \mathrm{SO}_{2}(g) $$ How many liters of \(\mathrm{SO}_{2}(d=2.60 \mathrm{~g} / \mathrm{L})\) are produced when \(1.00 \times 10^{4} \mathrm{~kg}\) of oil burns at the same temperature and pressure?

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