Question

How many atoms are contained in each of the following? (a) 2 molecules \(\mathrm{CH}_{3} \mathrm{COOH}\) (d) \(92.5 \mathrm{~g} \mathrm{Au}\) (b) \(0.75 \mathrm{~mol} \mathrm{C}_{2} \mathrm{H}_{6}\) (e) \(75 \mathrm{~g} \mathrm{PCl}_{3}\) (c) \(25 \mathrm{~mol} \mathrm{H}_{2} \mathrm{O}\) (f) \(15 \mathrm{~g} \mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}_{6}\)

Short answer

For 2 \(\text{CH}_3\text{COOH}\), 16 atoms. For 92.5 g Au, \(2.83 \times 10^{23}\) atoms. For 0.75 mol \(\text{C}_2\text{H}_6\), \(3.612 \times 10^{24}\) atoms. For 25 mol \(\text{H}_2\text{O}\), \(4.517 \times 10^{25}\) atoms. For 75 g \(\text{PCl}_3\), \(1.31 \times 10^{24}\) atoms. For 15 g \(\text{C}_6\text{H}_{12}\text{O}_6\), \(1.2 \times 10^{24}\) atoms.

Step-by-step solution

- Calculate the number of atoms in 2 molecules of \(\text{CH}_3\text{COOH}\)

Start by determining the number of each type of atom in one molecule of \(\text{CH}_3\text{COOH}\). \(\text{CH}_3\text{COOH}\) has 2 C atoms, 4 H atoms, and 2 O atoms. In total, that's 2 + 4 + 2 = 8 atoms per molecule. Therefore, for 2 molecules: \[ \text{Number of atoms} = 2 \times 8 = 16 \]

- Convert grams to moles for \(\text{Au}\)

First, find the molar mass of gold (Au), which is approximately 197 g/mol. Then, convert 92.5 grams of Au to moles using the formula: \[ \text{Moles} = \frac{92.5}{197} \approx 0.47 \]

- Calculate the number of atoms in \(\text{Au}\)

To find the number of atoms, multiply the number of moles by Avogadro's number \( 6.022 \times 10^{23} \). \[ \text{Number of atoms} = 0.47 \times 6.022 \times 10^{23} \approx 2.83 \times 10^{23} \]

- Calculate atoms in 0.75 moles of \(\text{C}_2\text{H}_6\)

First, determine the total number of atoms in one molecule of \(\text{C}_2\text{H}_6\): 2 C atoms and 6 H atoms, giving a total of 8 atoms. For 0.75 moles, use Avogadro's number: \[ \text{Number of atoms} = 0.75 \times 8 \times 6.022 \times 10^{23} = 3.612 \times 10^{24} \]

- Calculate the number of atoms in 25 moles of \(\text{H}_2\text{O}\)

A molecule of \(\text{H}_2\text{O}\) contains 2 H atoms and 1 O atom, totaling 3 atoms. For 25 moles: \[ \text{Number of atoms} = 25 \times 3 \times 6.022 \times 10^{23} = 4.517 \times 10^{25} \]

- Calculate atoms in 75 grams of \(\text{PCl}_3\)

First, find the molar mass of \(\text{PCl}_3\) (P: 31 g/mol, Cl: 35.5 g/mol, total: 31 + 3(35.5) = 137.5 g/mol). Convert grams to moles: \[ \text{Moles} = \frac{75}{137.5} \approx 0.545 \] Then, determine the total atoms in one molecule, which is 1 P atom + 3 Cl atoms = 4 atoms. \[ \text{Number of atoms} = 0.545 \times 4 \times 6.022 \times 10^{23} = 1.31 \times 10^{24} \]

- Calculate atoms in 15 grams of \(\text{C}_6\text{H}_{12}\text{O}_6\)

Determine the molar mass of \(\text{C}_6\text{H}_{12}\text{O}_6\) (C: 12 g/mol, H: 1 g/mol, O: 16 g/mol, total: 6(12) + 12(1) + 6(16) = 180 g/mol). Convert grams to moles: \[ \text{Moles} = \frac{15}{180} = 0.0833 \] Determine the total atoms in one molecule: 6 C atoms + 12 H atoms + 6 O atoms = 24 atoms. \[ \text{Number of atoms} = 0.0833 \times 24 \times 6.022 \times 10^{23} = 1.2 \times 10^{24} \]

Key concepts

molecules

A molecule is the smallest unit of a chemical compound that retains its chemical properties. Molecules are made up of atoms, which are bonded together. For example, a water molecule (H₂O) consists of two hydrogen atoms and one oxygen atom.

Each molecule of a compound has a fixed number of atoms. For instance, in the exercise, each molecule of acetic acid (CH₃COOH) has 8 atoms - 2 carbon, 4 hydrogen, and 2 oxygen atoms.

In chemistry problems, knowing the composition of molecules helps in determining the total number of atoms present in a given quantity of the compound. This is essential for understanding chemical reactions and stoichiometry.

moles to atoms conversion

Converting moles to atoms is a fundamental concept in chemistry. A mole is a unit that measures the amount of a substance and is equivalent to a number of particles (atoms, molecules, ions, etc.).

To convert moles to atoms, you use Avogadro's number, which is 6.022 x 10²³ - the number of particles in one mole.

For example, to find the number of atoms in 0.75 moles of ethane (C₂H₆), first calculate the total number of atoms per molecule of ethane. Then, multiply the number of moles by Avogadro's number and the number of atoms per molecule:

Number of atoms = 0.75 moles x 8 atoms/molecule x 6.022 x 10²³ atoms/mole = 3.612 x 10²⁴ atoms.

Avogadro's number

Avogadro's number (6.022 x 10²³) is a constant used to quantify the number of particles (atoms, molecules, ions) in one mole of a substance.

This number is fundamental because it allows chemists to count particles by weighing them. For instance, knowing Avogadro's number allows the conversion of moles to atoms and vice versa.

When dealing with 92.5 grams of gold (Au), you first convert the grams to moles by dividing by the molar mass (197 g/mol). Then, multiply the result by Avogadro's number to find how many atoms are in the gold sample:

Number of atoms = 0.47 moles x 6.022 x 10²³ atoms/mole ≈ 2.83 x 10²³ atoms.

molar mass calculation

Molar mass is the mass of one mole of a substance, expressed in grams per mole (g/mol). It is calculated by summing the atomic masses of all atoms in a molecule.

For a compound like glucose (C₆H₁₂O₆), you find the molar mass by adding together the masses of 6 carbon atoms, 12 hydrogen atoms, and 6 oxygen atoms. Each element's atomic mass can be found on the periodic table:

Molar mass of glucose = 6(12 g/mol) + 12(1 g/mol) + 6(16 g/mol) = 180 g/mol.

Knowing the molar mass helps in converting between grams and moles, which is crucial for many chemical calculations.

chemistry problem solving

Solving chemistry problems often involves several steps, including understanding the problem, identifying known and unknown quantities, and applying chemical principles to find the solution.

In problem-solving, it's crucial to:

• Understand the composition of molecules;
• Convert units (e.g., grams to moles);
• Use Avogadro's number for particle counts;
• Apply correct formulas and relationships (e.g., molar mass).

For example, to find the number of atoms in 75 grams of PCl₃, first calculate molar mass, convert grams to moles, and then find the number of atoms:

Number of atoms = 0.545 moles x 4 atoms/molecule x 6.022 x 10²³ atoms/mole = 1.31 x 10²⁴ atoms.