Question

Using the average atomic masses given inside the front cover of this text, calculate the number of atoms present in each of the following samples. a. \(2.89 \mathrm{g}\) of gold b. 0.000259 mol of platinum c. 0.000259 g of platinum d. 2.0 lb of magnesium e. \(1.90 \mathrm{mL}\) of liquid mercury (density \(=13.6 \mathrm{g} / \mathrm{mL}\) ) f. 4.30 mol of tungsten g. \(4.30 \mathrm{g}\) of tungsten

Short answer

a. \(8.85 \times 10^{22}\) atoms of gold b. \(1.56 \times 10^{20}\) atoms of platinum c. \(8.01 \times 10^{17}\) atoms of platinum d. \(2.25 \times 10^{25}\) atoms of magnesium e. \(7.75 \times 10^{22}\) atoms of mercury f. \(2.59 \times 10^{24}\) atoms of tungsten g. \(1.41 \times 10^{22}\) atoms of tungsten

Step-by-step solution

Calculate the number of moles

Calculate the number of moles using the formula: number of moles = mass of sample / molar mass number of moles = 2.89 g / 197.0 g/mol = 0.0147 mol of gold

Calculate the number of atoms

Multiply the number of moles by Avogadro's number to get the number of atoms: number of atoms = number of moles × Avogadro's number = 0.0147 × 6.022 × 10²³ = \(8.85 \times 10^{22}\) atoms of gold b. 0.000259 mol of platinum

Calculate the number of atoms

Multiply the number of moles by Avogadro's number to get the number of atoms: number of atoms = 0.000259 × 6.022 × 10²³ = \(1.56 \times 10^{20}\) atoms of platinum c. 0.000259 g of platinum

Calculate the number of moles

Calculate the number of moles using the formula: number of moles = 0.000259 g / 195.1 g/mol = 1.33 × 10⁻⁶ mol of platinum

Calculate the number of atoms

Multiply the number of moles by Avogadro's number to get the number of atoms: number of atoms = 1.33 × 10⁻⁶ × 6.022 × 10²³ = \(8.01 \times 10^{17}\) atoms of platinum d. 2.0 lb of magnesium

Convert pounds to grams

There are 1 lb = 454 g, so 2.0 lb = 2.0 × 454 = 908 g of magnesium

Calculate the number of moles

Calculate the number of moles using the formula: number of moles = 908 g / 24.31 g/mol = 37.36 mol of magnesium

Calculate the number of atoms

Multiply the number of moles by Avogadro's number to get the number of atoms: number of atoms = 37.36 × 6.022 × 10²³ = \(2.25 × 10^{25}\) atoms of magnesium e. 1.90 mL of liquid mercury (density = 13.6 g/mL)

Convert mL to grams

Multiply the volume by the density to get the mass: mass of mercury = 1.90 mL × 13.6 g/mL = 25.84 g

Calculate the number of moles

Calculate the number of moles using the formula: number of moles = 25.84 g / 200.6 g/mol = 0.1288 mol of mercury

Calculate the number of atoms

Multiply the number of moles by Avogadro's number to get the number of atoms: number of atoms = 0.1288 × 6.022 × 10²³ = \(7.75 \times 10^{22}\) atoms of mercury f. 4.30 mol of tungsten

Calculate the number of atoms

Multiply the number of moles by Avogadro's number to get the number of atoms: number of atoms = 4.30 × 6.022 × 10²³ = \(2.59 \times 10^{24}\) atoms of tungsten g. 4.30 g of tungsten

Calculate the number of moles

Calculate the number of moles using the formula: number of moles = 4.30 g / 183.8 g/mol = 0.0234 mol of tungsten

Calculate the number of atoms

Multiply the number of moles by Avogadro's number to get the number of atoms: number of atoms = 0.0234 × 6.022 × 10²³ = \(1.41 \times 10^{22}\) atoms of tungsten

Key concepts

Avogadro's Number

In the microscopic world of atoms, large numbers make it possible for us to communicate quantities. Avogadro's number, represented as \(6.022 \times 10^{23}\), is the number of atoms, molecules, or particles in one mole of a substance. This constant bridges the gap between the atomic scale and the macroscopic scale that we observe in the laboratory. When calculating the number of atoms in a sample, we often start by determining the moles and then use Avogadro's number to "scale up" from single atoms to a quantity that is substantial enough to measure. For instance, if you have 0.0147 moles of gold and you want to know how many atoms that corresponds to, you multiply by Avogadro's number: \[0.0147 \text{ moles} \times 6.022 \times 10^{23} \text{ atoms/mole} = 8.85 \times 10^{22} \text{ atoms}\] This number highlights the sheer number of atoms even in tiny samples of materials and is a key component in helping us quantify substances accurately.

Moles Calculation

Understanding moles is crucial to converting between mass and the number of particles. A mole is a unit that measures the amount of substance and is based on the number of atoms in 12 grams of carbon-12, which is Avogadro's number. To find the number of moles in a sample, you use the formula: \[\text{moles} = \frac{\text{mass of the sample (g)}}{\text{molar mass (g/mol)}}\] For example, if you want to find the number of moles in 4.30 g of tungsten, which has a molar mass of 183.8 g/mol: \[\text{moles of tungsten} = \frac{4.30 \text{ g}}{183.8 \text{ g/mol}} = 0.0234 \text{ mol}\] Through this calculation, we can then further determine how many particles are in the sample by using Avogadro's number.

Conversion Factors

Conversion factors are an essential tool in chemistry, allowing us to convert from one unit to another. These are ratios derived from the concepts being converted, such as grams to moles or moles to atoms. To use a conversion factor effectively, identify what units you start with and what units you need. For converting mass to moles, the molar mass (g/mol) serves as the conversion factor. To convert moles to atoms, Avogadro's number is the conversion factor: - Given mass in grams and want moles: \[ \text{moles} = \text{mass (g)} \times \left( \frac{1 \text{ mol}}{\text{molar mass (g/mol)}} \right) \] - Given moles and want number of atoms: \[ \text{atoms} = \text{moles} \times \left( 6.022 \times 10^{23} \right) \] By stringing these conversions together, you can bridge different types of measurements seamlessly.

Density and Volume Relation

Density is an important property that relates mass and volume, allowing one to be calculated if the other is known. It is expressed as mass per unit volume, typically in g/mL or g/cm³. The formula to determine mass from density and volume is:\[\text{mass} = \text{density} \times \text{volume}\] Take liquid mercury, for example, known to have a density of 13.6 g/mL. If you have a volume of 1.90 mL, the mass can be calculated as:\[\text{mass of mercury} = 1.90 \text{ mL} \times 13.6 \text{ g/mL} = 25.84 \text{ g}\] This relationship is particularly useful in solving problems that involve substances where a conversion between mass and volume is required to proceed further with calculations such as finding moles or atoms.