Question

Cosmone is a molecule used by fragrance manufacturers to provide a rich and elegant musky essence to many perfumes. Cosmone has the molecular formula \(\mathrm{C}_{15} \mathrm{H}_{26} \mathrm{O}\). (a) Calculate the molar mass of cosmone. (b) Calculate the mass of \(3.82\) moles of cosmone. (c) Calculate the number of molecules of cosmone in a sample containing \(8.36 \times 10^{-4} \mathrm{~mol}\) cosmone. (d) Calculate the number of moles of carbon in a \(8.35-\mathrm{mol}\) sample of cosmone. (e) Calculate the mass of oxygen in a 4.29-g sample of cosmone. (f) Calculate the mass of a sample of cosmone that contains \(6.58 \times 10^{19}\) atoms of hydrogen. (g) Calculate the mass of one molecule of cosmone. (h) Calculate the number of atoms of carbon in \(8.00-g\) of cosmone.

Short answer

Molar mass: 222.41 g/mol. Mass of 3.82 moles: 848.80 g. Molecules in 8.36 × 10^{-4} mol: 5.03 × 10^{20}. Moles of carbon in 8.35 mol: 125.25 mol. Mass of oxygen in 4.29 g: 0.308 g. Mass for 6.58 × 10^{19} H atoms: 9.32 × 10^{-3} g. Mass of one molecule: 3.69 × 10^{-22} g. Carbon atoms in 8.00 g: 3.25 × 10^{23}.

Step-by-step solution

Determine atomic masses - Step 1

First, identify the atomic masses of each element: Carbon (C) has an atomic mass of approximately 12.01 g/mol, Hydrogen (H) has an atomic mass of approximately 1.01 g/mol, and Oxygen (O) has an atomic mass of approximately 16.00 g/mol.

Calculate molar mass - Step 2

Using the molecular formula \(\mathrm{C}_{15} \mathrm{H}_{26} \mathrm{O}\), calculate the molar mass by multiplying the atomic mass of each element by its respective count and adding the results: \(15 \cdot 12.01 + 26 \cdot 1.01 + 16.00\).

Find molar mass - Step 3

Compute: \[ M = 15 \cdot 12.01 + 26 \cdot 1.01 + 16.00 = 180.15 + 26.26 + 16.00 = 222.41 \, \text{g/mol}\]

Calculate mass of 3.82 moles - Step 4

Using the molar mass calculated, multiply it by the number of moles: \[ m = n \times M = 3.82 \, \text{mol} \times 222.41 \, \text{g/mol} = 848.80 \, \text{g}\]

Number of molecules in 8.36 × 10^{-4} mol - Step 5

Use Avogadro's number \(6.022 \times 10^{23} \, \text{mol}^{-1}\) to find the number of molecules: \[N = n \times N_A = 8.36 \times 10^{-4} \, \text{mol} \times 6.022 \times 10^{23} \, \text{mol}^{-1} = 5.03 \times 10^{20} \, \text{molecules}\]

Number of moles of carbon - Step 6

Since each molecule of cosmone contains 15 atoms of carbon, the number of moles of carbon in 8.35 mol of cosmone is: \[ n_{C} = 8.35 \, \text{mol} \times 15 = 125.25 \, \text{mol}\]

Mass of oxygen - Step 7

Using the molar mass of cosmone, calculate the mass of cosmone that contains 4.29 g. Then find the fraction of the total mass that is due to oxygen: \[4.29 \, \text{g} \, \times\, \left( \frac{16.00 \, \text{g/mol}}{222.41 \, \text{g/mol}} \right) = 0.308 \, \text{g}\]

Mass of sample with hydrogen atoms - Step 8

Calculate the number of moles of hydrogen atoms, then find the mass of corresponding moles of cosmone: \[ n = \frac{6.58 \times 10^{19}}{6.022 \times 10^{23} / 26} \, \text{mol} = 4.19\times 10^{-5} \ \, mass = 4.19 \times 10^{-5} \, \text{mol} \times 222.41 \, \text{g/mol} = 9.32 \times 10^{-3} \, \text{g}\]

Mass of one molecule - Step 9

Divide the molar mass by Avogadro's number: \[ m = \frac{222.41 \, \text{g/mol}}{6.022 \times 10^{23} \, \text{mol}^{-1}} = 3.69 \times 10^{-22} \, \text{g}\]

Number of carbon atoms in 8.00 g of cosmone - Step 10

Find moles of cosmone, then multiply by atoms of C and Avogadro's number: \[ n = \frac{8.00 \, \text{g}}{222.41 \, \text{g/mol}} = 0.0360 \, \text{mol}, \C_{atoms} = 0.0360 \, \text{mol} \times 15 \times 6.022 \times 10^{23} = 3.25 \times 10^{23} \]

Key concepts

Molar Mass Calculation

To find the molar mass of a molecule, add together the atomic masses of all the atoms in its molecular formula. Let's break this down using Cosmone's formula, \(\text{C}_{15} \text{H}_{26} \text{O}\). First, find the atomic masses: Carbon (C) is approximately 12.01 g/mol, Hydrogen (H) is 1.01 g/mol, and Oxygen (O) is 16.00 g/mol. Multiply each atomic mass by the number of atoms in the molecule: for Carbon: \(\text{15} \times 12.01\), for Hydrogen: \(\text{26} \times 1.01\), and for Oxygen: \(\text{1} \times 16.00\). Add these values: 180.15 + 26.26 + 16.00 = 222.41 g/mol.

Molecular Formula

The molecular formula represents the actual number of each type of atom in a molecule. For Cosmone, the formula \(\text{C}_{15} \text{H}_{26} \text{O}\) tells us there are 15 Carbon atoms, 26 Hydrogen atoms, and 1 Oxygen atom in every molecule. This formula is essential for communicating the exact composition of the substance.

Stoichiometry

Stoichiometry involves calculating the quantitative relationships between reactants and products in a chemical reaction. Using the molecular formula and molar mass, stoichiometry helps determine how much of each substance is needed or produced. For example, to find the mass of 3.82 moles of Cosmone: multiply the amount in moles (3.82) by the molar mass (222.41 g/mol): 3.82 \(\times\) 222.41 = 848.80 g.

Avogadro's Number

Avogadro’s Number, \(\text{6.022} \times \text{10}^{23}\), represents the number of particles (atoms, molecules) in one mole of a substance. Using this number, you can find the quantity of molecules in a given mole amount. For Cosmone, the number of molecules in \(8.36 \times 10^{-4}\) moles is calculated by multiplying by Avogadro's number: \(8.36 \times 10^{-4} \times 6.022 \times 10^{23} \). This equals about \(\text{5.03} \times \text{10}^{20} \) molecules.

Moles to Molecules Conversion

Converting moles to molecules involves using Avogadro's Number. For any substance, multiply the number of moles by \(\text{6.022} \times \text{10}^{23}\) to get the number of molecules. For Cosmone, if you have 0.0360 moles, the number of molecules is \(\text{0.0360} \times 6.022 \times 10^{23} = 2.17 \times 10^{22}\) molecules.

Mass and Mole Relationships

Understanding the relationship between mass and moles is key in chemistry. First, use the molar mass to convert between mass and moles. For instance, for an 8.00-g sample of Cosmone, divide the mass by its molar mass (222.41 g/mol): \(8.00 \div 222.41 = 0.0360\) moles. Then, to find the number of carbon atoms, multiply the moles by the number of atoms in the molecule (15 for carbon), and then by Avogadro’s Number: \(0.0360 \times 15 \times 6.022 \times 10^{23} \). This equals \(\text{3.25} \times \text{10}^{23}\) carbon atoms.