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 mole of platinum c. \(0.000259 \mathrm{~g}\) 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 moles of tungsten g. \(4.30 \mathrm{~g}\) of tungsten

Short answer

Calculating the number of atoms for each sample, we get: a. \(8.88 \times 10^{21}\) gold atoms b. \(1.56 \times 10^{20}\) platinum atoms c. \(8.04 \times 10^{17}\) platinum atoms d. \(4.49 \times 10^{24}\) magnesium atoms e. \(6.11 \times 10^{22}\) mercury atoms f. \(2.59 \times 10^{24}\) tungsten atoms g. \(1.41 \times 10^{22}\) tungsten atoms

Step-by-step solution

Convert grams to moles

To convert the grams of gold to moles, we will use the molar mass of gold (Au), which is 196.97 g/mol: moles of gold = \( \frac{2.89\,\text{g}}{196.97\,\text{g/mol}} \)

Calculate the number of atoms

Now, we will use the Avogadro's constant to calculate the number of atoms in 2.89 g of gold: number of atoms = (moles of gold) * (Avogadro's constant) #b. 0.000259 mole of platinum# This is a straightforward problem. We only need to multiply the number of moles by the Avogadro's constant. #c. 0.000259 g of platinum#

Convert grams to moles

To convert the grams of platinum (Pt) to moles, we will use the molar mass of platinum, which is \(195.08\,\text{g/mol}\): moles of platinum = \( \frac{0.000259\,\text{g}}{195.08\,\text{g/mol}} \)

Calculate the number of atoms

Now, we will use the Avogadro's constant to calculate the number of atoms in 0.000259 g of platinum: number of atoms = (moles of platinum) * (Avogadro's constant) #d. 2.0 lb of magnesium#

Convert pounds to grams

To convert pounds to grams, we will use the conversion factor: \(1 \,\text{lb} = 453.59\,\text{g}\): mass in grams = \( 2.0\, \text{lb} \times \frac{453.59\,\text{g}}{1\,\text{lb}} \)

Convert grams to moles

Now, we will use the molar mass of magnesium (Mg), which is \(24.305\,\text{g/mol}\): moles of magnesium = \( \frac{\text{mass in grams}}{24.305\,\text{g/mol}} \)

Calculate the number of atoms

Now, we will use the Avogadro's constant to calculate the number of atoms in 2.0 lb of magnesium: number of atoms = (moles of magnesium) * (Avogadro's constant) #e. 1.90 mL of liquid mercury#

Calculate mass from volume and density

First, we will calculate the mass of mercury (Hg) using its density and volume: mass of mercury = volume * density = \(1.90\, \text{mL} \times 13.6\,\frac{\text{g}}{\text{mL}}\)

Convert grams to moles

To convert the grams of mercury to moles, we will use the molar mass of mercury, which is \(200.59\,\text{g/mol}\): moles of mercury = \( \frac{\text{mass of mercury}}{200.59\,\text{g/mol}} \)

Calculate the number of atoms

Now, we will use the Avogadro's constant to calculate the number of atoms in \(1.90\,\text{mL}\) of liquid mercury: number of atoms = (moles of mercury) * (Avogadro's constant) #f. 4.30 moles of tungsten# This is another simple problem. We only need to multiply the number of moles by Avogadro's constant. #g. 4.30 g of tungsten#

Convert grams to moles

To convert the grams of tungsten (W) to moles, we will use the molar mass of tungsten, which is \(183.84\,\text{g/mol}\): moles of tungsten = \( \frac{4.30\,\text{g}}{183.84\,\text{g/mol}} \)

Calculate the number of atoms

Now, we will use the Avogadro's constant to calculate the number of atoms in 4.30 g of tungsten: number of atoms = (moles of tungsten) * (Avogadro's constant)

Key concepts

Avogadro's constant

Avogadro's constant, often denoted as N_A, is the number of constituent particles, usually atoms or molecules, that are contained in one mole of a substance. The value of Avogadro's constant is approximately 6.022 x 10²³ entities per mole.

This constant is a bridge between the macroscopic scale, where we measure substance in grams or moles, and the microscopic scale of individual atoms and molecules. In calculations, it's crucial for converting moles of a substance to the actual number of atoms or molecules present. For instance, to find out how many atoms are in a given sample, you would multiply the number of moles by Avogadro's constant.

Molar mass

The molar mass is another fundamental concept in chemistry, which refers to the mass of one mole of a substance. It is expressed in units of grams per mole (g/mol) and is numerically equivalent to the average atomic mass of an element.

Each element has a unique molar mass, making this property important when converting between grams and moles of a substance. For example, the molar mass of carbon is 12.01 g/mol, meaning one mole of carbon has a mass of 12.01 grams. In practice, determining the molar mass of a compound involves summing the molar masses of its constituent elements, adjusted for the number of atoms of each element present in the compound.

Mole-to-atom conversion

Converting moles to atoms requires the use of Avogadro's constant. The process involves a straightforward multiplication: the number of moles of a substance is multiplied by 6.022 x 10²³ (Avogadro's constant) to yield the number of atoms present.

For example, to convert 2 moles of carbon to atoms, you'd compute the following: 2 moles of carbon x 6.022 x 10²³ atoms/mole = 1.2044 x 10²⁴ carbon atoms. This conversion is applicable to any element or compound as long as the number of moles is known.

Converting mass to moles

To convert the mass of a sample to the number of moles, you divide the mass by the molar mass of the substance. The formula for this conversion is: Number of moles = Mass of sample (g) / Molar mass (g/mol).

For instance, to convert 10 grams of water (H₂O) to moles, you would divide 10 grams by the molar mass of water (about 18.015 g/mol), resulting in approximately 0.555 moles of water. This conversion is the first step in many stoichiometric calculations, allowing chemists to relate the mass of a substance to the quantity of individual atoms or molecules it contains.